A Mathematical Model with Quarantine States for the ......2...

Research ArticleA Mathematical Model with Quarantine States forthe Dynamics of Ebola Virus Disease in Human Populations

Gideon A Ngwa1 and Miranda I Teboh-Ewungkem2

1Department of Mathematics University of Buea PO Box 63 Buea Cameroon2Department of Mathematics Lehigh University Bethlehem PA 18015 USA

Correspondence should be addressed to Miranda I Teboh-Ewungkem mit703lehighedu

Received 12 January 2016 Revised 30 May 2016 Accepted 8 June 2016

Academic Editor Chen Yanover

Copyright copy 2016 G A Ngwa and M I Teboh-EwungkemThis is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A deterministic ordinary differential equation model for the dynamics and spread of Ebola Virus Disease is derived and studiedThe model contains quarantine and nonquarantine states and can be used to evaluate transmission both in treatment centres andin the community Possible sources of exposure to infection including cadavers of Ebola Virus victims are included in the modelderivation and analysis Our modelrsquos results show that there exists a threshold parameter 119877

0 with the property that when its value

is above unity an endemic equilibrium exists whose value and size are determined by the size of this threshold parameter and whenits value is less than unity the infection does not spread into the community The equilibrium state when it exists is locally andasymptotically stable with oscillatory returns to the equilibrium point The basic reproduction number 119877

0 is shown to be strongly

dependent on the initial response of the emergency services to suspected cases of Ebola infection When intervention measuressuch as quarantining are instituted fully at the beginning the value of the reproduction number reduces and any further infectionscan only occur at the treatment centres Effective control measures to reduce 119877

0to values below unity are discussed

1 Introduction and Background

The world has been riveted by the 2014 outbreak of the EbolaVirus Disease (EVD) that affected parts of West Africa withGuinea Liberia and Sierra Leone being the most hard hitareas Isolated cases of the disease did spread by land to Sene-gal and Mali (localized transmission) and by air to NigeriaSome Ebola infected humans were transported to the US(except the one case that traveled to Texas and later on died)and other European countries for treatment An isolated caseoccurred in Spain another in Italy (a returning volunteerhealth care worker) and a few cases in the US and the UK[1ndash3] Though dubbed the West African Ebola outbreak themovement of patients and humans between countries if nothandled properly could have led to a global Ebola pandemicThere was also a separate Ebola outbreak affecting a remoteregion in theDemocratic Republic ofCongo (formerly Zaire)and it was only by November 21 2014 that the outbreak wasreported to have ended [2]

The Ebola Virus Disease (EVD) formally known as Ebolahaemorrhagic fever and caused by the Ebola Virus is verylethal with case fatalities ranging from 25 to 90 with amean of about 50 [2] The 2014 EVD outbreak though notthe first but one of many other EVD outbreaks that haveoccurred in Africa since the first recorded outbreak of 1976 isthe worst in terms of the numbers of Ebola cases and relateddeaths and the most complex [2] About 9 months after theidentification of a mysterious killer disease killing villagers ina small Guinean village as Ebola the 2014West African Ebolaoutbreak as of December 24 2014 had up to 19497 Ebolacases resulting in 7588 fatalities [1 3 4] a case fatality rate ofabout 389 By December 2015 the number of Ebola Viruscases (including suspected probable and confirmed) stood at28640 resulting in 11315 fatalities a case fatality rate of 395[3 5]

Ebola Virus the agent that causes EVD is hypothesisedto be introduced into the human population through contact

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2016 Article ID 9352725 29 pageshttpdxdoiorg10115520169352725

2 Computational and Mathematical Methods in Medicine

with the blood secretions fluids from organs and otherbody parts of dead or living animals infected with the virus(eg fruit bats primates and porcupines) [2 6] Human-to-human transmission can then occur through direct contact(via broken skin or mucous membranes such as eyes noseor mouth) with Ebola Virus infected blood secretions andfluids secreted through organs or other body parts in forexample saliva vomit urine faeces semen sweat and breastmilk Transmission can also be as a result of indirect contactwith surfaces and materials in for example bedding cloth-ing and floor areas or objects such as syringes contaminatedwith the aforementioned fluids [2 6]

When a healthy human (considered here to be suscepti-ble) who has no Ebola Virus in them is exposed to the virus(directly or indirectly) the human may become infected iftransmission is successful The risk of being infected with theEbola Virus is (i) very low or not recognizable where there iscasual contact with a feverish ambulant self-caring patientfor example sharing the same public place (ii) low wherethere is close face-to-face contact with a feverish and ambu-lant patient for example physical examination (iii) highwhere there is close face-to-face contact without appropriatepersonal protective equipment (including eye protection)with a patient who is coughing or vomiting has nose-bleeds or has diarrhea and (iv) very high where there ispercutaneous needle stick or mucosal exposure to virus-contaminated blood body fluids tissues or laboratory speci-mens in severely ill or knownpositive patientsThe symptomsof EVD may appear during the incubation period of thedisease estimated to be anywhere from 2 to 21 days [2 7ndash9] with an estimated 8- to 10-day average incubation periodalthough studies estimate it at 9ndash11 days for the 2014 EVDoutbreak inWest Africa [10] Studies have shown that duringthe asymptomatic part of the Ebola Virus Disease a humaninfected with the virus is not infectious and cannot transmitthe virus However with the onset of symptoms the humancan transmit the virus and is hence infectious [2 7]The onsetof symptoms commences the course of illness of the diseasewhich can lead to death 6ndash16 days later [8 9] or improvementof health with evidence of recovery 6ndash11 days later [8]

In the first few days of EVD illness (estimated at days 1ndash3[11]) a symptomatic patient may exhibit symptoms commonto those like the malaria disease or the flu (high feverheadache muscle and joint pains sore throat and generalweakness) Without effective disease management betweendays 4 and 5 to 7 the patient progresses to gastrointestinalsymptoms such as nausea watery diarrhea vomiting andabdominal pain [10 11] Some or many of the other symp-toms such as low blood pressure headaches chest painshortness of breath anemia exhibition of a rash abdominalpain confusion bleeding and conjunctivitis may develop[10 11] in some patients In the later phase of the course ofthe illness days 7ndash10 patients may present with confusionand may exhibit signs of internal andor visible bleedingprogressing towards coma shock and death [10 11]

Recovery from EVD can be achieved as evidenced by theless than 50 fatality rate for the 2014 EVD outbreak in WestAfrica With no known cure recovery is possible througheffective disease management the treatment of Ebola-related

symptoms and also the effective protection by the patientrsquosimmune response [7] Some of the disease managementstrategies include hydrating patients by administering intra-venous fluids and balancing electrolytes and maintaining thepatientrsquos blood pressure and oxygen levels Other schemesused include blood transfusion (using an Ebola survivorrsquosblood) and the use of experimental drugs on such patients(eg ZMAPP whose safety and efficacy have not yetbeen tested on humans) There are some other promisingdrugsvaccines under trials [2] Studies show that once apatient recovers from EVD they remain protected against thedisease and are immune to it at least for a projected periodbecause they develop antibodies that last for at least 10 years[7] Once recovered lifetime immunity is unknown orwhether a recovered individual can be infected with anotherEbola strain is unknown However after recovery a personcan potentially remain infectious as long as their blood andbody fluids including semen and breast milk contain thevirus In particular men can potentially transmit the virusthrough their seminal fluid within the first 7 to 12 weeksafter recovery from EVD [2] Table 1 shows the estimatedtime frames and projected progression of the infection in anaverage EVD patient

Given that there is no approved drug or vaccine outyet local control of the Ebola Virus transmission requiresa combined and coordinated control effort at the individuallevel the community level and the institutionalhealthgov-ernment level Institutions and governments need to educatethe public and raise awareness about risk factors proper handwashing proper handling of Ebola patients quick reportingof suspected Ebola cases safe burial practices use of publictransportation and so forth These education efforts needto be communicated with communitychief leaders who aretrusted by members of the communities they serve From aglobal perspective a good surveillance and contact tracingprogram followed by isolation and monitoring of probableand suspected cases with immediate commencement ofdisease management for patients exhibiting symptoms ofEVD is important if we must in the future elude a globalepidemic and control of EVD transmission locally andglobally [2] It was by effective surveillance contact tracingand isolation andmonitoring of probable and suspected casesfollowed by immediate supportive care for individuals andfamilies exhibiting symptoms that the EVD was broughtunder control in Nigeria [17] Senegal USA and Spain [1]

Efficient control and management of any future EVDoutbreaks can be achieved if new more economical and real-izable methods are used to target and manage the dynamicsof spread as well as the population sizes of those communitiesthat may be exposed to any future Ebola Virus Diseaseoutbreak More realistic mathematical models can play a rolein this regard since analyses of suchmodels can produce clearinsight to vulnerable spots on the Ebola transmission chainwhere control efforts can be concentrated Good modelscould also help in the identification of disease parameters thatcan possibly influence the size of the reproduction numberof EVD Existing mathematical models for Ebola [14 16 18ndash21] have been very instrumental in providing mathematicalinsight into the dynamics of Ebola Virus transmission Many

Computational and Mathematical Methods in Medicine 3

Table 1 A possible progression path of symptoms from exposure to the Ebola Virus to treatment or death Table shows a suggested transitionand time frame in humans of the virus from exposure to incubation to symptoms development and recovery or death This table is adaptedbased on the image in the Huffington post via [11] Superscript a for the 2014 epidemic the average incubation period is reported to bebetween 9 and 11 days [10] Superscript b other studies reported a mean of 4ndash10 days [8 9]

Exposure

Incubationperiod Course of illness Recovery or death

Range 2 to 21days fromexposure

Range 6 to 16 days from the end of the incubationperiod Recovery by the

end of days 6ndash11Death by the end

of days 6ndash16Probable Early symptomatic Late symptomaticDays 1ndash3 Days 4ndash7 Days 7ndash10

An individualcomes incontact with anEbola infectedindividual (deador alive) or havebeen in thevicinity ofsomeone whohas beenexposed

Average of 8ndash11adays before

symptoms areevidentAnother

estimate reportsan average of4ndash10b days

Patients exhibitmalaria-like or

flu-like symptomsfor example feverand weakness

Patients progress togastrointestinalsymptoms for

example nauseawatery diarrheavomiting andabdominal painOther symptomsmay include lowblood pressure

anemia headacheschest pain

shortness of breathexhibition of arash confusionbleeding andconjunctivitis

Patients maypresent with

confusion and mayexhibit signs ofinternal andorvisible bleeding

potentiallyprogressing

towards comashock and death

Some patients may recover whileothers will die

Recovery typically requires earlyintervention

of these models have also been helpful in that they haveprovided methods to derive estimates for the reproductionnumber for Ebola based on data from the previous outbreaksHowever few of the models have taken into account the factthat institution of quarantine states or treatment centres willaffect the course of the epidemic in the population [16] It isour understanding that the way the disease will spread will bedetermined by the initial and continual response of the healthservices in the event of the discovery of an Ebola diseasecase The objective of this paper is to derive a comprehensivemathematical model for the dynamics of Ebola transmissiontaking into consideration what is currently known of thedisease The primary objective is to derive a formula for thereproduction number for Ebola Virus Disease transmissionin view of providing a more complete and measurable indexfor the spread of Ebola and to investigate the level of impactof surveillance contact tracing isolation and monitoring ofsuspected cases in curbing disease transmission The modelis formulated in a way that it is extendable with appropriatemodifications to other disease outbreaks with similar char-acteristics to Ebola requiring such contact tracing strategiesOur model differs from other mathematical models that havebeen used to study the Ebola disease [14 15 18 20ndash22] in thatit captures the quarantined Ebola Virus Disease patients andprovides possibilities for those who escaped quarantine at theonset of the disease to enter quarantine at later stages To thebest of our knowledge this is the first integrated ordinarydifferential equation model for this kind of communicabledisease of humans Our final result would be a formula for

the basic reproduction number of Ebola that depends on thedisease parameters

The rest of the paper is divided up as follows In Section 2we outline the derivation of the model showing the statevariables and parameters used and how they relate togetherin a conceptual framework In Section 3 we present amathematical analysis of the derived model to ascertain thatthe results are physically realizable We then reparameterisethe model and investigate the existence and linear stability ofsteady state solution calculate the basic reproduction num-ber and present some special cases In Section 4 we presenta discussion on the parameters of the model In Section 5 wecarry out some numerical simulations based on the selectedfeasible parameters for the system and then round up thepaper with a discussion and conclusion in Section 6

2 The Mathematical Model

21 Description of Model Variables We divide the humanpopulation into 11 states representing disease status andquarantine state At any time 119905 there are the following(1) Susceptible Individuals Denoted by 119878 this class alsoincludes false probable cases that is all those individuals whowould have displayed early Ebola-like symptoms but whoeventually return a negative test for Ebola Virus infection(2) Suspected Ebola Cases The class of suspected EVDpatients comprises those who have come in contact with orbeen in the vicinity of anybody who is known to have been


sick or died of Ebola Individuals in this class may ormay not show symptoms Two types of suspected cases areincluded the quarantined suspected cases denoted by 119878

119876

and the nonquarantined suspected case denoted by 119878119873 Thus

a suspected case is either quarantined or not(3) Probable Cases The class of probable cases comprises allthose persons who at some point were considered suspectedcases and who now present with fever and at least three otherearly Ebola-like symptoms Two types of probable cases areincluded the quarantined probable cases denoted by 119875

119876

and the nonquarantined probable cases 119875119873 Thus a probable

case is either quarantined or not Since the early Ebola-likesymptoms of high fever headache muscle and joint painssore throat and general weakness can also be a result of otherinfectious diseases such asmalaria or fluwe cannot be certainat this stage whether or not the persons concerned haveEbola infection However since the class of probable personsis derived from suspected cases and to remove the uncer-tainties we will assume that probable cases may eventuallyturn out to be EVD patients and if that were to be the casesince they are already exhibiting some symptoms they canbe assumed to be mildly infectious(4) Confirmed Early Symptomatic Cases The class of con-firmed early asymptomatic cases comprises all those personswho at some point were considered probable cases and aconfirmatory laboratory test has been conducted to confirmthat there is indeed an infectionwith Ebola VirusThis class iscalled confirmed early symptomatic because all that they haveas symptoms are the early Ebola-like symptoms of high feverheadache muscle and joint pains sore throat and generalweakness Two types of confirmed early symptomatic casesare included the quarantined confirmed early symptomaticcases 119862

119876and the nonquarantined confirmed early symp-

tomatic cases 119862119873 Thus a confirmed early symptomatic case

is either quarantined or not The class of confirmed earlysymptomatic individuals may not be very infectious(5) Confirmed Late Symptomatic CasesThe class of confirmedlate symptomatic cases comprises all those persons who atsome point were considered confirmed early symptomaticcases and in addition the persons who now present withmostor all of the later Ebola-like symptoms of vomiting diarrheastomach pain skin rash red eyes hiccups internal bleedingand external bleeding Two types of confirmed late symp-tomatic cases are included the quarantined confirmed latesymptomatic cases 119868

119876and the nonquarantined confirmed late

symptomatic cases 119868119873 Thus a confirmed late symptomatic

case is either quarantined or not The class of confirmedlate symptomatic individuals may be very infectious and anybodily secretions from this class of persons can be infectiousto other humans

(6) Removed Individuals Three types of removals are consid-ered but only two are related to EVDThe removals related tothe EVD are confirmed individuals removed from the systemthrough disease induced death denoted by 119877

119863 or confirmed

cases that recover from the infection denoted by 119877119877 Now it

is known that unburied bodies or not yet cremated cadaversof EVD victims can infect other susceptible humans upon

contact [7] Therefore the cycle of infection really stops onlywhen a cadaver is properly buried or cremated Thus mem-bers from class 119877

119863 representing dead bodies or cadavers

of EVD victims are considered removed from the infectionchain and consequently from the system only when theyhave been properly disposed of The class 119877

119877 of individuals

who beat the odds and recover from their infection areconsidered removed because recovery is accompanied withthe acquisition of immunity so that this class of individualsare then protected against further infection [7] and they nolonger join the class of susceptible individuals The thirdtype of removal is obtained by considering individuals whodie naturally or due to other causes other than EVD Theseindividuals are counted as 119877

119873

The state variables are summarized in Notations

22 The Mathematical Model A compartmental frameworkis used to model the possible spread of EVD within apopulationThemodel accounts for contact tracing and quar-antining in which individuals who have come in contact orhave been associated with Ebola infected or Ebola-deceasedhumans are sought and quarantined They are monitored fortwenty-one days during which they may exhibit signs andsymptoms of the Ebola Virus or are cleared and declared freeWe assume thatmost of the quarantining occurs at designatedmakeshift temporal or permanent health facilities Howeverit has been documented that others do not get quarantinedbecause of fear of dying without a loved one near them or fearthat if quarantined theymay instead get infected at the centreas well as traditional practices and belief systems [14 16 22]Thus there may be many within communities who remainnonquarantined and we consider these groups in our modelIn all the living classes discussed we will assume that naturaldeath or death due to other causes occurs at constant rate 120583where 1120583 is approximately the life span of the human

221 The Susceptible Individuals The number of susceptibleindividuals in the population decreases when this populationis exposed by having come in contact with or being associatedwith any of the possibly infectious cases namely infectedprobable case confirmed case or the cadaver of a confirmedcase The density increases when some false suspected indi-viduals (a proportion of 1 minus 120579

2of nonquarantined and 1 minus 120579

6

of quarantined) and probable cases (a proportion of 1 minus 1205793

of nonquarantined individuals and 1 minus 1205797of quarantined

individuals) are eliminated from the suspected and probablecase listWe also assume a constant recruitment rateΠ as wellas natural death or death due to other causes Therefore theequation governing the rate of change with time within theclass of susceptible individuals may be written as

119889119878

119889119905= Π minus 120582119878 + (1 minus 120579

3) 120573119873119875119873+ (1 minus 120579

2) 120572119873119878119873

+ (1 minus 1205796) 120572119876119878119876+ (1 minus 120579

7) 120573119876119875119876minus 120583119878

(1)

where 120582 is the force of infection and the rest of the parametersare positive and are defined in Notations We identify twotypes of total populations at any time 119905 (i) the total living


population119867119871 and (ii) the total living population including

the cadavers of Ebola Virus victims that can take part in thespread of EVD119867 Thus at each time 119905 we have

119867119871 (119905) = (119878 + 119878119873 + 119878119876 + 119875119873 + 119875119876 + 119862119873 + 119862119876 + 119868119873

+ 119868119876+ 119877119877) (119905)

(2)

119867(119905) = (119878 + 119878119873+ 119878119876+ 119875119873+ 119875119876+ 119862119873+ 119862119876+ 119868119873+ 119868119876

+ 119877119877+ 119877119863) (119905)

(3)

Since the cadavers of EVD victims that have not beenproperly disposed of are very infectious the force of infectionmust then also take this fact into consideration and beweighted with 119867 instead of 119867

119871 The force of infection takes

the following form

120582 =1

119867(1205793120588119873119875119873+ 1205797120588119876119875119876+ 120591119873119862119873+ 120585119873119868119873+ 120591119876119862119876

+ 120585119876119868119876+ 119886119863119877119863)

(4)

where 119867 gt 0 is defined above and the parameters 120588119873

120588119876 120591119873 120591119876 120585119873 120585119876 and 119886

119863are positive constants as defined

in Notations There are no contributions to the force ofinfection from the 119877

119877class because it is assumed that

once a person recovers from EVD infection the recoveredindividual acquires immunity to subsequent infection withthe same strain of the virus Although studies have suggestedthat recovered men can potentially transmit the Ebola Virusthrough seminal fluids within the first 7ndash12 weeks of recovery[2] and mothers through breast milk we assume here thatwith education survivors who recover would have enoughinformation to practice safe sexual andor feeding habits toprotect their loved ones until completely clearThus recoveredindividuals are considered not to contribute to the force ofinfection

222 The Suspected Individuals A fraction 1 minus 1205791of the

exposed susceptible individuals get quarantined while theremaining fraction are not Also a fraction 120579

2(resp 120579

6) of the

nonquarantined (resp quarantined) suspected individualsbecome probable cases at rate 120572 while the remainder 1 minus1205792(resp 1 minus 120579

6) do not develop into probable cases and

return to the susceptible pool For the quarantined indi-viduals we assume that they are being monitored whilethe suspected nonquarantined individuals are not Howeveras they progress to probable cases (at rates 120572

119873and 120572

119876)

a fraction 1205792119887

of these humans will seek the health careservices as symptoms commence and become quarantinedwhile the remainder 120579

2119886remain nonquarantined Thus the

equation governing the rate of change within the two classesof suspected persons then takes the following form

119889119878119873

119889119905= 1205791120582119878 minus (1 minus 120579

2) 120572119873119878119873minus 1205792119886120572119873119878119873minus 1205792119887120572119873119878119873

minus 120583119878119873

119889119878119876

119889119905= (1 minus 120579

1) 120582119878 minus (1 minus 120579

6) 120572119876119878119876minus 1205796120572119876119878119876minus 120583119878119876

(5)

In the context of this model we make the assumptionthat once quarantined the individuals stay quarantined untilclearance and are released or they die of the infection Noticethat 120579

2= 1205792119886+ 1205792119887 so that 1 minus 120579

2+ 1205792119886+ 1205792119887= 1

223 The Probable Cases The fractions 1205792and 120579

6of sus-

pected cases that become probable cases increase the numberof individuals in the probable case class The population ofprobable cases is reduced (at rates 120573

119873and 120573

119876) when some

of these are confirmed to have the Ebola Virus throughlaboratory tests at rates 120572

119873and 120572

119876 For some proportions

1 minus 1205793and 1 minus 120579

7 the laboratory tests are negative and the

probable individuals revert to the susceptible class From theproportion 120579

3of nonquarantined probable cases whose tests

are positive for the Ebola Virus (ie confirmed for EVD) afraction 120579

3119887 become quarantined while the remainder 120579

3119886

remain nonquarantined So 1205793= 1205793119886+ 1205793119887Thus the equation

governing the rate of change within the classes of probablecases takes the following form

119889119875119873

119889119905= 1205792119886120572119873119878119873minus (1 minus 120579

3) 120573119873119875119873minus 1205793119886120573119873119875119873

minus 1205793119887120573119873119875119873minus 120583119875119873

119889119875119876

119889119905= 1205792119887120572119873119878119873+ 1205796120572119876119878119876minus (1 minus 120579

7) 120573119876119875119876minus 1205797120573119876119875119876

minus 120583119875119876

(6)

224 The Confirmed Early Symptomatic Cases The fractions1205793and 1205797of the probable cases become confirmed early symp-

tomatic cases thus increasing the number of confirmed caseswith early symptoms The population of early symptomaticindividuals is reduced when some recover at rates 119903

119864119873for

the nonquarantined cases and 119903119864119876

for the quarantined casesOthers may see their condition worsening and progress andbecome late symptomatic individuals in which case theyenter the full blown late symptomatic stages of the diseaseWe assume that this progression occurs at rates 120574

119873or 120574119876

respectively which are the reciprocal of themean time it takesfor the immune system to either be completely overwhelmedby the virus or be kept in check via supportive mechanismA fraction 1 minus 120579

4of the confirmed nonquarantined early

symptomatic cases will be quarantined as they become latesymptomatic cases while the remaining fraction 120579

4escape

quarantine due to lack of hospital space or fear and beliefcustoms [16 22] but become confirmed late symptomaticcases in the community Thus the equation governing therate of change within the two classes of confirmed earlysymptomatic cases takes the following form

119889119862119873

119889119905= 1205793119886120573119873119875119873minus 119903119864119873119862119873minus 1205794120574119873119862119873

minus (1 minus 1205794) 120574119873119862119873minus 120583119862119873

119889119862119876

119889119905= 1205793119887120573119873119875119873+ 1205797120573119876119875119876minus 119903119864119876119862119876minus 120574119876119862119876minus 120583119862119876

(7)


225 The Confirmed Late Symptomatic Cases The frac-tions 120579

4and 120579

8of confirmed early symptomatic cases who

progress to the late symptomatic stage increase the numberof confirmed late symptomatic cases The population of latesymptomatic individuals is reduced when some of theseindividuals are removed Removal could be as a result ofrecovery at rates proportional to 119903

119871119873and 119903119871119876

or as a resultof death because the EVD patientrsquos conditions worsen andthe Ebola Virus kills them The death rates are assumedproportional to 120575

119873and 120575

119876 Additionally as a control effort

or a desperate means towards survival some of the nonquar-antined late symptomatic cases are removed and quarantinedat rate 120590

119873 In our model we assume that Ebola-related

death only occurs at the late symptomatic stage Additionallywe assume that the confirmed late symptomatic individualswho are eventually put into quarantine at this late period(removing them from the community) may not have long tolive but may have a slightly higher chance at recovery thanwhen in the community and nonquarantined Since recoveryconfers immunity against the particular strain of the EbolaVirus individuals who recover become refractory to furtherinfection and hence are removed from the population ofsusceptible individuals Thus the equation governing therate of change within the two classes of confirmed latesymptomatic cases takes the following form

119889119868119873

119889119905= 1205794120574119873119862119873minus 120575119873119868119873minus 120590119873119868119873minus 119903119871119873119868119873minus 120583119868119873

119889119868119876

119889119905= (1 minus 120579

4) 120574119873119862119873+ 120590119873119868119873+ 120574119876119862119876minus 120575119876119868119876minus 119903119871119876119868119876

minus 120583119868119876

(8)

226 The Cadavers and the Recovered Persons The deadbodies of EVD victims are still very infectious and canstill infect susceptible individuals upon effective contact [2]Disease induced deaths from the class of confirmed latesymptomatic individuals occur at rates 120575

119873and 120575

119876and the

cadavers are disposed of via burial or cremation at rate 119887The recovered class contains all individuals who recover fromEVD Since recovery is assumed to confer immunity againstthe 2014 strain (the Zaire Virus) [7] of the Ebola Virusonce an individual recovers they become removed fromthe population of susceptible individuals Thus the equationgoverning the rate of change within the two classes ofrecovered persons and cadavers takes the following form

119889119877119877

119889119905= 119903119864119873119862119873+ 119903119871119873119868119873+ 119903119864119876119862119876+ 119903119871119876119868119876minus 120583119868119873

119889119877119863

119889119905= 120575119873119868119873+ 120575119876119868119876minus 119887119877119863

(9)

The population of humans who die either naturally or dueto other causes is represented by the variable 119877

119873and keeps

track of all natural deaths occurring at rate 120583 from all theliving population classes This is a collection class Anothercollection class is the class of disposed Ebola-related cadavers

disposed at rate 119887 These collection classes satisfy the equa-tions

119889119877119873

119889119905= 120583119867119871

119889119863119863

119889119905= 119887119877119863

(10)

Putting all the equations together we have

119889119878

119889119905= Π minus 120582119878 +

3120573119873119875119873+ 2120572119873119878119873+ 6120572119876119878119876

+ 7120573119876119875119876minus 120583119878

(11)

119889119878119873

119889119905= 1205791120582119878 minus (120572

119873+ 120583) 119878

119873 (12)

119889119878119876

119889119905= 1120582119878 minus (120572

119876+ 120583) 119878

119876 (13)

119889119875119873

119889119905= 1205792119886120572119873119878119873minus (120573119873+ 120583) 119875

119873 (14)

119889119875119876

119889119905= 1205792119887120572119873119878119873+ 1205796120572119876119878119876minus (120573119876+ 120583) 119875

119876 (15)

119889119862119873

119889119905= 1205793119886120573119873119875119873minus (119903119864119873+ 120574119873+ 120583)119862

119873 (16)

119889119862119876

119889119905= 1205793119887120573119873119875119873+ 1205797120573119876119875119876minus (119903119864119876+ 120574119876+ 120583)119862

119876 (17)

119889119868119873

119889119905= 1205794120574119873119862119873minus (119903119871119873+ 120575119873+ 120583) 119868

119873 (18)

119889119868119876

119889119905= 4120574119873119862119873+ 120590119873119868119873+ 120574119876119862119876

minus (119903119871119876+ 120575119876+ 120583) 119868

119876

(19)

119889119877119877

119889119905= 119903119864119873119862119873+ 119903119871119873119868119873+ 119903119864119876119862119876+ 119903119871119876119868119876minus 120583119877119877 (20)

119889119877119863

119889119905= 120575119873119868119873+ 120575119876119868119876minus 119887119877119863 (21)

119889119877119873

119889119905= 120583119867119871

(22)

119889119863119863

119889119905= 119887119877119863 (23)

where lowast= 1minus120579

lowastand all other parameters and state variables

are as in NotationsSuitable initial conditions are needed to completely spec-

ify the problem under consideration We can for exampleassume that we have a completely susceptible population and


a number of infectious persons are introduced into thepopulation at some point We can for example have that

119878 (0) = 1198780

119868119873(0) = 119868

0

119878119873 (0) = 119878119876 (0) = 119875119873 (0) = 0

119875119876(0) = 119862

119873(0) = 119862

119876(0) = 119868

119876(0) = 119877

119863(0) = 119877

119877(0)

= 119877119873(0) = 119863

119863(0) = 0

(24)

Class 119863119863is used to keep count of all the dead that are

properly disposed of class 119877119863is used to keep count of all

the deaths due to EVD and class 119877119873is used to keep count of

the deaths due to causes other than EVD infection The rateof change equation for the two groups of total populationsis obtained by using (2) and (3) and adding up the relevantequations from (11) to (23) to obtain

119889119867119871

119889119905= Π minus 120583119867

119871minus 120575119873119868119873minus 120575119876119868119876 (25)

119889119867

119889119905= Π minus 120583119867 minus (119887 minus 120583) 119877

119863 (26)

where 119867119871is the total living population and 119867 is the aug-

mented total population adjusted to account for nondisposedcadavers that are known to be very infectious On the otherhand if we keep count of all classes by adding up (11)ndash(23) thetotal human population (living and dead) will be constant ifΠ = 0 In what follows we will use the classes 119877

119863 119877119873 and

119863119863 comprising classes of already dead persons only as place

holders and study the problem containing the living humansand their possible interactions with cadavers of EVD victimsas often is the case in some cultures in Africa and so wecannot have a constant total population Note that (26) canalso be written as 119889119867119889119905 = Π minus 120583119867

119871minus 119887119877119863

227 Infectivity of Persons Infected with EVD Ebola is ahighly infectious disease and person to person transmissionis possible whenever a susceptible person comes in contactwith bodily fluids from an individual infected with the EbolaVirus We therefore define effective contact here generallyto mean contact with these fluids The level of infectivityof an infected person usually increases with duration ofthe infection and severity of symptoms and the cadaversof EVD victims are the most infectious [23] Thus we willassume in this paper that probable persons who indeed areinfected with the Ebola Virus are the least infectious whileconfirmed late symptomatic cases are very infectious and thelevel of infectivity will culminate with that of the cadaverof an EVD victim While under quarantine it is assumedthat contact between the persons in quarantine and thesusceptible individuals is minimalThus though the potentialinfectivity of the corresponding class of persons in quarantineincreases with disease progression their effective transmis-sion to members of the public is small compared to that fromthe nonquarantined class It is therefore reasonable to assumethat any transmission from persons under quarantine will

affect mostly health care providers and use that branch ofthe dynamics to study the effect of the transmission of theinfections to health care providers who are here consideredpart of the total population In what follows we do notexplicitly single out the infectivity of those in quarantine butstudy general dynamics as derived by the current modellingexercise

3 Mathematical Analysis

31 Well-Posedness Positivity and Boundedness of SolutionIn this subsection we discuss important properties of themodel such as well-posedness positivity and boundedness ofthe solutionsWe start by definingwhat wemean by a realisticsolution

Definition 1 (realistic solution) A solution of system (27) orequivalently system comprising (11)ndash(21) is called realistic ifit is nonnegative and bounded

It is evident that a solution satisfying Definition 1 isphysically realizable in the sense that its values can bemeasured through data collection For notational simplicitywe use vector notation as follows let x = (119878 119878

119873 119878119876

119875119873 119875119876 119862119873 119862119876 119868119873 119868119876 119877119877 119877119863)119879 be a column vector in R11

containing the 11 state variables so that in this nota-tion 119909

1= 119878 119909

2= 119878119873 119909

11= 119877119863 Let f(x) =

(1198911(x) 1198912(x) 119891

11(x))119879 be the vector valued function

defined inR11 so that in this notation 1198911(x) is the right-hand

side of the differential equation for first variable 1198781198912(x) is the

right-hand side of the equation for the second variable 1199092=

119878119873 and 119891

11(x) is the right side of the differential equation

for the 11th variable 11990911= 119877119863 and so is precisely system (11)ndash

(21) in that order with prototype initial conditions (24) Wethen write the system in the form

dx119889119905= f (x) x (0) = x

0 (27)

where x [0infin) rarr R11 is a column vector of state variablesand f R11 rarr R11 is the vector containing the right-hand sides of each of the state variables as derived fromcorresponding equations in (11)ndash(21) We can then have thefollowing result

Lemma 2 The function f in (27) is Lipschitz continuous in x

Proof Since all the terms in the right-hand side are linearpolynomials or rational functions of nonvanishing polyno-mial functions and since the state variables 119878 119878

lowast 119875lowast 119862lowast

119868lowast and 119877

lowast are continuously differentiable functions of 119905 the

components of the vector valued function f of (27) are allcontinuously differentiable Further let L(x y 120579) = x+120579(yminusx) 0 le 120579 le 1 Then L(x y 120579) is a line segment that joinspoints x to the point y as 120579 ranges on the interval [0 1] Weapply the mean value theorem to see that1003817100381710038171003817f (y) minus f (x)

1003817100381710038171003817infin=10038171003817100381710038171003817f1015840 (z y minus x)10038171003817100381710038171003817infin

z isin L (x y 120579) a mean value point(28)


where f1015840(z y minus x) is the directional derivative of the functionf at the mean value point z in the direction of the vector yminusxBut

10038171003817100381710038171003817f1015840 (z y minus x)10038171003817100381710038171003817infin =

1003817100381710038171003817100381710038171003817100381710038171003817

11

sum

119896=1

(nabla119891119896 (z) sdot (y minus x)) e119896

1003817100381710038171003817100381710038171003817100381710038171003817infin

le

1003817100381710038171003817100381710038171003817100381710038171003817

11

sum

119896=1

nabla119891119896 (z)1003817100381710038171003817100381710038171003817100381710038171003817infin

1003817100381710038171003817y minus x1003817100381710038171003817infin

(29)

where e119896is the 119896th coordinate unit vector in R11

+ It is now

a straightforward computation to verify that since R11+

is aconvex set and taking into consideration the nature of thefunctions 119891

119894 119894 = 1 11 all the partial derivatives are

bounded and so there exist119872 gt 0 such that1003817100381710038171003817100381710038171003817100381710038171003817

11

sum

119896=1

nabla119891119896 (z)1003817100381710038171003817100381710038171003817100381710038171003817infin

le 119872 forallz isin L (x y 120579) isin R11+ (30)

and so there exist119872 gt 0 such that1003817100381710038171003817f (y) minus f (x)

1003817100381710038171003817infinle 119872

1003817100381710038171003817y minus x1003817100381710038171003817infin (31)

and hence f is Lipschitz continuous

Theorem 3 (uniqueness of solutions) The differential equa-tion (27) has a unique solution

Proof By Lemma 2 the right-hand side of (27) is Lips-chitzian hence a unique solution exists by existence anduniqueness theorem of Picard See for example [24]

Theorem 4 (positivity) The region R11+

wherein solutionsdefined by (11)ndash(21) are defined is positively invariant under theflow defined by that system

Proof We show that each trajectory of the system starting inR11+will remain inR11

+ Assume for a contradiction that there

exists a point 1199051isin [0infin) such that 119878(119905

1) = 0 1198781015840(119905

1) lt 0

(where the prime denotes differentiationwith respect to time)but for 0 lt 119905 lt 119905

1 119878(119905) gt 0 and 119878

119873(119905) gt 0 119878

119876(119905) gt 0

119875119873(119905) gt 0 119875

119876(119905) gt 0 119862

119873(119905) gt 0 119862

119876(119905) gt 0 119868

119873(119905) gt 0

119868119876(119905) gt 0 119877

119877(119905) gt 0 and 119877

119863(119905) gt 0 So at the point 119905 = 119905

1

119878(119905) is decreasing from the value zero in which case it willgo negative If such an 119878 will satisfy the given differentialequation then we have

119889119878

119889119905

10038161003816100381610038161003816100381610038161003816119905=1199051

= Π minus 120582119878 (1199051) + (1 minus 120579

3) 120573119873119875119873(1199051)

+ (1 minus 1205792) 120572119873119878119873(1199051) + (1 minus 120579

6) 120572119876119878119876(1199051)

+ (1 minus 1205797) 120573119876119875119876(1199051) minus 120583119878 (119905

1)

= Π + (1 minus 1205793) 120573119873119875119873(1199051)

+ (1 minus 1205792) 120572119873119878119873(1199051) + (1 minus 120579

6) 120572119876119878119876(1199051)

+ (1 minus 1205797) 120573119876119875119876(1199051) gt 0

(32)

and so 1198781015840(1199051) gt 0 contradicting the assumption that 1198781015840(119905

1) lt

0 So no such 1199051exist The same argument can be made for all

the state variables It is now a simple matter to verify usingtechniques as explained in [24] thatwheneverwe start system(27) with nonnegative initial data in R11

+ the solution will

remain nonnegative for all 119905 gt 0 and that if x0= 0 the

solution will remain x = 0 forall119905 gt 0 and the region R11+

isindeed positively invariant

The last two theorems have established the fact that fromamathematical andphysical standpoint the differential equa-tion (27) is well-posed We next show that the nonnegativeunique solutions postulated byTheorem 3 are indeed realisticin the sense of Definition 1

Theorem 5 (boundedness) The nonnegative solutions char-acterized by Theorems 3 and 4 are bounded

Proof It suffices to prove that the total living population sizeis bounded for all 119905 gt 0 We show that the solutions lie in thebounded region

Ω119867119871

= 119867119871(119905) 0 le 119867

119871(119905) le

Π

120583 sub R

11

+ (33)

From the definition of 119867119871given in (2) if 119867

119871is bounded

the rest of the state variables that add up to 119867119871will also be

bounded From (25) we have

119889119867119871

119889119905= Π minus 120583119867

119871minus 120575119873119868119873minus 120575119876119868119876le Π minus 120583119867

119871997904rArr

119867119871(119905) le

Π

120583+ (119867119871(0) minus

Π

120583) 119890minus120583119905

(34)

Thus from (34) we see that whatever the size of119867119871(0)119867

119871(119905)

is bounded above by a quantity that converges to Π120583 as 119905 rarrinfin In particular if 120583119867

119871(0) lt Π then119867

119871(119905) is bounded above

by Π120583 and for all initial conditions

119867119871(119905) le lim119905rarrinfin

sup(Π120583+ (119867119871(0) minus

Π

120583) 119890minus120583119905) (35)

Thus119867119871(119905) is nonnegative and bounded

Remark 6 Starting from the premise that 119867119871(119905) ge 0 for

all 119905 gt 0 Theorem 5 establishes boundedness for the totalliving population and thus by extension verifies the positiveinvariance of the positive octant in R11 as postulated byTheorem 4 since each of the variables functions 119878 119878

lowast 119875lowast119862lowast

119868lowast and 119877

lowast where lowast isin 119873119876 119877 is a subset of119867

119871

32 Reparameterisation and Nondimensionalisation Theonly physical dimension in our system is that of time Butwe have state variables which depend on the density ofhumans and parameters which depend on the interactionsbetween the different classes of humans A state variable orparameter that measures the number of individuals of certaintype has dimension-like quantity associated with it [25] To


remove the dimension-like character on the parameters andvariables we make the following change of variables

119904 =119878

1198780

119904119899=119878119873

1198780

119873

119904119902=119878119876

1198780

119876

119901119899=119875119873

1198750

119873

119901119902=119875119876

1198750

119876

119888119899=119862119873

1198620

119873

119888119902=119862119876

1198620

119876

ℎ =119867

1198670

119894119899=119868119873

1198680

119873

119894119902=119868119876

1198680

119876

119903119903=119877119877

1198770

119877

119903119889=119877119863

1198770

119863

119903119899=119877119873

1198770

119873

119889119863=119863119863

1198630

119863

119905lowast=119905

1198790

ℎ119897=119867119871

1198670

119871

(36)

where

1198780=Π

120583

1198780

119873= 1198780

119876= 1198780

1198750

119873=12057921198861205721198731198780

119873

120573119873+ 120583

1198750

119876=12057921198871205721198731198780

119873

120573119876+ 120583

1198620

119873=12057931198861205731198731198750

119873

119903119864119873+ 120574119873+ 120583

1198620

119876=12057931198871205731198731198750

119873

119903119864119876+ 120574119876+ 120583

1198680

119873=

12057941205741198731198620

119873

119903119871119873+ 120575119873+ 120583

1198680

119876=(1 minus 120579

4) 1205741198731198620

119873

119903119871119876+ 120575119876+ 120583

1198770

119877= 1199031198641198731198620

1198731198790

1198770

119863=1205751198731198680

N119887

1198770

119873= 12058311987901198670

119871

1198630

119863= 11988711987901198770

119863

1198670= 1198670

119871=Π

120583= 1198780

1198790=1

120583

(37)

We then define the dimensionless parameter groupings

120588119899=12057931205881198731198750

119873

1205831198670

120588119902=

12057971205881198761198750

119876

1205831198670

120591119899=1205911198731198620

119873

1205831198670

120591119902=

1205911198761198620

119876

1205831198670

120585119899=1205851198731198680

119873

1205831198670

120585119902=

1205851198761198680

119876

1205831198670

119886119889=1198861198631198770

119863

1205831198670

120572119899= (120572119873+ 120583)119879

0

120572119902= (120572119876+ 120583)119879

0

120573119899= (120573119873+ 120583) 119879

0

120573119902= (120573119876+ 120583)119879

0


120583119889= 1198871198790

1198871=(1 minus 120579

3) 1205731198731198750

119873

1205831198780

1198872=(1 minus 120579

2) 1205721198731198780

119873

1205831198780

1198873=

(1 minus 1205796) 1205721198761198780

119876

1205831198780

1198874=

(1 minus 1205797) 1205731198761198750

119876

1205831198780

1198875=1205751198731198680

119873

1205831198670

1198876=

1205751198761198680

119876

1205831198670

1198878=(119887 minus 120583) 119877

0

119863

1205831198670

1198861=

12057961205721198761198780

119876

12057921198871205721198731198780

119873

1198862=

12057971205731198761198750

119876

12057931198871205731198731198750

119873

1198863=

1205901198731198680

119873

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198864=

1205741198761198620

119876

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198865=1199031198711198731198680

119873

1199031198641198731198620

119873

1198866=

1199031198641198761198620

119876

1199031198641198731198620

119873

1198867=

1199031198711198761198680

119876

1199031198641198731198620

119873

1198868=

1205751198761198680

119876

1205751198731198680

119873

120574119899= (119903119864119873+ 120574119873+ 120583) 119879

0

120574119902= (119903119864119876+ 120574119876+ 120583) 119879

0

120575119902= (119903119871119876+ 120575119876+ 120583) 119879

0

120575119899= (119903119871119873+ 120575119873+ 120583) 119879

0

(38)

The force of infection 120582 then takes the form

120582 = 120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)

(39)

This leads to the equivalent system of equations

119889119904

119889119905= 1 minus 120582119904 + 119887

1119901119899+ 1198872119904119899+ 1198873119904119902+ 1198874119901119902minus 119904 (40)

119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (41)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (42)

119889119901119899

119889119905= 120573119899(119904119899minus 119901119899) (43)

119889119901119902

119889119905= 120573119902(119904119899+ 1198861119904119902minus 119901119902) (44)

119889119888119899

119889119905= 120574119899(119901119899minus 119888119899) (45)

119889119888119902

119889119905= 120574119902(119901119899+ 1198862119901119902minus 119888119902) (46)

119889119894119899

119889119905= 120575119899(119888119899minus 119894119899) (47)

119889119894119902

119889119905= 120575119902(119888119899+ 1198863119894119899+ 1198864119888119902minus 119894119902) (48)

119889119903119903

119889119905= 119888119899+ 1198865119894119899+ 1198866119888119902+ 1198867119894119902minus 119903119903 (49)

119889119903119889

119889119905= 120583119889(119894119899+ 1198868119894119902minus 119903119889) (50)

119889119903119899

119889119905= ℎ119897 (51)

119889119889119863

119889119905= 119889119863 (52)

and the total populations satisfy the scaled equation

119889ℎ119897

119889119905= 1 minus ℎ

119897minus 1198875119894119899minus 1198876119894119902 (53)

119889ℎ

119889119905= 1 minus ℎ minus 119887

8119903119889 (54)


where 1198878gt 0 if it is assumed that the rate of disposal of Ebola

Virus Disease victims 119887 is larger than the natural humandeath rate120583The scaled or dimensionless parameters are thenas follows

120588119899=12057931205792119886120588119873120572119873

(120573119873+ 120583) 120583

120588119902=12057971205792119887120588119876120572119873

(120573119876+ 120583) 120583

120591119899=

12057931198861205792119886120591119873120572119873120573119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) 120583

120591119902=

12057931198871205792119886120591119876120572119873120573119873

(120573119873+ 120583) (119903

119864119876+ 120574119876+ 120583) 120583

120585119899=

120579311988612057921198861205794120585119873120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 120583

120585119902=

12057931198861205792119886(1 minus 120579

4) 120585119876120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119876+ 120575119876+ 120583) 120583

119886119889=

120579211988612057931198861205794120575119873120574119873120573119873120572119873119886119863

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 119887120583

1198871=(1 minus 120579

3) 1205792119886120573119873120572119873

(120573119873+ 120583) 120583

1198872=(1 minus 120579

2) 120572119873

120583

1198873=(1 minus 120579

6) 120572119876

120583

1198874=(1 minus 120579

7) 1205792119887120573119876120572119873

(120573119876+ 120583) 120583

1198875=

120579412057921198861205793119886120575119873120574119873120573119873120572119873

(119903119871119873+ 120575119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198876=

(1 minus 1205794) 12057921198861205793119886120575119876120574119873120573119873120572119873

(119903119871119876+ 120575119876+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198878=

(119887 minus 120583) 120579412057921198861205793119886120572119873120573119873120574119873120575119873

119887120583 (120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583)

120575119899=119903119871119873+ 120575119873+ 120583

120583

120572119899=120572119873+ 120583

120583

120572119902=120572119876+ 120583

120583

120573119899=120573119873+ 120583

120583

120573119902=120573119876+ 120583

120583

120574119899=119903119864119873+ 120574119873+ 120583

120583

120574119902=119903119864119876+ 120574119876+ 120583

120583

120575119902=119903119871119876+ 120575119876+ 120583

120583

1198861=1205796120572119876

1205792119887120572119873

1198862=12057971205792119887120573119876(120573119873+ 120583)

12057921198861205793119887(120573119876+ 120583) 120573

119873

1198863=

1205794120590119873

(1 minus 1205794) (119903119871119873+ 120575119873+ 120583)

1198864=

1205793119887120574119876(119903119864119873+ 120574119873+ 120583)

(1 minus 1205794) 1205793119886120574119873(119903119864119876+ 120574119876+ 120583)

1198865=

1199031198711198731205794120574119873

119903119864119873(119903119871119873+ 120575119873+ 120583)

120583119889=119887

120583

1198866=1205793119887119903119864119876(119903119864119873+ 120574119873+ 120583)

1205793119886119903119864119873(119903119864119876+ 120574119876+ 120583)

1198867=119903119871119876(1 minus 120579

4) 120574119873

119903119864119873(119903119871119876+ 120575119876+ 120583)

1198868=(1 minus 120579

4) 120575119876(119903119871119873+ 120575119873+ 120583)

1205794120575119873(119903119871119876+ 120575119876+ 120583)

(55)

33The Steady State Solutions andLinear Stability Thesteadystate of the system is obtained by setting the right-hand side ofthe scaled system to zero and solving for the scalar equationsLet xlowast = (119904lowast 119904lowast

119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast

119889 ℎlowast) be a steady

state solution of the systemThen (43) (45) and (47) indicatethat

119904lowast

119899= 119901lowast

119899= 119888lowast

119899= 119894lowast

119899(56)


andwe can use any of these as a parameter to derive the valuesof the other steady state variablesWe use the variables119901lowast

119899and

119904lowast

119902as parameters to obtain the expressions

119901lowast

119902(119901lowast

119899 119904lowast

119902) = 119901lowast

119899+ 1198861119904lowast

119902

119888lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

1119901lowast

119899+ 1198602119904lowast

119902

119894lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

3119901lowast

119899+ 1198604119904lowast

119902

119903lowast

119903(119901lowast

119899 119904lowast

119902) = 119860

5119901lowast

119899+ 1198606119904lowast

119902

119903lowast

119889(119901lowast

119899 119904lowast

119902) = 119860

7119901lowast

119899+ 1198608119904lowast

119902

ℎlowast(119901lowast

119899 119904lowast

119902) = 1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902

119904lowast(119901lowast

119899 119904lowast

119902) = 1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902

ℎlowast

119897(119901lowast

119899 119904lowast

119902) = 1 minus (119887

5+ 11988761198603) 119901lowast

119899minus 11988761198604119904lowast

119902

(57)

where

1198601= 1 + 119886

2

1198602= 11988611198862

1198603= 1 + 119886

3+ 11988641198601

1198604= 11988641198602

1198605= 1 + 119886

5+ 11988661198601+ 11988671198603

1198606= 11988661198602+ 11988671198604

1198607= 1 + 119886

81198603

1198608= 11988681198604

1198611= 11988781198607

1198612= 11988781198608

1198613= 120572119899minus 1198871minus 1198872minus 1198874

1198614= 120572119902minus 1198873minus 11988741198861

(58)

Here the solution for the scaled total living (ℎ119897) and scaled

living and Ebola-deceased (ℎ) populations is respectivelyobtained by equating the right-hand sides of (53) and (54) tozero while that for 119904lowast is obtained by adding up (40) (41) and

(42) It is easy to verify from reparameterisation (38) that theparameter groupings 119861

3and 119861

4are both nonnegative In fact

1198613= 1

+120572119873

120583(1205731198731205792119886(1205793minus 1)

120573119873+ 120583

+1205731198761205792119887(1205797minus 1)

120573119876+ 120583

+ 1205792)

gt 0 since 1205792= 1205792119886+ 1205792119887

1198614=1205721198761205796(1205731198761205797+ 120583) + 120583 (120573

119876+ 120583)

120583 (120573119876+ 120583)

gt 0

(59)

showing that 1198613gt 0 and 119861

4gt 0

To obtain a value for119901lowast119899and 119904lowast119902 we substitute all computed

steady state values (56) and (57) into (41) and (42) Theexpression for 120582lowast119904lowast in terms of 119901lowast

119899and 119904lowast119902is obtained from

(39) Performing the aforementioned procedures leads to thetwo equations

1205791(1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119899119901lowast

119899(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(60)

(1 minus 1205791) (1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119902119904lowast

119902(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(61)

where

1198615= 120588119899+ 120588119902+ 120591119899+ 1205911199021198601+ 120585119899+ 1205851199021198603+ 1198861198891198607

1198616= 1205881199021198861+ 1205911199021198602+ 1205851199021198604+ 1198861198891198608

(62)

Next we solve (60) and (61) simultaneously which clearlydiffer in some of their coefficients to obtain the expressionsfor 119901lowast119899and 119904lowast119902 Quickly observe that the two equations may be

reduced to one such that

(1 minus 1198611119901lowast

119899minus 1198612119904lowast

119902) (120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast

1199021205791) = 0 (63)

Two possibilities arise either (i) 1 minus 1198611119901lowast

119899minus 1198612119904lowast

119902= 0 or (ii)

120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast

1199021205791= 0 The first condition leads to the

system

1 minus 1198613119901lowast

119899minus 1198614119904lowast

119902= 0


119899minus 1198612119904lowast

119902= 0

(64)

However the two equations are equivalent to ℎlowast = 0 and119904lowast= 0 (see (57)) which are unrealistic based on our constant

population recruitment model Hence we only consider thesecond possibility which yields the relation

119901lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902 (65)


so that substituting (65) into (60) yields

119904lowast

119902= 0

or 119904lowast119902=1198617

1198618

=(1 minus 120579

1) 120572119899(1198770minus 1)

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612) (R minus 1)

= 119909 (1198770minus 1

R minus 1)

(66)

where

1198617= 120572119902120572119899(1 minus 120579

1) ((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

) minus 1)

= 120572119902120572119899(1 minus 120579

1) (1198770minus 1)

1198618= 120572119902[(12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614)

minus (12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612)] = 120572

119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612)((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (

12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614

12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612

) minus 1) = 120572119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612) (1198770119911 minus 1)

(67)

with

119909 =(1 minus 120579

1) 120572119899

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612)

119910 =

1205791120572119902119909

(1 minus 1205791) 120572119899

119911 =

(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

(68)

1198770=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

R =1198770(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

= 1199111198770

(69)

Remark 7 It can be shown that 1198612lt 1198614 In fact

1198612= 11988781198868119886411988611198862=1205796120572119876

120583

1205797120573119876

(120573119876+ 120583)

((1 minus120583

119887)

sdot120575119876

(119903119871119876+ 120575119876+ 120583)

120574119876

(119903119864119876+ 120574119876+ 120583)) lt

1205796120572119876

120583

sdot1205797120573119876

(120573119876+ 120583)

lt1205796120572119876

120583

(1205797120573119876+ 120583)

(120573119876+ 120583)

+ 1 = 1198614

(70)

Thus if (1 minus 1205791)120572119899(1198614minus 1198612) gt 1205791120572119902(1198611minus 1198613) then 119911 gt 1

This will hold if 1198613gt 1198611 In the case where 119861

3lt 1198611 we will

require that1198614minus1198612be greater than (120579

1120572119902(1minus120579

1)120572119899)(1198611minus1198613)

We identify 1198770as the unique threshold parameter of the

system as follows

Lemma 8 The parameter 1198770defined in (69) is the unique

threshold parameter of the system whenever 119911 gt 1

Proof If 119911 gt 1 thenR = 1199111198770gt 1 whenever 119877

0gt 1 and the

existence or nonexistence of a realistic solution of the form of(66) is determined solely by the size of 119877

0

The rest of the steady states are then obtained by usingthese values for 119901lowast

119899and 119904lowast119902given by (66) in (65) and (57) to

obtain the following

119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119894lowast

119899= 119888lowast

119899= 119904lowast

119899= 119901lowast

119899= 119910(

1198770minus 1

R minus 1)

119901lowast

119902= (119910 + 119886

1119909) (1198770minus 1

R minus 1)

119888lowast

119902= (1198601119910 + 119860

2119909) (1198770minus 1

R minus 1)

119894lowast

119902= (1198603119910 + 119860

4119909) (1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

119903lowast

119889= (1198607119910 + 119860

8119909) (1198770minus 1

R minus 1)

119904lowast= 1 minus (119861

3119910 + 1198614119909) (1198770minus 1

R minus 1)

ℎlowast= 1 minus (119861

1119910 + 1198612119909) (1198770minus 1

R minus 1)

(71)

where 119909 119910 and 119911 are as defined in (68) We have proved thefollowing result

Theorem 9 (on the existence of equilibrium solutions)System (40)ndash(52) has at least two equilibriumsolutions the disease-free equilibrium xlowast = 119864

119889119891119890=

(1 0 0 0 0 0 0 0 0 0 0 1) and an endemic equilibriumxlowast = 119864

119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast

119889 ℎlowast) The

endemic equilibrium 119864119890119890 exists and is realistic only when

the threshold parameters 1198770and R given by (69) are of

appropriate magnitude

The stability of the steady states is governed by the sign ofthe eigenvalues of the linearizing matrix near the steady statesolutions If 119869(xlowast) is the Jacobianmatrix at the steady state xlowastthen we have


119869dfe =

((((((((((((((((((((((

(

minus1 11988721198873(1198871minus 120588119899) (1198874minus 120588119902) minus120591119899minus120591119902minus120585119899minus1205851199020 minus119886

1198890

0 minus1205721198990 120579

1120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 0 minus120572119902

1205791120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 120573119899

0 minus120573119899

0 0 0 0 0 0 0 0

0 1205731199021198861120573119902

0 minus120573119902

0 0 0 0 0 0 0

0 0 0 120574119899

0 minus1205741198990 0 0 0 0 0

0 0 0 120574119902

1198862120574119902

0 minus1205741199020 0 0 0 0

0 0 0 0 0 120575119899

0 minus1205751198990 0 0 0

0 0 0 0 0 12057511990211988641205751199021198863120575119902minus1205751199020 0 0

0 0 0 0 0 1 11988661198865

1198867minus1 0 0

0 0 0 0 0 0 0 12058311988912058311988911988680 minus120583

1198890

0 0 0 0 0 0 0 0 0 0 minus1198878minus1

))))))))))))))))))))))

)

(72)

where 1205791= 1 minus 120579

1 Thus if 120577 is an eigenvalue of the linearized

system at the disease-free state then 120577 is obtained by thesolvability condition

119875 (120577) =1003816100381610038161003816119869dfe minus 120577I

1003816100381610038161003816 = (120577 + 1)31198759(120577) = 0 (73)

an equation involving a polynomial of degree 12 in 120577 where1198759(120577) is a polynomial of degree 9 in 120577 given by

1198759 (120577) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus120572119899minus 120577 0 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

0 minus120572119902minus 120577 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

120573119899

0 minus120573119899minus 120577 0 0 0 0 0 0

120573119902

1198861120573119902

0 minus120573119902minus 120577 0 0 0 0 0

0 0 120574119899

0 minus120574119899minus 120577 0 0 0 0

0 0 120574119902

1198862120574119902

0 minus120574119902minus 120577 0 0 0

0 0 0 0 120575119899

0 minus120575119899minus 120577 0 0

0 0 0 0 120575119902

1198864120575119902

1198863120575119902minus120575119902minus 120577 0

0 0 0 0 0 0 120583119889

1205831198891198868minus120583119889minus 120577

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(74)

Now all we need to know at this stage is whether there issolution of (73) for 120577 with positive real part which will thenindicate the existence of unstable perturbations in the linearregime The coefficients of polynomial (73) can give us vitalinformation about the stability or instability of the disease-free equilibrium For example by Descartesrsquo rule of signsa sign change in the sequence of coefficients indicates thepresence of a positive real root which in the linear regimesignifies the presence of exponentially growing perturbationsWe can write polynomial equation (73) in the form

119875 (120577) = (120577 + 1)3

9

sum

119896=0

119888119894120577119894 (75)

where1198889= 1

1198888= 120572119899+ 120572119902+ 120573119899+ 120573119902+ 120574119899+ 120574119902+ 120575119899+ 120575119902+ 120583119889

1198880= 1205731198991205731199021205741198991205741199021205751198991205751199021205831198891205721198991205721199021 minus

12057911198615

120572119899

minus(1 minus 120579

1) 1198616

120572119902

= 120573119899120573119902120574119899120574119902120575119899120575119902120583119889120572119899120572119902(1 minus 119877

0)

(76)

and we can see that 1198880changes sign from positive to negative

when 1198770increases from values of 119877

0lt 1 through 119877

0= 1 to

values of1198770gt 1 indicating a change in stability of the disease-

free equilibrium as 1198770increases from unity


34 The Basic Reproduction Number A threshold parameterthat is of essential importance to infectious disease trans-mission is the basic reproduction number denoted by 119877

0 1198770

measures the average number of secondary clinical cases ofinfection generated in an absolutely susceptible populationby a single infectious individual throughout the periodwithin which the individual is infectious [26ndash29] Generallythe disease eventually disappears from the community if1198770lt 1 (and in some situations there is the occurrence

of backward bifurcation) and may possibly establish itselfwithin the community if 119877

0gt 1 The critical case 119877

0= 1

represents the situation in which the disease reproduces itselfthereby leaving the community with a similar number ofinfection cases at any time The definition of 119877

0specifically

requires that initially everybody but the infectious individualin the population be susceptible Thus this definition breaksdown within a population in which some of the individ-uals are already infected or immune to the disease underconsideration In such a case the notion of reproductionnumberR becomes useful Unlike 119877

0which is fixedRmay

vary considerably with disease progression However R isbounded from above by 119877

0and it is computed at different

points depending on the number of infected or immune casesin the population

One way of calculating 1198770is to determine a threshold

condition for which endemic steady state solutions to thesystem under study exist (as we did to derive (69)) orfor which the disease-free steady state is unstable Anothermethod is the next-generation approach where 119877

0is the

spectral radius of the next-generation matrix [26] Using thenext-generation approach we identify all state variables forthe infection process 119901

119899 119901119902 119888119899 119888119902 119894119899 119894119902 119903119903 119903119889 and ℎ The

transitions from 119904119899 119904119902to 119901119899 119901119902are not considered new

infections but rather a progression of the infected individualsthrough the different stages of disease compartments Hencewe identify terms representing new infections from the aboveequations and rewrite the system as the difference of twovectors F and V where F consists of all new infections andV consists of the remaining terms or transitions betweenstates That is we set x = F minus V where x is the vectorof state variables corresponding to new infections x =

(119904 119904119899 119904119902 119901119899 119901119902 119888119899 119888119902 119894119899 119894119902 119903119903 119903119889 ℎ)119879 This gives rise to

F =

(((((((((((((((((

(

0

1205791120582119904

(1 minus 1205791) 120582119904

0

0

0

0

0

0

0

0

0

)))))))))))))))))

)

V =

((((((((((((((((((((((((((((((((

(

minus1 + 120582119904 minus 1198871119901119899minus 1198872119904119899minus 1198873119904119902minus 1198874119901119902+ 119904

120572119899119904119899

120572119902119904119902

120573119899(minus119904119899+ 119901119899)

120573119902(minus119904119899minus 1198861119904119902+ 119901119902)

120574119899(minus119901119899+ 119888119899)

120574119902(minus119901119899minus 1198862119901119902+ 119888119902)

120575119899(minus119888119899+ 119894119899)

120575119902(minus119888119899minus 1198863119894119899minus 1198864119888119902+ 119894119902)

minus119888119899minus 1198865119894119899minus 1198866119888119902minus 1198867119894119902+ 119903119903

minus120583119889119894119899minus 1205831198891198868119894119902+ 120583119889119903119889

minus1 + ℎ + 1198878119903119889

))))))))))))))))))))))))))))))))

)

(77)

where the force of infection 120582 is given by (39) To obtain thenext-generation operator 119865119881minus1 we must calculate (119865)

119894119895=

120597F119894120597119909119895and (119881)

119894119895= 120597V

119894120597119909119895evaluated at the disease-free

equilibrium position where 119904 = 1 = ℎ 119904119899= 119904119902= 119901119899=

119901119902= 119888119899= 119888119902= 119894119899= 119894119902= 119903119903= 119903119889= 0 The basic

reproduction number is then the spectral radius of the next-generationmatrix119865119881minus1Thus if 984858(119865119881minus1) is the spectral radiusof the matrix 119865119881minus1 then

1198770= 984858 (119865119881

minus1) =

1

120572119899120572119902

[1205721199021205791(1

+ (1 + 1198863+ (1 + 119886

2) 1198864) 119886119889

+ 120585119902(11988621198864+ 1198863+ 1198864+ 1) + 119886

2120591119902+ 120585119899+ 120588119899+ 120588119902

+ 120591119899+ 120591119902) minus 120572

119899(1205791minus 1) (119886

1120588119902

+ 1198861[11988621198864(1198868119886119889+ 120585119902) + 1198862120591119902])]

=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

(78)

as computed beforeThe expression for 119877

0has two parts The first part

measures the number of new EVD cases generated byan infected nonquarantined human It is the product of1205791(the proportion of susceptible individuals who become

suspected but remain nonquarantined) 1198615(which indicates

the contacts from this proportion of individuals with infectedindividuals at various stages of the disease) and 1120572

119899(which

is the average duration a human remains as a suspectednonquarantined individual) In the sameway the second partcan be interpreted likewise

The stability of the endemic steady state is obtained bycalculating the eigenvalues of the linearized matrix evaluated


at the endemic state The computations soon become verycomplicated because of the size of the system and we proceedwith a simplification of the system

35 Pseudo-Steady StateApproximation EbolaVirusDiseaseis a very deadly infection that normally kills most of its vic-tims within about 21 days of exposure to the infection Thuswhen compared with the life span of the human the elapsedtime representing the progression of the infection from firstexposure to death is short when compared to the total timerequired as the life span of the human Thus we set 120583 asymp1life span of human so that the rates 120572

lowast 120573lowast and so forth

will all be such that 1rate asymp resident time in given statesome of which will be short compared with the life span ofthe human It is therefore reasonable to assume that

120583

120583 + rateasymp small 997904rArr

120583 + rate120583

asymp large (79)

so that the scaling above renders some of the state variablesessentially at equilibrium That is the quantities 1120573

119899 1120573119902

1120574119899 1120574119902 1120575119899 1120575119902 and 119887120583 may be regarded as small

parameters so that in the corresponding equations (43)ndash(48)and (50) the state variables modelled by these equations areessentially in equilibrium and we can evoke the Michaelis-Menten pseudo-steady state hypothesis [30] To proceed wemake the pseudoequilibrium approximation

119901119899= 119888119899= 119894119899= 119904119899

119901119902= 119904119899+ 1198861119904119902

119888119902= 1198601119904119899+ 1198602119904119902

119894119902= 1198603119904119899+ 1198604119904119902

119903119889= 1198607119904119899+ 1198608119904119902

(80)

to have the reduced system119889119904

119889119905= 1 minus 120582119904 + (120572

119899minus 1198613) 119904119899+ (120572119902minus 1198614) 119904119902minus 119904 (81)

119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (82)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (83)

119889119903119903

119889119905= 1198605119904119899+ 1198606119904119902minus 119903119903 (84)

119889ℎ

119889119905= 1 minus ℎ minus 119861

1119904119899minus 1198612119904119902 (85)

and the total population and the force of infection also reduceaccordingly In particular we have

120582119904 = (120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)) 119904 = (119861

5(119904119899

ℎ)

+ 1198616(

119904119902

ℎ)) 119904

(86)

where the variables 119904 and the augmented population ℎ satisfythe differential equations (81) and (85) respectively

System (81)ndash(85) has the same steady states solutions asthe original system if we combine it with (80) However onits own it represents a pseudo-steady state approximation[30] of the original system Clearly the reduced system hastwo realistic steady states 119864dfe and 119864

119890119890 so that if 119864xlowast =

(119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast) is a steady state solution then

119864dfe = (1 0 0 0 1)

119864119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast)

(87)

where following the same method as was done in the fullsystem

119904lowast

119902=1198617

1198618

119904lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902

119903lowast

119903= 1198605119904lowast

119899+ 1198606119904lowast

119902

119904lowast= 1 minus 119861

3119904lowast

119899minus 1198614119904lowast

119902

ℎlowast= 1 minus 119861

1119904lowast

119899minus 1198612119904lowast

119902

(88)

All coefficients are as defined in (58) When steady states (88)are rendered in parameters of the reduced system taking intoconsideration the fact that from (68) 119911 = 119861

3119910 + 119861

4119909 and

1198611119910 + 1198612119909 = 1 we get

119904lowast= (119911 minus 1

R minus 1)

119904lowast

119899= 119910(

1198770minus 1

R minus 1)

119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

ℎlowast= ((119911 minus 1) 119877

0

R minus 1)

(89)

The stability of the steady states is determined by theeigenvalues of the linearized matrix of the reduced systemevaluated at the steady state xlowast = (119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast) If 119869(xlowast) is

the Jacobian of the system near the steady state xlowast then


119869 (xlowast) =((

(

minus1198623(xlowast) minus 1 minus119862

4(xlowast) + 120572

119899minus 1198613minus1198625(xlowast) + 120572

119902minus 11986140 minus119862

6(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) minus 120572

11989912057911198625(xlowast) 0 120579

11198626(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) 120579


1199020 12057911198626(xlowast)

0 1198605

1198606

minus1 0

0 minus1198611

minus1198612

0 minus1

))

)

(90)

where 1205791= 1 minus 120579

1and

1198623(xlowast) = 119861

5

119904lowast

119899

ℎlowast+ 1198616

119904lowast

119902

ℎlowast=120572119899119910 (1198770minus 1)

1205791(119911 minus 1)

1198624(xlowast) = 119861

5

119904lowast

ℎlowast=1198615

1198770

1198625(xlowast) = 119861

6

119904lowast

ℎlowast=1198616

1198770

1198626(xlowast) = minus(119861

5

119904lowast119904lowast

119899

ℎlowast2+ 1198616


119902

ℎlowast2) = minus

120572119899119910 (1198770minus 1)

1205791 (119911 minus 1) 1198770

(91)

The asterisk is used to indicate that the quantities so cal-culated are evaluated at the steady state We can perform astability analysis on the reduced system by noting that if 120577 isan eigenvalue of (90) then 120577 satisfies the polynomial equation

1198755(120577 xlowast)

= (120577 + 1)2(1205773+ 1198762(xlowast) 1205772 + 119876

1(xlowast) 120577 + 119876

0(xlowast))

= 0

(92)

where

1198762(xlowast) = 120572

119899+ 120572119902+ 1198623(xlowast) minus 119862

4(xlowast) 120579

1minus 1198625(xlowast) 120579

1

+ 1

1198761(xlowast) = 120572

119902

minus 1205791(minus11986111198626(xlowast) + 120572

1199021198624(xlowast) + 119862

4(xlowast))

+ 11986121198626(xlowast) 120579

1+ 1198623(xlowast) (120572

1199021205791+ 11986131205791+ 11986141205791)

+ 120572119899(120572119902+ 12057911198623(xlowast) minus 119862

5(xlowast) 120579

1+ 1) minus 119862

5(xlowast)

sdot 1205791

1198760(xlowast) = 120572

1199021205791(11986111198626(xlowast) + 119861


4(xlowast))

+ 120572119899(1205791(11986121198626(xlowast) + 119861


5(xlowast)) + 120572

119902)

(93)

Now the signs of the zeros of (92) will depend on the signs ofthe coefficients 119876

119894 119894 isin 0 1 2 We now examine these

At the disease-free state where 119904lowast = 1 = ℎlowast 119904lowast119899= 119904lowast

119902=

119903lowast

119903= 0 or equivalently 119877

0= 1 we have xlowast = xdfe =

(1 0 0 0 1) so that 1198623(xdfe) = 0 1198624(xdfe) = 1198615 1198625(xdfe) =

1198616 1198626(xdfe) = 0 and (92) becomes

1198755(120577 xdfe) = (120577 + 1)

3(1205772+ 119876dfe120577 + 119877dfe) = 0 (94)

where

119876dfe = 120572119899 + 120572119902 minus 11986151205791 minus 1198616 (1 minus 1205791)

= 120572119899(1 minus 119877

0) + 120572119902+ (1 minus 120579

1) 1198616(

120572119899minus 120572119902

120572119902

)

119877dfe = 120572119902 (120572119899 minus 11986151205791) + 1205721198991198616 (1205791 minus 1)

= 120572119902120572119899(1 minus 119877

0)

(95)

The roots of (94) are minus1 minus1 minus1 and (minus119876dfe plusmn

radic1198762

dfe minus 4120572119902120572119899(1 minus 1198770))2 showing that there is onepositive real solution as 119877

0increases beyond unity and the

disease-free equilibrium loses stability at 1198770= 1 For the

local stability when 1198770le 1 the additional requirement

119876dfe gt 0 is necessaryAt the endemic steady state and in the original scaled

parameter groupings of the system xlowast = x119890119890

=


119899 119904lowast

119902 119903lowast

119903 ℎlowast) the coefficients of (92) simplify accordingly

and we have

1198755(120577 xlowast) = (120577 + 1)2 (1205773 + 119875x

119890119890

1205772+ 119876x

119890119890

120577 + 119877x119890119890

) = 0 (96)

where


119875x119890119890

=

119861612057911205791 (119911 minus 1) (120572119899 minus 120572119902) + 1205721199021198770 (120572119899 (1198770 minus 1) 119910 + (120572119902 + 1) 1205791 (119911 minus 1))

12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)

Now the necessary and sufficient conditions that will guaran-tee the stability of the nontrivial steady state x

119890119890will be the

Routh-Hurwitz criteria which in the present parameteriza-tions are

119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)

With this characterization we can then explore special casesof intervention

36 Some Special Cases All initial suspected cases are quar-antined that is 120579

1= 0 In this case we see that 119904

119899= 0 and we

have only the right branch of our flow chart in Figure 1 Wehave here a problem involving infections only at the treatmentcentres Mathematically we then have

1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)

showing that all solutions of the equation 1198755(120577 xlowast) = 0

are negative or have negative real parts whenever they arecomplex indicating that the nontrivial steady state is stableto small perturbations whenever 119877

0gt 1 In this case we

can regard an increase in 1198770as an increase in the parameter

grouping 1198616

All initial suspected cases escape quarantine that is 1205791=

1 In this case we see that 119904119902= 0 and initially we will be on the

left branch of our flow chart in Figure 1 The strength of thepresent model is that based on its derivation it is possiblefor some individuals to eventually enter quarantine as thesystems wake up from sleep and control measures kick intoplace Mathematically we then have

1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)


Suspected nonquarantinedcasesSN

Probable nonquarantinedcasesPN

Confirmed nonquarantinedearly symptomatic cases

CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN

Confirmed nonquarantinedlate symptomatic cases

IN

Susceptible false suspected and probable casesSusceptibles (S)

120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN

rENCN Removal byrecovery (RR)

rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ

Suspectedquarantined cases

SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ

Probablequarantined cases

PQ

1205797120573QPQ

Confirmed quarantinedearly symptomatic cases

CQ

120574QCQ

Confirmed quarantinedlate symptomatic cases

(IQ)

(1 minus 1205793)120573NPN

Removal by deathdue to EVD (RD)

bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π

Figure 1 Conceptual framework showing the relationships between the different compartments that make up the different population ofindividuals and actors in the case of an EVD outbreak Susceptible individuals include false suspected and probable cases True suspectsand probable cases are confirmed by a laboratory test and the confirmed cases can later develop symptoms and die of the infection orrecover to become immune to the infection Humans can also die naturally or due to other causes Nonquarantined cases can becomequarantined through intervention strategies Others run the course of the illness from infection to death without being quarantined Flowfrom compartment to compartment is as explained in the text

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)

again showing that all solutions of the equation 1198755(120577 xlowast) =

0 are negative or have negative real parts whenever theyare complex indicating that the nontrivial steady state isstable to small perturbations whenever 119877

0gt 1 In this

case we can regard an increase in 1198770as an increase in the

parameter grouping 1198615 1198770in this case appears larger than

in the previous casesThe rate at which suspected individuals become probable

cases is the same that is 120572119873= 120572119876 In this case the flow

from being a suspected case to a probable case is the samein all circumstances irrespective of whether or not one is

quarantined or nonquarantined Mathematically we havethat 120572

119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)

In this case as well all solutions of the equation 1198755(120577 xlowast) =

0 either are negative or have negative real parts whenever1198770gt 1 and 119911 gt 1 showing that in this case again the steady

state is stable to small perturbations In this particular casean increase in 119877

0can be regarded as an increase in the two

parameter groupings 1198615and 119861

6

4 Parameter Discussion

Some parameter values were chosen based on estimates in[15 20] on the 2014 Ebola outbreak while others wereselected from past estimates (see [9 14 18]) and are sum-marized in Table 2 In [20] an estimate based on dataprimarily from March to August 20th yielded the followingaverage transmission rates and 95 confidence intervals 027


(027 027) per day in Guinea 045 (043 048) per day inSierra Leone and 028 (028 029) per day in Liberia In [15]the number of cases of the 2014 Ebola outbreak data (upuntil early October) was fitted to a discrete mathematicalmodel yielding estimates for the contact rates (per day) in thecommunity and hospital (considered quarantined) settings as0128 in Sierra Leone for nonquarantined cases (and 0080for quarantined cases about a 61 reduction) while the ratesin Liberia were 0160 for nonquarantined cases (and 0062for quarantined cases close to a 375 drop) The models in[15 20] did not separate transmission based on early or latesymptomatic EVD cases which was considered in ourmodelBased on the information we will assume that effectivecontacts between susceptible humans and late symptomaticEVD patients in the communities fall in the range [012 048]per day which contains the range cited in [16] Thus 120585

119873isin

[012 048] However wewill consider scenarios inwhich thisparameter varies Furthermore if we assume about a 375to 61 reduction in effective contact rates in the quarantinedsettings then we can assume that 120585

119876= 120601119876120585119873 where

120601119876isin [00375 0062] However to understand how effective

quarantining Ebola patients is general values of 120601119876isin [0 1]

can be considered Under these assumptions a small value of120601119876will indicate that quarantining was effective while values

of 120601119876close to 1 will indicate that quarantining patients had no

effect in minimizing contacts and reducing transmissionsSince patients with EVD at the onset of symptoms are less

infectious than EVD patients in the later stages of symptoms[2 10] we assume that the effective contact rate betweenconfirmed nonquarantined early symptomatic individualsand susceptible individuals denoted by 120591

119873 is proportional

to 120585119873with proportionality constant 119902

119873isin [0 1] Likewise

we assume that the effective contact rate between confirmedquarantined early symptomatic individuals and susceptibleindividuals is proportional to 120585

119902with proportionality con-

stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1

The range for the parameter 119886119863 of the effective con-

tact rate between cadavers of confirmed late symptomaticindividuals and susceptible individuals was chosen to be[0111 0489] per day where 0111 is the rate estimated in [15]for Sierra Leone and 0489 is that for Liberia With controlmeasures and education in place these rates can be muchlower

The incubation period of EVD is estimated to be between2 and 21 days [2 7ndash9] with a mean of 4ndash10 days reported in[8 9] In [10] a mean incubation period of 9ndash11 days wasreported for the 2014 EVD Here we will consider a rangefrom 4 to 11 days with a mean of about 10 days used as thebaseline valueThuswewill consider that120572

119873and120572119876are in the

range [111 14] At the end of the incubation period earlysymptoms may emerge 1ndash3 days later [10] with a mean of 2days Thus the mean rate at which nonquarantined (120573

119873) and

quarantined (120573119876) suspected cases become probable cases lies

in [13 1] [12] About 1 or 2 to 4 days later after the earlysymptoms more severe symptoms may develop so that therates at which nonquarantined (120574

119873) and quarantined (120574

119876)

probable cases become confirmed cases lie in [14 12]

The parameter 120574119873

measures the rate at which earlysymptomatic individuals leave that class This could be as aresult of recovery or due to increase and spread of the viruswithin the human It takes about 2 to 4 days to progress fromthe early symptomatic stage to the late symptomatic stageso that 120574

119873 120574119876isin [14 12] which can be assumed to be the

reciprocal of the mean time it takes from when the immunesystem is either completely overwhelmed by the virus or keptin check via supportive mechanism Severe symptoms arefollowed either by death after about an average of two tofour days beyond entering the late symptomatic stage or byrecovery [12] and thus we can assume that 120575

119873and 120575

119876are

in the range [14 12] If on the other hand the EVD patientrecovers then it will take longer for patient to be completelyclear of the virus Notice that based on the ranges givenabove the time frames are [6 16] days from the onset ofsymptoms of Ebola to death or recovery The range from theonset of symptoms which commences the course of illnessto death was given as 6ndash16 days in [8] while the rangefor recovery was cited as 6ndash11 days [8] For our baselineparameters the mean time from the onset of the illness todeath or recovery will be in the range of 6ndash11 days

Here we will consider that the recovery rate for quar-antined early symptomatic EVD patients lies in the range[04829 05903] per day with a baseline value of 05366 perday as cited in [16]Thus 119903

119864119876isin [04829 05903] If we assume

that patients quarantined in the hospital have a better chanceof surviving than those in the community or at home withoutthe necessary expert care that some of the quarantined EVDpatients may get then we can consider that the recovery ratefor nonquarantined EVD patients in the community wouldbe slightly lower Thus we scale 119903

119864119876by some proportion 120596 isin

[0 1] so that 119903119864119873= 120596119903119864119876 Since late symptomatic patients

have a much lower recovery chance then both 119903119871119873

and 119903119871119876

will be lower than 119903119864119873

and 119903119864119876 respectively Hence we will

consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1

Sometimes late symptomatic EVD patients are removed fromthe community and quarantined Here we assume a mean of2 days so that 120590

119873= 05 per day

The fractions 120579119894measure the proportions of individu-

als moving into various compartments If we assume thatmembers in the quarantined classes are not left uncheckedbut have medical professionals checking them and givingthem supportive remedies to boost their immune system tofight the Ebola Virus or enable their recovery then it will bereasonable to assume that 120579

6 1205797 1205794 and 120579

5are all greater than

05 If such an assumption is not made then the parameterscan be chosen to be equal or close to each other

The parameter Π is chosen to be 555 per day as in [16]Furthermore the parameters 120588

119873and 120588

119876and the effective

contact rates between probable nonquarantined and respec-tively quarantined individuals and susceptible individualswill be varied to see their effects on the model dynamicsHowever the values chosen will be such that the value of 119877

0

computed is within realistic reported rangesThe parameter 120583 the natural death rate for humans

is chosen based on estimates from [13] The parameter 119887measures the time it takes from death to burial of EVDpatients A mean value of 2 days was cited in [14] for the 1995


Table 2 Parameters baseline values and ranges of baseline values with references

Parameters Baseline values Range of values ReferenceΠ 3 555 Varies120588119873

Varies Varies120588119876


027 [012 048] Estimatedlowast

120591119873

119902119873120585119873 119902119873= 075 119902

119873isin [0 1] Variable

120585119876

120601119876120585119873 120601119876= 05 120601

119876isin [0 1] Estimatedlowast

120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876

13 [14 12] [10 12]120583 1(60 times 365) [1(40 times 365) 1(70 times 365)] dayminus1 [13]119887 125 [1450 12] dayminus1 [14 15]119886119863

03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873

05 [13 1] EstimatelowastEstimates discussed in Section 4

and 2000 Ebola outbreak epidemics in the DemocraticRepublic of Congo and Uganda respectively For the 2014West African Outbreak the estimates were 201 days inLiberia and 450 days in Sierra Leone [15]

5 Numerical Simulation of the ScaledReduced Model

The parameter values given in Table 2 were used to carryout some numerical simulations for the reducedmodel (81)ndash(85) when the constant recruitment term is 555 persons perday The varying parameters were chosen so that 119877

0would

be within ranges of reported values which are typically lessthan 25 (see eg [16 22]) In Figure 2 we show a timeseries solution for a representative choice of values for theparameters 120588

119873and 120588119876 Figures 2(a)ndash2(e) show the long term

solutions to the reduced model exhibiting convergence to thestable nontrivial equilibrium in the case where 119877

0gt 1 the

case with sustained infection in the community Figure 2(f)then shows an example of convergence to the trivial steadystate when 119877

0lt 1 the case where the disease is eradicated In

that example we notice that as 1198770lt 1 119904119899rarr 0 as 119899 rarr infin and

this in turn implies that 119904119902rarr 0 as 119899 rarr infin and eventually the

system relaxes to the trivial state (119904 119904119899 119904119902 119903119903 ℎ) = (1 0 0 0 1)

for large timeWe note here that the computed value of119877

0can be shown

to be linear in the variables 120588119876and 120588

119873 when eventually all

parameters have been assigned it may be written as 1198770=

1199030+1199031120588119876+1199032120588119873 where 119903

119894 119894 = 0 1 2 are positive constants that

can be shown to be dependent on the other parameters Thus1198770will increase linearlywith increase in any of the parameters

120588119876and 120588

119873for fixed given values of the other parameters

Though we have theoretically found for example that 119904lowastbecomes infinite when R is near one equivalent to 119877

0being

near 1119911 this case does not arise because we have assumed inthe analysis that 119911 gt 1 Thus the case 0 le 119911 lt 1 is linked withthe trivial steady state

The situation shown in Figure 3 has important conse-quences for control strategies While 119904lowast

119899varies sharply for

a narrow band of reproduction numbers its values do notchange much for larger values of 119877

0 Referring to Figure 3

an application of a control measure that will reduce 1198770say

from a high value of 10 down to 5 a 50 reduction will notappreciably affect the rest of the disease transmission Thusthe system is best controlledwhen119877

0is small which can occur

in the early stages of the infection or late in the infectionwhensome effective control measures have already been institutedsuch as effective quarantining or prompt removal of EVDdeceased individuals Notice that as 119877

0further increases the

number of susceptible individuals continues to dropWenotehowever that typical values of119877

0computed for the 2014 Ebola

outbreak are less than 25Next we investigate the effect of 120585

119873on 1198770and the

model dynamics In Figure 2(f) we showed an example of


00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)

Figure 2 (a)ndash(e) Time series showing convergence of the solutions to the steady states for the nondimensional reduced model when theconstant recruitment term is 555 persons per day In this example 120579

1= 085 and 120588

119873= 115 and 120588

119876= 085 and all the other parameters are

as in Table 2 giving values of 1198770= 1102 In this case the nonzero steady state is stable and the solution converges to the steady state value

as given by (89) as 119905 rarr infin (f) Time series showing the long term behaviour of the variable 119904119899in the reduced model for 120588

119873= 120588119876= 08 and

all other values of the parameters are as given in Table 2 In this case 1198770= 088 and so the only steady state is the trivial steady state which is

stable

convergence to the trivial steady state for 120588119873= 08 = 120588

119876

for the nondimensional reduced model when the constantrecruitment term is 555 persons per dayThemodel dynamicsyielded an 119877

0value of 088 lt 1 when all other parameters

were as given in Table 2 From this scenario we increasedonly 120585

119873from its baseline Table 2 value of 027 to 0453

This yields an increase in 1198770to 100038 and we see from

Figures 4(a) and 4(b) that the disease begins to propagateand stabilize within the community There is a major peakwhich starts to decay as EVD deaths begin to rise Anestimated size of the epidemic can be computed as thedifference between 119904 and ℎ when the disease dynamics settlesto its equilibrium state As more and more persons becomeinfected 119877

0increases and the estimated size of the epidemic


10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)

Figure 3 Graph showing the behaviour of the steady states 119904lowast and 119904lowast119899as a function of 119877

0 (a) This graph shows the form of the steady state 119904lowast

as a function of 1198770 (b) This graph shows the form of the steady state 119904lowast

119899as a function of 119877

0 The steady state solution 119904lowast

119899varies greatly only in

a narrow band of reproduction numbers but saturates for large values of 1198770 On the other hand 119904lowast continues to drop to zero as 119877

0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)

Figure 4 (a)ndash(d) Time series plot showing the propagation and stabilization of EVD to a stable nontrivial steady state for the nondimensionalreduced model when the constant recruitment term is 555 persons per day and with 120588

119873= 120588119876= 08 and 120579

1= 085 as used in Figure 2(f)

Except for 120585119873that is increased from its baseline value of 027 all other parameters are as in Table 2 In graphs (a) and (b) 120585

119873is increased

to 0453 This yields 1198770= 100038 slightly bigger than 1 The graphs show that there is a major peak which starts to decay as EVD deaths

begin to rise The size of the epidemic can be estimated as the difference in the areas between the 119904 and ℎ curves as the disease settles to itssteady state Graphs (c) and (d) show the model dynamics when 120585

119873is further increased to 048 which yields 119877

0= 101765 In graphs (c) and

(d) the oscillations are more pronounced and the size of the epidemic is larger due to the increased effective contacts with late symptomaticindividuals


09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)

Figure 5 (a)-(b) Time series plot showing the propagation and stabilization of EVD to a stable nontrivial steady state for the nondimensionalreduced model when the constant recruitment term is 555 persons per day and with 120588

119873= 115 120588

119876= 085 and 120579

1= 085 as in Figures 2(a)ndash

2(e) Except for 120585119873that is increased from 027 to 036 all the other parameters are as given in Table 2 and the corresponding 119877

0value is

1198770= 1159 Notice that in this case the disease has a higher frequency of oscillations and the difference between the areas under ℎ and 119904 is

considerably larger indicating that the size of the disease burden is considerably larger in this case

also increases as is expected In particular increasing 120585119873

further to 048 increases 1198770to 101765 and from Figures

4(c) and 4(d) the difference between 119904 and ℎ is visiblylarger compared to the difference from Figures 4(a) and 4(b)Moreover the oscillatory dynamics becoming more pro-nounced indicated a higher back and forthmovement activitybetween the 119904 and 119904

119899and 119904ℎclasses

In Figures 2(a)ndash2(e) we showed an example of the longterm dynamics of the solutions of the reduced model for120588119873= 115 and 120588

119876= 085 For this case we obtained

1198770= 1102 gt 1 and the model dynamics show how the

reduced model converges to a stable nontrivial equilibriumwhen all other parameters are as stated as in Table 2 Fromthis point if we increase only 120585

119873from its default value of 027

to 036 we see that 1198770increases to approximately 1159 and

the model exhibits irregular random oscillations with higherfrequency but eventually stabilizes (Figures 5(a) and 5(b))The size of the epidemic is considerably larger in this caseThis highlights the importance of reducing contacts betweenEVD patients and susceptible humans in controlling the sizeof the disease burden and lowering the impact of the disease

51 Fade-Outs and Epidemics in Ebola Models Our modelresults as highlighted in Figures 2(a)ndash2(e) 4 and 5 indicatethat it is possible to have a long term endemic situationfor Ebola transmission if conditions are right In particularin our model for the case where we have a relatively largeconstant recruitment term of 555 persons per day and withavailable resources to sustain the quarantine efforts then aslong as there are people in the community (nonquarantined)with the possibility to come in contact with infectiousEVD fluids then the disease can be sustained as long as1198770gt 1 (Figures 2(a)ndash2(e) 4 and 5) Increasing control

by reducing contacts early enough between suspected andprobable individuals with susceptible individuals can bringdown the size of 119877

0to a value lt1 which eventually leads to

the eradication of the disease Thus control which includesquarantining has to be comprehensive and sustained untileradication is achieved

The 2014 Ebola outbreak did not show sustained diseasestates The disease dynamics exhibited epidemic fade-outsHere we show that such fade-outs are possible with ourmodel To investigate the epidemic-like fade-outs we firstnote that due to the scaling adopted in our model our timescales are large However any epidemic-like EVD behaviourwould be expected to occur over a shorter timescale Thusfor the results illustrated here we plot the model dynamicsin terms of the original variable by simulating the equationsthat make up the system (11)ndash(23) To illustrate that the largetime scales do not affect the long term dynamics of the modelresults we first present a graph of the original system in thecase where the parameters are maintained as those used inFigure 6 with Π = 555 persons per day 120579

1= 085 and

120588119873= 115 and 120588

119876= 085 and with all the other parameters

as given in Table 2The 1198770value was 1102 and so a sustained

disease with no other effort is possible over a long time frameof more than 10000 days

When the number of individuals recruited daily reducesthen we can show that for the case where Π = 3 personsper day 120579

1= 085 and 120588

119873= 2 and 120588

119876= 1 and all the

other parameters remain as in Table 2 then an epidemic-likebehaviour is obtained (see Figure 7)

The dynamics of Figures 6 and 7 indicate that quaran-tining alone is not sufficient to eradicate the Ebola epidemicespecially when there is a relative high number of dailyrecruitment In fact quarantining can instead serve as a bufferzone allowing the possibility of sustained disease dynamicswhen there are a reasonable number of people recruitedeach day However when the daily recruitment is controlledreduced to a value of 3 per day then the number of new dailyinfections is reduced to a low value as depicted in Figures7(a)ndash7(c) However the estimated cumulative number ofinfections increases daily (see Figure 7(d))


S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)

Figure 6 (a)ndash(c) Time series showing convergence of the solutions to the steady states for the full model in dimensional form when theconstant recruitment term is 555 persons per day In this example 120579

1= 085 and 120588

119873= 115 and 120588


as in Table 2 giving a value of 1198770= 1102 The graph shows the short scale dynamics as well as the long term behaviour showing stability of

the nonzero steady state

6 Discussion and Conclusion

In this paper we set out to derive a comprehensive modelfor the dynamics of Ebola Virus Disease transmission in acomplex environment where quarantining is not effectivemeaning that some suspected cases escape quarantine whileothers do not When the West African countries of LiberiaGuinea and Sierra Leone came face to face with EbolaVirus Disease infection in 2014 it took the internationalcommunity some time to react to the crises As a resultmost of the initial cases of EBV infection escapedmonitoringand entered the community African belief systems and othertraditional practices further compounded the situation andbefore long large number of cases of EBV infections werein the community Even when the international communityreacted and started putting in place treatment centres it stilltook some time for people to be sensitized on the dangers theyare facing The consequence was that infections continuedin families during funerals and even in hospitals Peoplechecked into hospitals and would not tell the truth abouttheir case histories and as a result somemedical practitionersgot exposed to the infection A case in point is that of DrStella Ameyo Adadevoh an Ebola victim and everyday hero[31] who prevented the spread of Ebola in Nigeria and paidwith her life We still pay tribute and honour to her and the

other health workers whose dedication was inspirational andhelpful in curbing the 2014 Ebola outbreak in Africa

As we now look forward with optimism for a better andEbola-free tomorrow there is work going on in the scientificcommunity to develop vaccines [2] Mathematical modellingof the dynamics and transmission of Ebola provides uniqueavenues for exploration of possible management scenarios inthe event of an EVD outbreak since during an outbreakmanagement of the cases is crucial for containment of thespread of the infection within the community In this paperwe have presented a comprehensive ordinary differentialequation model that handles management issues of EVDinfection Our model takes care of quarantine and nonquar-antine cases and therefore can be used to predict progressionof disease dynamics in the population Our analysis hasshown that the initial response to all suspected cases of EVDinfection is crucial This is captured through the parameter1205791which measures the initial fraction of suspected cases

that are put into quarantine We have shown that the basicreproduction number can be indexed by this parameter in thesense that when all cases are initially quarantined the spreadof the infection can only take place at the treatment centresbut in cases where all suspected cases escape quarantine thereproduction number can be large Our model has been ableto quantify the densities of infected and recovered individuals


S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)

Figure 7 (a)-(b) Time series showing epidemic-like behaviours of the solutions to the steady states for the full model in dimensional formwhen the constant recruitment term is 3 persons per day over a short time scale In this example 120579

1= 085 and 120588

119873= 2 and 120588


other parameters are as in Table 2 The value of 1198770= 163455 The graph shows the short scale dynamics exhibiting epidemic-like behaviour

that fades out

within the population based on baseline parameters identi-fied during the 2014 Ebola Virus Disease outbreak in Africa

The basic reproduction number in our model dependson the initial exposure rates including exposure to cadaversof EVD victims The provision of scope for further quar-antining during the progression of the infection means thatthese exposure rates are weighted accordingly dependingon whether or not the system woke up from slumber andpicked up those personswho initially escaped quarantine Forexample the parameter 120591

119876which measures the effective con-

tact rate between confirmed quarantined early symptomaticindividuals and susceptible individuals is eventually scaledby the proportions 120579

3119886and 1205792119886

which respectively are thoseproportions of suspected and probable cases that eventuallyprogress to become EVD patients and have escaped quar-antine Thus our framework can progressively be used ateach stage to manage the progression of infections in thecommunity

Our results show that eventually the system settles downto a nonzero fixed point when there is constant recruitmentinto the population of 555 persons per day and for 119877

0gt 1

The values of the steady states are completely determined interms of the parameters in this case Our analysis also shows

that it is possible to control EVD infection in the communityprovided we reduce and maintain the reproduction numberto below unity Such control measures are possible if there iseffective contact tracing and identification of EVD patientsand effective quarantining since a reduction of the propor-tion of cases that escape quarantine reduces the value of 119877

0

Additionally our model results indicate that when thereare a high constant number of recruitment into an EVDcommunity quarantining alone may not be sufficient toeradicate the disease It may serve as a buffer enhancinga sustained epidemic However reducing the number ofpersons recruited per day can bring the diseases to very lowvalues

To demonstrate the feasibility of our results we per-formed a pseudoequilibrium approximation to the systemderived based on the assumption that the duration of man-ifestation of EVD infection in the community per individualis short when compared with the natural life span of anaverage human The reduced model was used to show thatall steady state solutions are stable to small perturbationsand that there can be oscillatory returns to the equilibriumsolutionThese results were confirmed with numerical simu-lations Given the size of the system we have not been able to


perform a detailed nonlinear analysis on themodel Howeverthe discussion on the nature of the parameters for themodel isbased on the statistics gathered from the 2014 EVD outbreakin Africa and we believe that ourmodel can be useful in char-acterizing and studying a class of epidemics of Ebola-typeWe have not yet carried out a complete sensitivity analysis onall the parameters to determine the most crucial parametersin our model This is under consideration Furthermore theeffect of stochasticity seems relevant to study This and otheraspects of the model are under consideration

Notations

State Variables and Their Descriptions

119867119871(119905) Total population density of living humans at

any time 119905119867(119905) The total population density at time 119905 of

living humans together with Ebola-relatedcadavers that is those humans who havedied of EVD and have not yet been disposedat time 119905

119878(119905) Population density at time 119905 of all susceptiblehumans in the population

119878119873(119905) 119878119876(119905) Population densities at time 119905 of all humans

that are known to have been in contact withor have a history of association with anyperson known to have once had or died ofEVD these are suspected Ebola Virus patientcases that are either not quarantined 119878

119873or

quarantined 119878119876and who are not yet showing

any Ebola-like symptoms119875119873(119905) 119875119876(119905) Population densities at time 119905 of all persons

suspected of having EVD infection and whopresent with fever and at least three otherEbola-like symptoms these probable casesare either not quarantined 119875

119873or quarantined

119875119876

119862119873(119905) 119862119876(119905) Population densities at time 119905 of all probable

Ebola Virus infected humans who after a labtest have been confirmed to indeed haveEVD infection and who still present onlyearly Ebola-like symptoms of fever achestiredness and so forth they are calledconfirmed early symptomatic in the senseexplained in the text these confirmed EbolaVirus carriers are either not quarantined 119862

119873

or quarantined 119862119876

119868119873(119905) 119868119876(119905) Population densities at time 119905 of all

confirmed EVD patients who now presentwith full later stage Ebola-like symptomsthey are called confirmed late symptomaticin the sense explained in the text theseconfirmed EVD patients with full blownsymptoms are either not quarantined 119868

119873or

quarantined 119868119876

119877119877(119905) Population density at time 119905 of all humans

who were once infected with EVD infectionand who have recovered from the infectionthis class of persons are then immune to anyfurther infection and are removed from thesusceptible pool


who were once infected with EVD and whohave died because of the EVD infection thisclass of persons though dead are stillinfectious


who died naturally or due to other causesthis is just a collection class

119863119863(119905) Population density at time 119905 of all

Ebola-related dead humans removed fromthe infection cycle because they receivedproper burial or were cremated

Parameters Their Descriptions and Their CorrespondingQuasidimension

Π Net constant migration rate of humans119867119871119879minus1

120588119873 Effective contact rate between probablenonquarantined individuals and susceptibleindividuals a fraction 120579

3of these contacts

are potentially infectious to the susceptiblehumans 119879minus1

120588119876 Effective contact rate between probablequarantined individuals and susceptibleindividuals a fraction 120579

7of these contacts


120591119873 Effective contact rate between confirmednonquarantined early symptomaticindividuals and susceptible individuals 119879minus1

120585119873 Effective contact rate between confirmednonquarantined late symptomaticindividuals and susceptible individuals 119879minus1

120591119876 Effective contact rate between confirmedquarantined early symptomatic individualsand susceptible individuals 119879minus1

120585119876 Effective contact rate between confirmedquarantined late symptomatic individualsand susceptible individuals 119879minus1

119886119863 Effective contact rate between cadavers ofconfirmed late symptomatic individuals andsusceptible individuals 119879minus1

120582 A real function depending on the activemembers of the population representing theforce of infection 119879minus1

120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1

120572119873 Rate at which nonquarantined suspectedcases become nonquarantined probablecases 119879minus1

120572119876 Rate at which quarantined suspected casesbecome quarantined probable cases 119879minus1


120573119873 Rate at which nonquarantined probablecases become nonquarantined confirmedearly symptomatic cases 119879minus1

120573119876 Rate at which quarantined probable casesbecome quarantined confirmed earlysymptomatic cases 119879minus1

120574119873 Rate at which nonquarantined confirmedearly symptomatic cases becomenonquarantined confirmed late symptomaticcases 119879minus1

120574119876 Rate at which quarantined confirmed early

symptomatic cases become quarantinedconfirmed late symptomatic cases 119879minus1

120575119873 Rate at which nonquarantined confirmed latesymptomatic cases die due to the EVD 119879minus1

120575119876 Rate at which quarantined confirmed late

symptomatic cases die due to the EVD 119879minus1120583 Constant natural death rate for humans 119879minus1119887 Rate at which cadavers are removed and

buried 119879minus1119903119864119873

Rate of recovery of nonquarantinedconfirmed early symptomatic cases 119879minus1

119903119871119873

Rate of recovery of nonquarantinedconfirmed late symptomatic cases 119879minus1

119903119864119876 Rate at which quarantined confirmed earlysymptomatic cases recover 119879minus1

119903119871119876 Rate at which quarantined confirmed latesymptomatic cases recover 119879minus1

120590119873 Rate at which nonquarantined confirmedlate symptomatic cases are removed andquarantined 119879minus1

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Gideon A Ngwa acknowledges the grants and support ofthe Cameroon Ministry of Higher Education through theinitiative for the modernization of research in CameroonrsquosHigher Education

References

[1] Center for Disease Control and Prevention (CDC) 2014 EbolaOutbreak in West Africa Center for Disease Control andPrevention (CDC) 2014

[2] World Health Organisation (WHO) ldquoEbola virus diseaserdquo FactSheet 103 WHO Press 2015 httpwwwwhointmediacentrefactsheetsfs103en

[3] World Health Organisation Ebola Response RoadmapmdashSituation Report WHO Press 2015

[4] World Health Organisation Emergencies Preparedness Respo-nsemdashOrigins of the 2014 Ebola Epidemic WHO Press 2015

[5] Center for Disease Control and Prevention The 2014 EbolaOutbreak in West AfricamdashCase Count CDC 2015

[6] Center forDiseaseControl and PreventionEbolaVirusDiseaseTransmission CDC 2014

[7] Center for Disease Control and Prevention Ebola Virus Dis-ease Signs and Symptoms CDC 2014 httpwwwcdcgovvhfebolasymptoms

[8] H Feldmann and T W Geisbert ldquoEbola haemorrhagic feverrdquoThe Lancet vol 377 no 9768 pp 849ndash862 2011

[9] M Goeijenbier J J A van Kampen C B E M ReuskenM P G Koopmans and E C M van Gorp ldquoEbola virusdisease a review on epidemiology symptoms treatment andpathogenesisrdquoThe Netherlands Journal of Medicine vol 72 no9 pp 442ndash448 2014

[10] Center for Disease Control and Prevention Clinical Presenta-tion and Clinical Course CDC 2014

[11] A L Chan What Actually Happens when a Person is Infectedwith the Ebola Virus The Huffington Post 2014

[12] WHO ER Team ldquoEbola virus disease in West Africamdashthe first9 months of the epidemic and forward projectionsrdquo The NewEngland Journal ofMedicine vol 371 no 16 pp 1481ndash1495 2014

[13] Central Intelligence Agency Country Comparison LifeExpectancy at Birth The World Fact Book 2014

[14] J Legrand R F Grais P Y Boelle A J Valleron and AFlahault ldquoUnderstanding the dynamics of Ebola epidemicsrdquoEpidemiology and Infection vol 135 no 4 pp 610ndash621 2007

[15] C M Rivers E T Lofgren M Marathe S Eubank and L LBryan ldquoModeling the impact of interventions on an epidemic ofebola in sierra leone and liberiardquo PLoS Currents vol 16 article6 2014

[16] F B Agusto M I Teboh-Ewungkem and A B GumelldquoMathematical assessment of the effect of traditional beliefsand customs on the transmission dynamics of the 2014 Ebolaoutbreaksrdquo BMCMedicine vol 13 no 1 article 96 2015

[17] FO Fasina A ShittuD Lazarus et al ldquoTransmission dynamicsand control of Ebola virus disease outbreak in Nigeria July toseptember 2014rdquo Eurosurveillance vol 19 no 40 pp 1ndash7 2014

[18] G Chowell N W Hengartner C Castillo-Chavez P WFenimore and J M Hyman ldquoThe basic reproductive numberof Ebola and the effects of public health measures the cases ofCongo and Ugandardquo Journal of Theoretical Biology vol 229 no1 pp 119ndash126 2004

[19] J Astacio D M Briere M Guillen J Martinez F RodriguezandN Valenzuela-Campos ldquoMathematical models to study theoutbreaks of ebolardquo Tech Rep BU-1365-M 2015

[20] C L Althaus ldquoEstimating the reproduction number of ebolavirus (ebov) during the 2014 outbreak in west africardquo PLoSCurrents vol 10 2014

[21] S Towers O Patterson-Lomba and C Castillo-Chavez ldquoTem-poral variations in the effective reproduction number of the2014 west Africa Ebola outbreakrdquo PLoS Currents Outbreaks2014

[22] G Chowell and H Nishiura ldquoTransmission dynamics andcontrol of Ebola virus disease (EVD) a reviewrdquo BMCMedicinevol 12 article 196 2014

[23] A Sifferlin ldquoEbola bodies are infectious a week after deathstudy showsrdquo Sifferlin Times 2015

[24] H K HaleOrdinary Differential Equations JohnWiley amp SonsNew York NY USA 1969

[25] I NasellHybrid Models of Tropical Infections vol 59 of LectureNotes in Biomathematics Springer Berlin Germany 1985


[26] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 119877

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[27] G A Ngwa andW S Shu ldquoA mathematical model for endemicmalaria with variable human andmosquito populationsrdquoMath-ematical and Computer Modelling vol 32 no 7-8 pp 747ndash7632000

[28] M I Teboh-Ewungkem C N Podder and A B GumelldquoMathematical study of the role of gametocytes and an imper-fect vaccine on malaria transmission dynamicsrdquo Bulletin ofMathematical Biology vol 72 no 1 pp 63ndash93 2010

[29] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[30] L Michaelis and M I Menten ldquoDie kinetik der invertin-wirkungrdquo Biochemische Zeitschrift vol 49 Article ID 3333691913

[31] TOgunlesi ldquoDr stella ameyo adadevoh ebola victim and every-day herordquo The Guardian 2014 httpwwwtheguardiancomlifeandstylewomens-blog2014oct20dr-stella-ameyo-ada-devoh-ebola-doctor-nigeria-hero


with the blood secretions fluids from organs and otherbody parts of dead or living animals infected with the virus(eg fruit bats primates and porcupines) [2 6] Human-to-human transmission can then occur through direct contact(via broken skin or mucous membranes such as eyes noseor mouth) with Ebola Virus infected blood secretions andfluids secreted through organs or other body parts in forexample saliva vomit urine faeces semen sweat and breastmilk Transmission can also be as a result of indirect contactwith surfaces and materials in for example bedding cloth-ing and floor areas or objects such as syringes contaminatedwith the aforementioned fluids [2 6]

When a healthy human (considered here to be suscepti-ble) who has no Ebola Virus in them is exposed to the virus(directly or indirectly) the human may become infected iftransmission is successful The risk of being infected with theEbola Virus is (i) very low or not recognizable where there iscasual contact with a feverish ambulant self-caring patientfor example sharing the same public place (ii) low wherethere is close face-to-face contact with a feverish and ambu-lant patient for example physical examination (iii) highwhere there is close face-to-face contact without appropriatepersonal protective equipment (including eye protection)with a patient who is coughing or vomiting has nose-bleeds or has diarrhea and (iv) very high where there ispercutaneous needle stick or mucosal exposure to virus-contaminated blood body fluids tissues or laboratory speci-mens in severely ill or knownpositive patientsThe symptomsof EVD may appear during the incubation period of thedisease estimated to be anywhere from 2 to 21 days [2 7ndash9] with an estimated 8- to 10-day average incubation periodalthough studies estimate it at 9ndash11 days for the 2014 EVDoutbreak inWest Africa [10] Studies have shown that duringthe asymptomatic part of the Ebola Virus Disease a humaninfected with the virus is not infectious and cannot transmitthe virus However with the onset of symptoms the humancan transmit the virus and is hence infectious [2 7]The onsetof symptoms commences the course of illness of the diseasewhich can lead to death 6ndash16 days later [8 9] or improvementof health with evidence of recovery 6ndash11 days later [8]

In the first few days of EVD illness (estimated at days 1ndash3[11]) a symptomatic patient may exhibit symptoms commonto those like the malaria disease or the flu (high feverheadache muscle and joint pains sore throat and generalweakness) Without effective disease management betweendays 4 and 5 to 7 the patient progresses to gastrointestinalsymptoms such as nausea watery diarrhea vomiting andabdominal pain [10 11] Some or many of the other symp-toms such as low blood pressure headaches chest painshortness of breath anemia exhibition of a rash abdominalpain confusion bleeding and conjunctivitis may develop[10 11] in some patients In the later phase of the course ofthe illness days 7ndash10 patients may present with confusionand may exhibit signs of internal andor visible bleedingprogressing towards coma shock and death [10 11]

Recovery from EVD can be achieved as evidenced by theless than 50 fatality rate for the 2014 EVD outbreak in WestAfrica With no known cure recovery is possible througheffective disease management the treatment of Ebola-related

symptoms and also the effective protection by the patientrsquosimmune response [7] Some of the disease managementstrategies include hydrating patients by administering intra-venous fluids and balancing electrolytes and maintaining thepatientrsquos blood pressure and oxygen levels Other schemesused include blood transfusion (using an Ebola survivorrsquosblood) and the use of experimental drugs on such patients(eg ZMAPP whose safety and efficacy have not yetbeen tested on humans) There are some other promisingdrugsvaccines under trials [2] Studies show that once apatient recovers from EVD they remain protected against thedisease and are immune to it at least for a projected periodbecause they develop antibodies that last for at least 10 years[7] Once recovered lifetime immunity is unknown orwhether a recovered individual can be infected with anotherEbola strain is unknown However after recovery a personcan potentially remain infectious as long as their blood andbody fluids including semen and breast milk contain thevirus In particular men can potentially transmit the virusthrough their seminal fluid within the first 7 to 12 weeksafter recovery from EVD [2] Table 1 shows the estimatedtime frames and projected progression of the infection in anaverage EVD patient

Given that there is no approved drug or vaccine outyet local control of the Ebola Virus transmission requiresa combined and coordinated control effort at the individuallevel the community level and the institutionalhealthgov-ernment level Institutions and governments need to educatethe public and raise awareness about risk factors proper handwashing proper handling of Ebola patients quick reportingof suspected Ebola cases safe burial practices use of publictransportation and so forth These education efforts needto be communicated with communitychief leaders who aretrusted by members of the communities they serve From aglobal perspective a good surveillance and contact tracingprogram followed by isolation and monitoring of probableand suspected cases with immediate commencement ofdisease management for patients exhibiting symptoms ofEVD is important if we must in the future elude a globalepidemic and control of EVD transmission locally andglobally [2] It was by effective surveillance contact tracingand isolation andmonitoring of probable and suspected casesfollowed by immediate supportive care for individuals andfamilies exhibiting symptoms that the EVD was broughtunder control in Nigeria [17] Senegal USA and Spain [1]

Efficient control and management of any future EVDoutbreaks can be achieved if new more economical and real-izable methods are used to target and manage the dynamicsof spread as well as the population sizes of those communitiesthat may be exposed to any future Ebola Virus Diseaseoutbreak More realistic mathematical models can play a rolein this regard since analyses of suchmodels can produce clearinsight to vulnerable spots on the Ebola transmission chainwhere control efforts can be concentrated Good modelscould also help in the identification of disease parameters thatcan possibly influence the size of the reproduction numberof EVD Existing mathematical models for Ebola [14 16 18ndash21] have been very instrumental in providing mathematicalinsight into the dynamics of Ebola Virus transmission Many



Exposure





























119876



119876










119863 or confirmed








119873





6




119889119878

119889119905= Π minus 120582119878 + (1 minus 120579

3) 120573119873119875119873+ (1 minus 120579

2) 120572119873119878119873

+ (1 minus 1205796) 120572119876119878119876+ (1 minus 120579

7) 120573119876119875119876minus 120583119878

(1)





119867119871 (119905) = (119878 + 119878119873 + 119878119876 + 119875119873 + 119875119876 + 119862119873 + 119862119876 + 119868119873

+ 119868119876+ 119877119877) (119905)

(2)

119867(119905) = (119878 + 119878119873+ 119878119876+ 119875119873+ 119875119876+ 119862119873+ 119862119876+ 119868119873+ 119868119876

+ 119877119877+ 119877119863) (119905)

(3)



the following form

120582 =1

119867(1205793120588119873119875119873+ 1205797120588119876119875119876+ 120591119873119862119873+ 120585119873119868119873+ 120591119876119862119876

+ 120585119876119868119876+ 119886119863119877119863)

(4)


120588119876 120591119873 120591119876 120585119873 120585119876 and 119886







2(resp 120579

6) of the




119873and 120572

119876)

a fraction 1205792119887




119889119878119873

119889119905= 1205791120582119878 minus (1 minus 120579

2) 120572119873119878119873minus 1205792119886120572119873119878119873minus 1205792119887120572119873119878119873

minus 120583119878119873

119889119878119876

119889119905= (1 minus 120579

1) 120582119878 minus (1 minus 120579

6) 120572119876119878119876minus 1205796120572119876119878119876minus 120583119878119876

(5)


2= 1205792119886+ 1205792119887 so that 1 minus 120579

2+ 1205792119886+ 1205792119887= 1


6of sus-


119873and 120573

119876) when some


119873and 120572


1 minus 1205793and 1 minus 120579






3119886



119889119875119873

119889119905= 1205792119886120572119873119878119873minus (1 minus 120579

3) 120573119873119875119873minus 1205793119886120573119873119875119873

minus 1205793119887120573119873119875119873minus 120583119875119873

119889119875119876

119889119905= 1205792119887120572119873119878119873+ 1205796120572119876119878119876minus (1 minus 120579

7) 120573119876119875119876minus 1205797120573119876119875119876

minus 120583119875119876

(6)



119864119873for



119873or 120574119876




4escape


119889119862119873

119889119905= 1205793119886120573119873119875119873minus 119903119864119873119862119873minus 1205794120574119873119862119873


119889119862119876


(7)



4and 120579



119871119873and 119903119871119876


119873and 120575





119889119868119873


119889119868119876

119889119905= (1 minus 120579

4) 120574119873119862119873+ 120590119873119868119873+ 120574119876119862119876minus 120575119876119868119876minus 119903119871119876119868119876

minus 120583119868119876

(8)


119873and 120575

119876and the


119889119877119877

119889119905= 119903119864119873119862119873+ 119903119871119873119868119873+ 119903119864119876119862119876+ 119903119871119876119868119876minus 120583119868119873

119889119877119863

119889119905= 120575119873119868119873+ 120575119876119868119876minus 119887119877119863

(9)


119873and keeps



119889119877119873

119889119905= 120583119867119871

119889119863119863

119889119905= 119887119877119863

(10)


119889119878

119889119905= Π minus 120582119878 +

3120573119873119875119873+ 2120572119873119878119873+ 6120572119876119878119876

+ 7120573119876119875119876minus 120583119878

(11)

119889119878119873

119889119905= 1205791120582119878 minus (120572

119873+ 120583) 119878

119873 (12)

119889119878119876

119889119905= 1120582119878 minus (120572

119876+ 120583) 119878

119876 (13)

119889119875119873

119889119905= 1205792119886120572119873119878119873minus (120573119873+ 120583) 119875

119873 (14)

119889119875119876

119889119905= 1205792119887120572119873119878119873+ 1205796120572119876119878119876minus (120573119876+ 120583) 119875

119876 (15)

119889119862119873

119889119905= 1205793119886120573119873119875119873minus (119903119864119873+ 120574119873+ 120583)119862

119873 (16)

119889119862119876

119889119905= 1205793119887120573119873119875119873+ 1205797120573119876119875119876minus (119903119864119876+ 120574119876+ 120583)119862

119876 (17)

119889119868119873

119889119905= 1205794120574119873119862119873minus (119903119871119873+ 120575119873+ 120583) 119868

119873 (18)

119889119868119876

119889119905= 4120574119873119862119873+ 120590119873119868119873+ 120574119876119862119876

minus (119903119871119876+ 120575119876+ 120583) 119868

119876

(19)

119889119877119877

119889119905= 119903119864119873119862119873+ 119903119871119873119868119873+ 119903119864119876119862119876+ 119903119871119876119868119876minus 120583119877119877 (20)

119889119877119863

119889119905= 120575119873119868119873+ 120575119876119868119876minus 119887119877119863 (21)

119889119877119873

119889119905= 120583119867119871

(22)

119889119863119863

119889119905= 119887119877119863 (23)







119878 (0) = 1198780

119868119873(0) = 119868

0

119878119873 (0) = 119878119876 (0) = 119875119873 (0) = 0

119875119876(0) = 119862

119873(0) = 119862

119876(0) = 119868

119876(0) = 119877

119863(0) = 119877

119877(0)

= 119877119873(0) = 119863

119863(0) = 0

(24)





119889119867119871

119889119905= Π minus 120583119867

119871minus 120575119873119868119873minus 120575119876119868119876 (25)

119889119867


119863 (26)



119863 119877119873 and



119871minus 119887119877119863







119873 119878119876



1= 119878 119909

2= 119878119873 119909

11= 119877119863 Let f(x) =

(1198911(x) 1198912(x) 119891





119878119873 and 119891




dx119889119905= f (x) x (0) = x

0 (27)





119868lowast and 119877







10038171003817100381710038171003817f1015840 (z y minus x)10038171003817100381710038171003817infin =

1003817100381710038171003817100381710038171003817100381710038171003817

11

sum

119896=1


1003817100381710038171003817100381710038171003817100381710038171003817infin

le

1003817100381710038171003817100381710038171003817100381710038171003817

11

sum

119896=1

nabla119891119896 (z)1003817100381710038171003817100381710038171003817100381710038171003817infin

1003817100381710038171003817y minus x1003817100381710038171003817infin

(29)


+ It is now





11

sum

119896=1

nabla119891119896 (z)1003817100381710038171003817100381710038171003817100381710038171003817infin



1003817100381710038171003817infinle 119872

1003817100381710038171003817y minus x1003817100381710038171003817infin (31)









1) = 0 1198781015840(119905

1) lt 0


1 119878(119905) gt 0 and 119878

119873(119905) gt 0 119878

119876(119905) gt 0

119875119873(119905) gt 0 119875

119876(119905) gt 0 119862

119873(119905) gt 0 119862

119876(119905) gt 0 119868

119873(119905) gt 0

119868119876(119905) gt 0 119877

119877(119905) gt 0 and 119877

119863(119905) gt 0 So at the point 119905 = 119905

1


119889119878

119889119905

10038161003816100381610038161003816100381610038161003816119905=1199051

= Π minus 120582119878 (1199051) + (1 minus 120579

3) 120573119873119875119873(1199051)

+ (1 minus 1205792) 120572119873119878119873(1199051) + (1 minus 120579

6) 120572119876119878119876(1199051)

+ (1 minus 1205797) 120573119876119875119876(1199051) minus 120583119878 (119905

1)

= Π + (1 minus 1205793) 120573119873119875119873(1199051)

+ (1 minus 1205792) 120572119873119878119873(1199051) + (1 minus 120579

6) 120572119876119878119876(1199051)

+ (1 minus 1205797) 120573119876119875119876(1199051) gt 0

(32)


1) lt



+ the solution will







Ω119867119871

= 119867119871(119905) 0 le 119867

119871(119905) le

Π

120583 sub R

11

+ (33)


119871is bounded



119889119867119871

119889119905= Π minus 120583119867


119871997904rArr

119867119871(119905) le

Π

120583+ (119867119871(0) minus

Π

120583) 119890minus120583119905

(34)


119871(119905)


119871(0) lt Π then119867



119867119871(119905) le lim119905rarrinfin

sup(Π120583+ (119867119871(0) minus

Π

120583) 119890minus120583119905) (35)





119868lowast and 119877


119871




119904 =119878

1198780

119904119899=119878119873

1198780

119873

119904119902=119878119876

1198780

119876

119901119899=119875119873

1198750

119873

119901119902=119875119876

1198750

119876

119888119899=119862119873

1198620

119873

119888119902=119862119876

1198620

119876

ℎ =119867

1198670

119894119899=119868119873

1198680

119873

119894119902=119868119876

1198680

119876

119903119903=119877119877

1198770

119877

119903119889=119877119863

1198770

119863

119903119899=119877119873

1198770

119873

119889119863=119863119863

1198630

119863

119905lowast=119905

1198790

ℎ119897=119867119871

1198670

119871

(36)

where

1198780=Π

120583

1198780

119873= 1198780

119876= 1198780

1198750

119873=12057921198861205721198731198780

119873

120573119873+ 120583

1198750

119876=12057921198871205721198731198780

119873

120573119876+ 120583

1198620

119873=12057931198861205731198731198750

119873

119903119864119873+ 120574119873+ 120583

1198620

119876=12057931198871205731198731198750

119873

119903119864119876+ 120574119876+ 120583

1198680

119873=

12057941205741198731198620

119873

119903119871119873+ 120575119873+ 120583

1198680

119876=(1 minus 120579

4) 1205741198731198620

119873

119903119871119876+ 120575119876+ 120583

1198770

119877= 1199031198641198731198620

1198731198790

1198770

119863=1205751198731198680

N119887

1198770

119873= 12058311987901198670

119871

1198630

119863= 11988711987901198770

119863

1198670= 1198670

119871=Π

120583= 1198780

1198790=1

120583

(37)


120588119899=12057931205881198731198750

119873

1205831198670

120588119902=

12057971205881198761198750

119876

1205831198670

120591119899=1205911198731198620

119873

1205831198670

120591119902=

1205911198761198620

119876

1205831198670

120585119899=1205851198731198680

119873

1205831198670

120585119902=

1205851198761198680

119876

1205831198670

119886119889=1198861198631198770

119863

1205831198670

120572119899= (120572119873+ 120583)119879

0

120572119902= (120572119876+ 120583)119879

0

120573119899= (120573119873+ 120583) 119879

0

120573119902= (120573119876+ 120583)119879

0


120583119889= 1198871198790

1198871=(1 minus 120579

3) 1205731198731198750

119873

1205831198780

1198872=(1 minus 120579

2) 1205721198731198780

119873

1205831198780

1198873=

(1 minus 1205796) 1205721198761198780

119876

1205831198780

1198874=

(1 minus 1205797) 1205731198761198750

119876

1205831198780

1198875=1205751198731198680

119873

1205831198670

1198876=

1205751198761198680

119876

1205831198670

1198878=(119887 minus 120583) 119877

0

119863

1205831198670

1198861=

12057961205721198761198780

119876

12057921198871205721198731198780

119873

1198862=

12057971205731198761198750

119876

12057931198871205731198731198750

119873

1198863=

1205901198731198680

119873

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198864=

1205741198761198620

119876

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198865=1199031198711198731198680

119873

1199031198641198731198620

119873

1198866=

1199031198641198761198620

119876

1199031198641198731198620

119873

1198867=

1199031198711198761198680

119876

1199031198641198731198620

119873

1198868=

1205751198761198680

119876

1205751198731198680

119873

120574119899= (119903119864119873+ 120574119873+ 120583) 119879

0

120574119902= (119903119864119876+ 120574119876+ 120583) 119879

0

120575119902= (119903119871119876+ 120575119876+ 120583) 119879

0

120575119899= (119903119871119873+ 120575119873+ 120583) 119879

0

(38)


120582 = 120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)

(39)


119889119904

119889119905= 1 minus 120582119904 + 119887

1119901119899+ 1198872119904119899+ 1198873119904119902+ 1198874119901119902minus 119904 (40)

119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (41)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (42)

119889119901119899

119889119905= 120573119899(119904119899minus 119901119899) (43)

119889119901119902

119889119905= 120573119902(119904119899+ 1198861119904119902minus 119901119902) (44)

119889119888119899

119889119905= 120574119899(119901119899minus 119888119899) (45)

119889119888119902

119889119905= 120574119902(119901119899+ 1198862119901119902minus 119888119902) (46)

119889119894119899

119889119905= 120575119899(119888119899minus 119894119899) (47)

119889119894119902

119889119905= 120575119902(119888119899+ 1198863119894119899+ 1198864119888119902minus 119894119902) (48)

119889119903119903

119889119905= 119888119899+ 1198865119894119899+ 1198866119888119902+ 1198867119894119902minus 119903119903 (49)

119889119903119889

119889119905= 120583119889(119894119899+ 1198868119894119902minus 119903119889) (50)

119889119903119899

119889119905= ℎ119897 (51)

119889119889119863

119889119905= 119889119863 (52)


119889ℎ119897

119889119905= 1 minus ℎ

119897minus 1198875119894119899minus 1198876119894119902 (53)

119889ℎ

119889119905= 1 minus ℎ minus 119887

8119903119889 (54)




120588119899=12057931205792119886120588119873120572119873

(120573119873+ 120583) 120583

120588119902=12057971205792119887120588119876120572119873

(120573119876+ 120583) 120583

120591119899=

12057931198861205792119886120591119873120572119873120573119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) 120583

120591119902=

12057931198871205792119886120591119876120572119873120573119873

(120573119873+ 120583) (119903

119864119876+ 120574119876+ 120583) 120583

120585119899=

120579311988612057921198861205794120585119873120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 120583

120585119902=

12057931198861205792119886(1 minus 120579

4) 120585119876120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119876+ 120575119876+ 120583) 120583

119886119889=

120579211988612057931198861205794120575119873120574119873120573119873120572119873119886119863

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 119887120583

1198871=(1 minus 120579

3) 1205792119886120573119873120572119873

(120573119873+ 120583) 120583

1198872=(1 minus 120579

2) 120572119873

120583

1198873=(1 minus 120579

6) 120572119876

120583

1198874=(1 minus 120579

7) 1205792119887120573119876120572119873

(120573119876+ 120583) 120583

1198875=

120579412057921198861205793119886120575119873120574119873120573119873120572119873

(119903119871119873+ 120575119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198876=

(1 minus 1205794) 12057921198861205793119886120575119876120574119873120573119873120572119873

(119903119871119876+ 120575119876+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198878=

(119887 minus 120583) 120579412057921198861205793119886120572119873120573119873120574119873120575119873

119887120583 (120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583)

120575119899=119903119871119873+ 120575119873+ 120583

120583

120572119899=120572119873+ 120583

120583

120572119902=120572119876+ 120583

120583

120573119899=120573119873+ 120583

120583

120573119902=120573119876+ 120583

120583

120574119899=119903119864119873+ 120574119873+ 120583

120583

120574119902=119903119864119876+ 120574119876+ 120583

120583

120575119902=119903119871119876+ 120575119876+ 120583

120583

1198861=1205796120572119876

1205792119887120572119873

1198862=12057971205792119887120573119876(120573119873+ 120583)

12057921198861205793119887(120573119876+ 120583) 120573

119873

1198863=

1205794120590119873

(1 minus 1205794) (119903119871119873+ 120575119873+ 120583)

1198864=

1205793119887120574119876(119903119864119873+ 120574119873+ 120583)

(1 minus 1205794) 1205793119886120574119873(119903119864119876+ 120574119876+ 120583)

1198865=

1199031198711198731205794120574119873

119903119864119873(119903119871119873+ 120575119873+ 120583)

120583119889=119887

120583

1198866=1205793119887119903119864119876(119903119864119873+ 120574119873+ 120583)

1205793119886119903119864119873(119903119864119876+ 120574119876+ 120583)

1198867=119903119871119876(1 minus 120579

4) 120574119873

119903119864119873(119903119871119876+ 120575119876+ 120583)

1198868=(1 minus 120579

4) 120575119876(119903119871119873+ 120575119873+ 120583)

1205794120575119873(119903119871119876+ 120575119876+ 120583)

(55)


119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast



119904lowast

119899= 119901lowast

119899= 119888lowast

119899= 119894lowast

119899(56)



119899and

119904lowast


119901lowast

119902(119901lowast

119899 119904lowast

119902) = 119901lowast

119899+ 1198861119904lowast

119902

119888lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

1119901lowast

119899+ 1198602119904lowast

119902

119894lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

3119901lowast

119899+ 1198604119904lowast

119902

119903lowast

119903(119901lowast

119899 119904lowast

119902) = 119860

5119901lowast

119899+ 1198606119904lowast

119902

119903lowast

119889(119901lowast

119899 119904lowast

119902) = 119860

7119901lowast

119899+ 1198608119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902

ℎlowast

119897(119901lowast

119899 119904lowast

119902) = 1 minus (119887

5+ 11988761198603) 119901lowast

119899minus 11988761198604119904lowast

119902

(57)

where

1198601= 1 + 119886

2

1198602= 11988611198862

1198603= 1 + 119886

3+ 11988641198601

1198604= 11988641198602

1198605= 1 + 119886

5+ 11988661198601+ 11988671198603

1198606= 11988661198602+ 11988671198604

1198607= 1 + 119886

81198603

1198608= 11988681198604

1198611= 11988781198607

1198612= 11988781198608


1198614= 120572119902minus 1198873minus 11988741198861

(58)




3and 119861


1198613= 1

+120572119873

120583(1205731198731205792119886(1205793minus 1)

120573119873+ 120583

+1205731198761205792119887(1205797minus 1)

120573119876+ 120583

+ 1205792)

gt 0 since 1205792= 1205792119886+ 1205792119887

1198614=1205721198761205796(1205731198761205797+ 120583) + 120583 (120573

119876+ 120583)

120583 (120573119876+ 120583)

gt 0

(59)


4gt 0





1205791(1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119899119901lowast

119899(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(60)

(1 minus 1205791) (1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119902119904lowast

119902(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(61)

where

1198615= 120588119899+ 120588119902+ 120591119899+ 1205911199021198601+ 120585119899+ 1205851199021198603+ 1198861198891198607

1198616= 1205881199021198861+ 1205911199021198602+ 1205851199021198604+ 1198861198891198608

(62)



(1 minus 1198611119901lowast

119899minus 1198612119904lowast

119902) (120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast

1199021205791) = 0 (63)


119899minus 1198612119904lowast

119902= 0 or (ii)

120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast


system


119899minus 1198614119904lowast

119902= 0


119899minus 1198612119904lowast

119902= 0

(64)



119901lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902 (65)



119904lowast

119902= 0

or 119904lowast119902=1198617

1198618

=(1 minus 120579

1) 120572119899(1198770minus 1)

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612) (R minus 1)

= 119909 (1198770minus 1

R minus 1)

(66)

where

1198617= 120572119902120572119899(1 minus 120579

1) ((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

) minus 1)

= 120572119902120572119899(1 minus 120579

1) (1198770minus 1)

1198618= 120572119902[(12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614)

minus (12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612)] = 120572

119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612)((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (

12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614

12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612

) minus 1) = 120572119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612) (1198770119911 minus 1)

(67)

with

119909 =(1 minus 120579

1) 120572119899

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612)

119910 =

1205791120572119902119909

(1 minus 1205791) 120572119899

119911 =

(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

(68)

1198770=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

R =1198770(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

= 1199111198770

(69)


1198612= 11988781198868119886411988611198862=1205796120572119876

120583

1205797120573119876

(120573119876+ 120583)

((1 minus120583

119887)

sdot120575119876

(119903119871119876+ 120575119876+ 120583)

120574119876

(119903119864119876+ 120574119876+ 120583)) lt

1205796120572119876

120583

sdot1205797120573119876

(120573119876+ 120583)

lt1205796120572119876

120583

(1205797120573119876+ 120583)

(120573119876+ 120583)

+ 1 = 1198614

(70)



3lt 1198611 we will


1120572119902(1minus120579

1)120572119899)(1198611minus1198613)


system as follows




0gt 1 and the


0




119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119894lowast

119899= 119888lowast

119899= 119904lowast

119899= 119901lowast

119899= 119910(

1198770minus 1

R minus 1)

119901lowast

119902= (119910 + 119886

1119909) (1198770minus 1

R minus 1)

119888lowast

119902= (1198601119910 + 119860

2119909) (1198770minus 1

R minus 1)

119894lowast

119902= (1198603119910 + 119860

4119909) (1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

119903lowast

119889= (1198607119910 + 119860

8119909) (1198770minus 1

R minus 1)

119904lowast= 1 minus (119861

3119910 + 1198614119909) (1198770minus 1

R minus 1)


1119910 + 1198612119909) (1198770minus 1

R minus 1)

(71)



119889119891119890=


119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast







119869dfe =

((((((((((((((((((((((

(


1198890

0 minus1205721198990 120579

1120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 0 minus120572119902

1205791120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 120573119899

0 minus120573119899

0 0 0 0 0 0 0 0

0 1205731199021198861120573119902

0 minus120573119902

0 0 0 0 0 0 0

0 0 0 120574119899

0 minus1205741198990 0 0 0 0 0

0 0 0 120574119902

1198862120574119902

0 minus1205741199020 0 0 0 0

0 0 0 0 0 120575119899

0 minus1205751198990 0 0 0

0 0 0 0 0 12057511990211988641205751199021198863120575119902minus1205751199020 0 0

0 0 0 0 0 1 11988661198865

1198867minus1 0 0

0 0 0 0 0 0 0 12058311988912058311988911988680 minus120583

1198890

0 0 0 0 0 0 0 0 0 0 minus1198878minus1

))))))))))))))))))))))

)

(72)

where 1205791= 1 minus 120579



119875 (120577) =1003816100381610038161003816119869dfe minus 120577I

1003816100381610038161003816 = (120577 + 1)31198759(120577) = 0 (73)


1198759 (120577) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus120572119899minus 120577 0 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

0 minus120572119902minus 120577 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

120573119899

0 minus120573119899minus 120577 0 0 0 0 0 0

120573119902

1198861120573119902

0 minus120573119902minus 120577 0 0 0 0 0

0 0 120574119899

0 minus120574119899minus 120577 0 0 0 0

0 0 120574119902

1198862120574119902

0 minus120574119902minus 120577 0 0 0

0 0 0 0 120575119899

0 minus120575119899minus 120577 0 0

0 0 0 0 120575119902

1198864120575119902

1198863120575119902minus120575119902minus 120577 0

0 0 0 0 0 0 120583119889

1205831198891198868minus120583119889minus 120577

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(74)


119875 (120577) = (120577 + 1)3

9

sum

119896=0

119888119894120577119894 (75)

where1198889= 1

1198888= 120572119899+ 120572119902+ 120573119899+ 120573119902+ 120574119899+ 120574119902+ 120575119899+ 120575119902+ 120583119889

1198880= 1205731198991205731199021205741198991205741199021205751198991205751199021205831198891205721198991205721199021 minus

12057911198615

120572119899


1) 1198616

120572119902

= 120573119899120573119902120574119899120574119902120575119899120575119902120583119889120572119899120572119902(1 minus 119877

0)

(76)




0= 1 to





0 1198770




0= 1


0specifically


0which is fixedRmay






0is the


119899 119901119902 119888119899 119888119902 119894119899 119894119902 119903119903 119903119889 and ℎ The




F =

(((((((((((((((((

(

0

1205791120582119904

(1 minus 1205791) 120582119904

0

0

0

0

0

0

0

0

0

)))))))))))))))))

)

V =

((((((((((((((((((((((((((((((((

(


120572119899119904119899

120572119902119904119902

120573119899(minus119904119899+ 119901119899)

120573119902(minus119904119899minus 1198861119904119902+ 119901119902)

120574119899(minus119901119899+ 119888119899)

120574119902(minus119901119899minus 1198862119901119902+ 119888119902)

120575119899(minus119888119899+ 119894119899)



minus120583119889119894119899minus 1205831198891198868119894119902+ 120583119889119903119889

minus1 + ℎ + 1198878119903119889

))))))))))))))))))))))))))))))))

)

(77)


119894119895=

120597F119894120597119909119895and (119881)

119894119895= 120597V



119901119902= 119888119899= 119888119902= 119894119899= 119894119902= 119903119903= 119903119889= 0 The basic


1198770= 984858 (119865119881

minus1) =

1

120572119899120572119902

[1205721199021205791(1

+ (1 + 1198863+ (1 + 119886

2) 1198864) 119886119889

+ 120585119902(11988621198864+ 1198863+ 1198864+ 1) + 119886

2120591119902+ 120585119899+ 120588119899+ 120588119902

+ 120591119899+ 120591119902) minus 120572

119899(1205791minus 1) (119886

1120588119902

+ 1198861[11988621198864(1198868119886119889+ 120585119902) + 1198862120591119902])]

=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

(78)






119899(which








120583


120583 + rate120583

asymp large (79)


119899 1120573119902



119901119899= 119888119899= 119894119899= 119904119899

119901119902= 119904119899+ 1198861119904119902

119888119902= 1198601119904119899+ 1198602119904119902

119894119902= 1198603119904119899+ 1198604119904119902

119903119889= 1198607119904119899+ 1198608119904119902

(80)


119889119905= 1 minus 120582119904 + (120572


119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (82)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (83)

119889119903119903

119889119905= 1198605119904119899+ 1198606119904119902minus 119903119903 (84)

119889ℎ

119889119905= 1 minus ℎ minus 119861

1119904119899minus 1198612119904119902 (85)


120582119904 = (120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)) 119904 = (119861

5(119904119899

ℎ)

+ 1198616(

119904119902

ℎ)) 119904

(86)





119899 119904lowast

119902 119903lowast


119864dfe = (1 0 0 0 1)

119864119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast)

(87)


119904lowast

119902=1198617

1198618

119904lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902

119903lowast

119903= 1198605119904lowast

119899+ 1198606119904lowast

119902

119904lowast= 1 minus 119861

3119904lowast

119899minus 1198614119904lowast

119902


1119904lowast

119899minus 1198612119904lowast

119902

(88)


3119910 + 119861

4119909 and

1198611119910 + 1198612119909 = 1 we get

119904lowast= (119911 minus 1

R minus 1)

119904lowast

119899= 119910(

1198770minus 1

R minus 1)

119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

ℎlowast= ((119911 minus 1) 119877

0

R minus 1)

(89)


119899 119904lowast

119902 119903lowast




119869 (xlowast) =((

(


4(xlowast) + 120572


119902minus 11986140 minus119862

6(xlowast)

12057911198623(xlowast) 120579


11989912057911198625(xlowast) 0 120579

11198626(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) 120579


1199020 12057911198626(xlowast)

0 1198605

1198606

minus1 0

0 minus1198611

minus1198612

0 minus1

))

)

(90)

where 1205791= 1 minus 120579

1and

1198623(xlowast) = 119861

5

119904lowast

119899

ℎlowast+ 1198616

119904lowast

119902

ℎlowast=120572119899119910 (1198770minus 1)

1205791(119911 minus 1)

1198624(xlowast) = 119861

5

119904lowast

ℎlowast=1198615

1198770

1198625(xlowast) = 119861

6

119904lowast

ℎlowast=1198616

1198770

1198626(xlowast) = minus(119861

5


119899

ℎlowast2+ 1198616


119902

ℎlowast2) = minus

120572119899119910 (1198770minus 1)

1205791 (119911 minus 1) 1198770

(91)


1198755(120577 xlowast)

= (120577 + 1)2(1205773+ 1198762(xlowast) 1205772 + 119876

1(xlowast) 120577 + 119876

0(xlowast))

= 0

(92)

where

1198762(xlowast) = 120572

119899+ 120572119902+ 1198623(xlowast) minus 119862

4(xlowast) 120579


1

+ 1

1198761(xlowast) = 120572

119902


1199021198624(xlowast) + 119862

4(xlowast))

+ 11986121198626(xlowast) 120579

1+ 1198623(xlowast) (120572

1199021205791+ 11986131205791+ 11986141205791)

+ 120572119899(120572119902+ 12057911198623(xlowast) minus 119862

5(xlowast) 120579

1+ 1) minus 119862

5(xlowast)

sdot 1205791

1198760(xlowast) = 120572

1199021205791(11986111198626(xlowast) + 119861


4(xlowast))

+ 120572119899(1205791(11986121198626(xlowast) + 119861


5(xlowast)) + 120572

119902)

(93)




119902=

119903lowast




1198616 1198626(xdfe) = 0 and (92) becomes

1198755(120577 xdfe) = (120577 + 1)

3(1205772+ 119876dfe120577 + 119877dfe) = 0 (94)

where


= 120572119899(1 minus 119877

0) + 120572119902+ (1 minus 120579

1) 1198616(

120572119899minus 120572119902

120572119902

)

119877dfe = 120572119902 (120572119899 minus 11986151205791) + 1205721198991198616 (1205791 minus 1)

= 120572119902120572119899(1 minus 119877

0)

(95)


radic1198762







=


119899 119904lowast

119902 119903lowast


and we have

1198755(120577 xlowast) = (120577 + 1)2 (1205773 + 119875x

119890119890

1205772+ 119876x

119890119890

120577 + 119877x119890119890

) = 0 (96)

where


119875x119890119890

=


12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)


119890119890will be the


119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)




119899= 0 and we


1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)





grouping 1198616




1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)





CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References





































Exposure





























119876



119876










119863 or confirmed








119873





6




119889119878

119889119905= Π minus 120582119878 + (1 minus 120579

3) 120573119873119875119873+ (1 minus 120579

2) 120572119873119878119873

+ (1 minus 1205796) 120572119876119878119876+ (1 minus 120579

7) 120573119876119875119876minus 120583119878

(1)





119867119871 (119905) = (119878 + 119878119873 + 119878119876 + 119875119873 + 119875119876 + 119862119873 + 119862119876 + 119868119873

+ 119868119876+ 119877119877) (119905)

(2)

119867(119905) = (119878 + 119878119873+ 119878119876+ 119875119873+ 119875119876+ 119862119873+ 119862119876+ 119868119873+ 119868119876

+ 119877119877+ 119877119863) (119905)

(3)



the following form

120582 =1

119867(1205793120588119873119875119873+ 1205797120588119876119875119876+ 120591119873119862119873+ 120585119873119868119873+ 120591119876119862119876

+ 120585119876119868119876+ 119886119863119877119863)

(4)


120588119876 120591119873 120591119876 120585119873 120585119876 and 119886







2(resp 120579

6) of the




119873and 120572

119876)

a fraction 1205792119887




119889119878119873

119889119905= 1205791120582119878 minus (1 minus 120579

2) 120572119873119878119873minus 1205792119886120572119873119878119873minus 1205792119887120572119873119878119873

minus 120583119878119873

119889119878119876

119889119905= (1 minus 120579

1) 120582119878 minus (1 minus 120579

6) 120572119876119878119876minus 1205796120572119876119878119876minus 120583119878119876

(5)


2= 1205792119886+ 1205792119887 so that 1 minus 120579

2+ 1205792119886+ 1205792119887= 1


6of sus-


119873and 120573

119876) when some


119873and 120572


1 minus 1205793and 1 minus 120579






3119886



119889119875119873

119889119905= 1205792119886120572119873119878119873minus (1 minus 120579

3) 120573119873119875119873minus 1205793119886120573119873119875119873

minus 1205793119887120573119873119875119873minus 120583119875119873

119889119875119876

119889119905= 1205792119887120572119873119878119873+ 1205796120572119876119878119876minus (1 minus 120579

7) 120573119876119875119876minus 1205797120573119876119875119876

minus 120583119875119876

(6)



119864119873for



119873or 120574119876




4escape


119889119862119873

119889119905= 1205793119886120573119873119875119873minus 119903119864119873119862119873minus 1205794120574119873119862119873


119889119862119876


(7)



4and 120579



119871119873and 119903119871119876


119873and 120575





119889119868119873


119889119868119876

119889119905= (1 minus 120579

4) 120574119873119862119873+ 120590119873119868119873+ 120574119876119862119876minus 120575119876119868119876minus 119903119871119876119868119876

minus 120583119868119876

(8)


119873and 120575

119876and the


119889119877119877

119889119905= 119903119864119873119862119873+ 119903119871119873119868119873+ 119903119864119876119862119876+ 119903119871119876119868119876minus 120583119868119873

119889119877119863

119889119905= 120575119873119868119873+ 120575119876119868119876minus 119887119877119863

(9)


119873and keeps



119889119877119873

119889119905= 120583119867119871

119889119863119863

119889119905= 119887119877119863

(10)


119889119878

119889119905= Π minus 120582119878 +

3120573119873119875119873+ 2120572119873119878119873+ 6120572119876119878119876

+ 7120573119876119875119876minus 120583119878

(11)

119889119878119873

119889119905= 1205791120582119878 minus (120572

119873+ 120583) 119878

119873 (12)

119889119878119876

119889119905= 1120582119878 minus (120572

119876+ 120583) 119878

119876 (13)

119889119875119873

119889119905= 1205792119886120572119873119878119873minus (120573119873+ 120583) 119875

119873 (14)

119889119875119876

119889119905= 1205792119887120572119873119878119873+ 1205796120572119876119878119876minus (120573119876+ 120583) 119875

119876 (15)

119889119862119873

119889119905= 1205793119886120573119873119875119873minus (119903119864119873+ 120574119873+ 120583)119862

119873 (16)

119889119862119876

119889119905= 1205793119887120573119873119875119873+ 1205797120573119876119875119876minus (119903119864119876+ 120574119876+ 120583)119862

119876 (17)

119889119868119873

119889119905= 1205794120574119873119862119873minus (119903119871119873+ 120575119873+ 120583) 119868

119873 (18)

119889119868119876

119889119905= 4120574119873119862119873+ 120590119873119868119873+ 120574119876119862119876

minus (119903119871119876+ 120575119876+ 120583) 119868

119876

(19)

119889119877119877

119889119905= 119903119864119873119862119873+ 119903119871119873119868119873+ 119903119864119876119862119876+ 119903119871119876119868119876minus 120583119877119877 (20)

119889119877119863

119889119905= 120575119873119868119873+ 120575119876119868119876minus 119887119877119863 (21)

119889119877119873

119889119905= 120583119867119871

(22)

119889119863119863

119889119905= 119887119877119863 (23)







119878 (0) = 1198780

119868119873(0) = 119868

0

119878119873 (0) = 119878119876 (0) = 119875119873 (0) = 0

119875119876(0) = 119862

119873(0) = 119862

119876(0) = 119868

119876(0) = 119877

119863(0) = 119877

119877(0)

= 119877119873(0) = 119863

119863(0) = 0

(24)





119889119867119871

119889119905= Π minus 120583119867

119871minus 120575119873119868119873minus 120575119876119868119876 (25)

119889119867


119863 (26)



119863 119877119873 and



119871minus 119887119877119863







119873 119878119876



1= 119878 119909

2= 119878119873 119909

11= 119877119863 Let f(x) =

(1198911(x) 1198912(x) 119891





119878119873 and 119891




dx119889119905= f (x) x (0) = x

0 (27)





119868lowast and 119877







10038171003817100381710038171003817f1015840 (z y minus x)10038171003817100381710038171003817infin =

1003817100381710038171003817100381710038171003817100381710038171003817

11

sum

119896=1


1003817100381710038171003817100381710038171003817100381710038171003817infin

le

1003817100381710038171003817100381710038171003817100381710038171003817

11

sum

119896=1

nabla119891119896 (z)1003817100381710038171003817100381710038171003817100381710038171003817infin

1003817100381710038171003817y minus x1003817100381710038171003817infin

(29)


+ It is now





11

sum

119896=1

nabla119891119896 (z)1003817100381710038171003817100381710038171003817100381710038171003817infin



1003817100381710038171003817infinle 119872

1003817100381710038171003817y minus x1003817100381710038171003817infin (31)









1) = 0 1198781015840(119905

1) lt 0


1 119878(119905) gt 0 and 119878

119873(119905) gt 0 119878

119876(119905) gt 0

119875119873(119905) gt 0 119875

119876(119905) gt 0 119862

119873(119905) gt 0 119862

119876(119905) gt 0 119868

119873(119905) gt 0

119868119876(119905) gt 0 119877

119877(119905) gt 0 and 119877

119863(119905) gt 0 So at the point 119905 = 119905

1


119889119878

119889119905

10038161003816100381610038161003816100381610038161003816119905=1199051

= Π minus 120582119878 (1199051) + (1 minus 120579

3) 120573119873119875119873(1199051)

+ (1 minus 1205792) 120572119873119878119873(1199051) + (1 minus 120579

6) 120572119876119878119876(1199051)

+ (1 minus 1205797) 120573119876119875119876(1199051) minus 120583119878 (119905

1)

= Π + (1 minus 1205793) 120573119873119875119873(1199051)

+ (1 minus 1205792) 120572119873119878119873(1199051) + (1 minus 120579

6) 120572119876119878119876(1199051)

+ (1 minus 1205797) 120573119876119875119876(1199051) gt 0

(32)


1) lt



+ the solution will







Ω119867119871

= 119867119871(119905) 0 le 119867

119871(119905) le

Π

120583 sub R

11

+ (33)


119871is bounded



119889119867119871

119889119905= Π minus 120583119867


119871997904rArr

119867119871(119905) le

Π

120583+ (119867119871(0) minus

Π

120583) 119890minus120583119905

(34)


119871(119905)


119871(0) lt Π then119867



119867119871(119905) le lim119905rarrinfin

sup(Π120583+ (119867119871(0) minus

Π

120583) 119890minus120583119905) (35)





119868lowast and 119877


119871




119904 =119878

1198780

119904119899=119878119873

1198780

119873

119904119902=119878119876

1198780

119876

119901119899=119875119873

1198750

119873

119901119902=119875119876

1198750

119876

119888119899=119862119873

1198620

119873

119888119902=119862119876

1198620

119876

ℎ =119867

1198670

119894119899=119868119873

1198680

119873

119894119902=119868119876

1198680

119876

119903119903=119877119877

1198770

119877

119903119889=119877119863

1198770

119863

119903119899=119877119873

1198770

119873

119889119863=119863119863

1198630

119863

119905lowast=119905

1198790

ℎ119897=119867119871

1198670

119871

(36)

where

1198780=Π

120583

1198780

119873= 1198780

119876= 1198780

1198750

119873=12057921198861205721198731198780

119873

120573119873+ 120583

1198750

119876=12057921198871205721198731198780

119873

120573119876+ 120583

1198620

119873=12057931198861205731198731198750

119873

119903119864119873+ 120574119873+ 120583

1198620

119876=12057931198871205731198731198750

119873

119903119864119876+ 120574119876+ 120583

1198680

119873=

12057941205741198731198620

119873

119903119871119873+ 120575119873+ 120583

1198680

119876=(1 minus 120579

4) 1205741198731198620

119873

119903119871119876+ 120575119876+ 120583

1198770

119877= 1199031198641198731198620

1198731198790

1198770

119863=1205751198731198680

N119887

1198770

119873= 12058311987901198670

119871

1198630

119863= 11988711987901198770

119863

1198670= 1198670

119871=Π

120583= 1198780

1198790=1

120583

(37)


120588119899=12057931205881198731198750

119873

1205831198670

120588119902=

12057971205881198761198750

119876

1205831198670

120591119899=1205911198731198620

119873

1205831198670

120591119902=

1205911198761198620

119876

1205831198670

120585119899=1205851198731198680

119873

1205831198670

120585119902=

1205851198761198680

119876

1205831198670

119886119889=1198861198631198770

119863

1205831198670

120572119899= (120572119873+ 120583)119879

0

120572119902= (120572119876+ 120583)119879

0

120573119899= (120573119873+ 120583) 119879

0

120573119902= (120573119876+ 120583)119879

0


120583119889= 1198871198790

1198871=(1 minus 120579

3) 1205731198731198750

119873

1205831198780

1198872=(1 minus 120579

2) 1205721198731198780

119873

1205831198780

1198873=

(1 minus 1205796) 1205721198761198780

119876

1205831198780

1198874=

(1 minus 1205797) 1205731198761198750

119876

1205831198780

1198875=1205751198731198680

119873

1205831198670

1198876=

1205751198761198680

119876

1205831198670

1198878=(119887 minus 120583) 119877

0

119863

1205831198670

1198861=

12057961205721198761198780

119876

12057921198871205721198731198780

119873

1198862=

12057971205731198761198750

119876

12057931198871205731198731198750

119873

1198863=

1205901198731198680

119873

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198864=

1205741198761198620

119876

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198865=1199031198711198731198680

119873

1199031198641198731198620

119873

1198866=

1199031198641198761198620

119876

1199031198641198731198620

119873

1198867=

1199031198711198761198680

119876

1199031198641198731198620

119873

1198868=

1205751198761198680

119876

1205751198731198680

119873

120574119899= (119903119864119873+ 120574119873+ 120583) 119879

0

120574119902= (119903119864119876+ 120574119876+ 120583) 119879

0

120575119902= (119903119871119876+ 120575119876+ 120583) 119879

0

120575119899= (119903119871119873+ 120575119873+ 120583) 119879

0

(38)


120582 = 120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)

(39)


119889119904

119889119905= 1 minus 120582119904 + 119887

1119901119899+ 1198872119904119899+ 1198873119904119902+ 1198874119901119902minus 119904 (40)

119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (41)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (42)

119889119901119899

119889119905= 120573119899(119904119899minus 119901119899) (43)

119889119901119902

119889119905= 120573119902(119904119899+ 1198861119904119902minus 119901119902) (44)

119889119888119899

119889119905= 120574119899(119901119899minus 119888119899) (45)

119889119888119902

119889119905= 120574119902(119901119899+ 1198862119901119902minus 119888119902) (46)

119889119894119899

119889119905= 120575119899(119888119899minus 119894119899) (47)

119889119894119902

119889119905= 120575119902(119888119899+ 1198863119894119899+ 1198864119888119902minus 119894119902) (48)

119889119903119903

119889119905= 119888119899+ 1198865119894119899+ 1198866119888119902+ 1198867119894119902minus 119903119903 (49)

119889119903119889

119889119905= 120583119889(119894119899+ 1198868119894119902minus 119903119889) (50)

119889119903119899

119889119905= ℎ119897 (51)

119889119889119863

119889119905= 119889119863 (52)


119889ℎ119897

119889119905= 1 minus ℎ

119897minus 1198875119894119899minus 1198876119894119902 (53)

119889ℎ

119889119905= 1 minus ℎ minus 119887

8119903119889 (54)




120588119899=12057931205792119886120588119873120572119873

(120573119873+ 120583) 120583

120588119902=12057971205792119887120588119876120572119873

(120573119876+ 120583) 120583

120591119899=

12057931198861205792119886120591119873120572119873120573119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) 120583

120591119902=

12057931198871205792119886120591119876120572119873120573119873

(120573119873+ 120583) (119903

119864119876+ 120574119876+ 120583) 120583

120585119899=

120579311988612057921198861205794120585119873120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 120583

120585119902=

12057931198861205792119886(1 minus 120579

4) 120585119876120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119876+ 120575119876+ 120583) 120583

119886119889=

120579211988612057931198861205794120575119873120574119873120573119873120572119873119886119863

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 119887120583

1198871=(1 minus 120579

3) 1205792119886120573119873120572119873

(120573119873+ 120583) 120583

1198872=(1 minus 120579

2) 120572119873

120583

1198873=(1 minus 120579

6) 120572119876

120583

1198874=(1 minus 120579

7) 1205792119887120573119876120572119873

(120573119876+ 120583) 120583

1198875=

120579412057921198861205793119886120575119873120574119873120573119873120572119873

(119903119871119873+ 120575119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198876=

(1 minus 1205794) 12057921198861205793119886120575119876120574119873120573119873120572119873

(119903119871119876+ 120575119876+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198878=

(119887 minus 120583) 120579412057921198861205793119886120572119873120573119873120574119873120575119873

119887120583 (120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583)

120575119899=119903119871119873+ 120575119873+ 120583

120583

120572119899=120572119873+ 120583

120583

120572119902=120572119876+ 120583

120583

120573119899=120573119873+ 120583

120583

120573119902=120573119876+ 120583

120583

120574119899=119903119864119873+ 120574119873+ 120583

120583

120574119902=119903119864119876+ 120574119876+ 120583

120583

120575119902=119903119871119876+ 120575119876+ 120583

120583

1198861=1205796120572119876

1205792119887120572119873

1198862=12057971205792119887120573119876(120573119873+ 120583)

12057921198861205793119887(120573119876+ 120583) 120573

119873

1198863=

1205794120590119873

(1 minus 1205794) (119903119871119873+ 120575119873+ 120583)

1198864=

1205793119887120574119876(119903119864119873+ 120574119873+ 120583)

(1 minus 1205794) 1205793119886120574119873(119903119864119876+ 120574119876+ 120583)

1198865=

1199031198711198731205794120574119873

119903119864119873(119903119871119873+ 120575119873+ 120583)

120583119889=119887

120583

1198866=1205793119887119903119864119876(119903119864119873+ 120574119873+ 120583)

1205793119886119903119864119873(119903119864119876+ 120574119876+ 120583)

1198867=119903119871119876(1 minus 120579

4) 120574119873

119903119864119873(119903119871119876+ 120575119876+ 120583)

1198868=(1 minus 120579

4) 120575119876(119903119871119873+ 120575119873+ 120583)

1205794120575119873(119903119871119876+ 120575119876+ 120583)

(55)


119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast



119904lowast

119899= 119901lowast

119899= 119888lowast

119899= 119894lowast

119899(56)



119899and

119904lowast


119901lowast

119902(119901lowast

119899 119904lowast

119902) = 119901lowast

119899+ 1198861119904lowast

119902

119888lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

1119901lowast

119899+ 1198602119904lowast

119902

119894lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

3119901lowast

119899+ 1198604119904lowast

119902

119903lowast

119903(119901lowast

119899 119904lowast

119902) = 119860

5119901lowast

119899+ 1198606119904lowast

119902

119903lowast

119889(119901lowast

119899 119904lowast

119902) = 119860

7119901lowast

119899+ 1198608119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902

ℎlowast

119897(119901lowast

119899 119904lowast

119902) = 1 minus (119887

5+ 11988761198603) 119901lowast

119899minus 11988761198604119904lowast

119902

(57)

where

1198601= 1 + 119886

2

1198602= 11988611198862

1198603= 1 + 119886

3+ 11988641198601

1198604= 11988641198602

1198605= 1 + 119886

5+ 11988661198601+ 11988671198603

1198606= 11988661198602+ 11988671198604

1198607= 1 + 119886

81198603

1198608= 11988681198604

1198611= 11988781198607

1198612= 11988781198608


1198614= 120572119902minus 1198873minus 11988741198861

(58)




3and 119861


1198613= 1

+120572119873

120583(1205731198731205792119886(1205793minus 1)

120573119873+ 120583

+1205731198761205792119887(1205797minus 1)

120573119876+ 120583

+ 1205792)

gt 0 since 1205792= 1205792119886+ 1205792119887

1198614=1205721198761205796(1205731198761205797+ 120583) + 120583 (120573

119876+ 120583)

120583 (120573119876+ 120583)

gt 0

(59)


4gt 0





1205791(1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119899119901lowast

119899(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(60)

(1 minus 1205791) (1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119902119904lowast

119902(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(61)

where

1198615= 120588119899+ 120588119902+ 120591119899+ 1205911199021198601+ 120585119899+ 1205851199021198603+ 1198861198891198607

1198616= 1205881199021198861+ 1205911199021198602+ 1205851199021198604+ 1198861198891198608

(62)



(1 minus 1198611119901lowast

119899minus 1198612119904lowast

119902) (120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast

1199021205791) = 0 (63)


119899minus 1198612119904lowast

119902= 0 or (ii)

120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast


system


119899minus 1198614119904lowast

119902= 0


119899minus 1198612119904lowast

119902= 0

(64)



119901lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902 (65)



119904lowast

119902= 0

or 119904lowast119902=1198617

1198618

=(1 minus 120579

1) 120572119899(1198770minus 1)

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612) (R minus 1)

= 119909 (1198770minus 1

R minus 1)

(66)

where

1198617= 120572119902120572119899(1 minus 120579

1) ((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

) minus 1)

= 120572119902120572119899(1 minus 120579

1) (1198770minus 1)

1198618= 120572119902[(12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614)

minus (12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612)] = 120572

119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612)((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (

12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614

12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612

) minus 1) = 120572119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612) (1198770119911 minus 1)

(67)

with

119909 =(1 minus 120579

1) 120572119899

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612)

119910 =

1205791120572119902119909

(1 minus 1205791) 120572119899

119911 =

(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

(68)

1198770=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

R =1198770(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

= 1199111198770

(69)


1198612= 11988781198868119886411988611198862=1205796120572119876

120583

1205797120573119876

(120573119876+ 120583)

((1 minus120583

119887)

sdot120575119876

(119903119871119876+ 120575119876+ 120583)

120574119876

(119903119864119876+ 120574119876+ 120583)) lt

1205796120572119876

120583

sdot1205797120573119876

(120573119876+ 120583)

lt1205796120572119876

120583

(1205797120573119876+ 120583)

(120573119876+ 120583)

+ 1 = 1198614

(70)



3lt 1198611 we will


1120572119902(1minus120579

1)120572119899)(1198611minus1198613)


system as follows




0gt 1 and the


0




119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119894lowast

119899= 119888lowast

119899= 119904lowast

119899= 119901lowast

119899= 119910(

1198770minus 1

R minus 1)

119901lowast

119902= (119910 + 119886

1119909) (1198770minus 1

R minus 1)

119888lowast

119902= (1198601119910 + 119860

2119909) (1198770minus 1

R minus 1)

119894lowast

119902= (1198603119910 + 119860

4119909) (1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

119903lowast

119889= (1198607119910 + 119860

8119909) (1198770minus 1

R minus 1)

119904lowast= 1 minus (119861

3119910 + 1198614119909) (1198770minus 1

R minus 1)


1119910 + 1198612119909) (1198770minus 1

R minus 1)

(71)



119889119891119890=


119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast







119869dfe =

((((((((((((((((((((((

(


1198890

0 minus1205721198990 120579

1120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 0 minus120572119902

1205791120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 120573119899

0 minus120573119899

0 0 0 0 0 0 0 0

0 1205731199021198861120573119902

0 minus120573119902

0 0 0 0 0 0 0

0 0 0 120574119899

0 minus1205741198990 0 0 0 0 0

0 0 0 120574119902

1198862120574119902

0 minus1205741199020 0 0 0 0

0 0 0 0 0 120575119899

0 minus1205751198990 0 0 0

0 0 0 0 0 12057511990211988641205751199021198863120575119902minus1205751199020 0 0

0 0 0 0 0 1 11988661198865

1198867minus1 0 0

0 0 0 0 0 0 0 12058311988912058311988911988680 minus120583

1198890

0 0 0 0 0 0 0 0 0 0 minus1198878minus1

))))))))))))))))))))))

)

(72)

where 1205791= 1 minus 120579



119875 (120577) =1003816100381610038161003816119869dfe minus 120577I

1003816100381610038161003816 = (120577 + 1)31198759(120577) = 0 (73)


1198759 (120577) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus120572119899minus 120577 0 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

0 minus120572119902minus 120577 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

120573119899

0 minus120573119899minus 120577 0 0 0 0 0 0

120573119902

1198861120573119902

0 minus120573119902minus 120577 0 0 0 0 0

0 0 120574119899

0 minus120574119899minus 120577 0 0 0 0

0 0 120574119902

1198862120574119902

0 minus120574119902minus 120577 0 0 0

0 0 0 0 120575119899

0 minus120575119899minus 120577 0 0

0 0 0 0 120575119902

1198864120575119902

1198863120575119902minus120575119902minus 120577 0

0 0 0 0 0 0 120583119889

1205831198891198868minus120583119889minus 120577

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(74)


119875 (120577) = (120577 + 1)3

9

sum

119896=0

119888119894120577119894 (75)

where1198889= 1

1198888= 120572119899+ 120572119902+ 120573119899+ 120573119902+ 120574119899+ 120574119902+ 120575119899+ 120575119902+ 120583119889

1198880= 1205731198991205731199021205741198991205741199021205751198991205751199021205831198891205721198991205721199021 minus

12057911198615

120572119899


1) 1198616

120572119902

= 120573119899120573119902120574119899120574119902120575119899120575119902120583119889120572119899120572119902(1 minus 119877

0)

(76)




0= 1 to





0 1198770




0= 1


0specifically


0which is fixedRmay






0is the


119899 119901119902 119888119899 119888119902 119894119899 119894119902 119903119903 119903119889 and ℎ The




F =

(((((((((((((((((

(

0

1205791120582119904

(1 minus 1205791) 120582119904

0

0

0

0

0

0

0

0

0

)))))))))))))))))

)

V =

((((((((((((((((((((((((((((((((

(


120572119899119904119899

120572119902119904119902

120573119899(minus119904119899+ 119901119899)

120573119902(minus119904119899minus 1198861119904119902+ 119901119902)

120574119899(minus119901119899+ 119888119899)

120574119902(minus119901119899minus 1198862119901119902+ 119888119902)

120575119899(minus119888119899+ 119894119899)



minus120583119889119894119899minus 1205831198891198868119894119902+ 120583119889119903119889

minus1 + ℎ + 1198878119903119889

))))))))))))))))))))))))))))))))

)

(77)


119894119895=

120597F119894120597119909119895and (119881)

119894119895= 120597V



119901119902= 119888119899= 119888119902= 119894119899= 119894119902= 119903119903= 119903119889= 0 The basic


1198770= 984858 (119865119881

minus1) =

1

120572119899120572119902

[1205721199021205791(1

+ (1 + 1198863+ (1 + 119886

2) 1198864) 119886119889

+ 120585119902(11988621198864+ 1198863+ 1198864+ 1) + 119886

2120591119902+ 120585119899+ 120588119899+ 120588119902

+ 120591119899+ 120591119902) minus 120572

119899(1205791minus 1) (119886

1120588119902

+ 1198861[11988621198864(1198868119886119889+ 120585119902) + 1198862120591119902])]

=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

(78)






119899(which








120583


120583 + rate120583

asymp large (79)


119899 1120573119902



119901119899= 119888119899= 119894119899= 119904119899

119901119902= 119904119899+ 1198861119904119902

119888119902= 1198601119904119899+ 1198602119904119902

119894119902= 1198603119904119899+ 1198604119904119902

119903119889= 1198607119904119899+ 1198608119904119902

(80)


119889119905= 1 minus 120582119904 + (120572


119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (82)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (83)

119889119903119903

119889119905= 1198605119904119899+ 1198606119904119902minus 119903119903 (84)

119889ℎ

119889119905= 1 minus ℎ minus 119861

1119904119899minus 1198612119904119902 (85)


120582119904 = (120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)) 119904 = (119861

5(119904119899

ℎ)

+ 1198616(

119904119902

ℎ)) 119904

(86)





119899 119904lowast

119902 119903lowast


119864dfe = (1 0 0 0 1)

119864119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast)

(87)


119904lowast

119902=1198617

1198618

119904lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902

119903lowast

119903= 1198605119904lowast

119899+ 1198606119904lowast

119902

119904lowast= 1 minus 119861

3119904lowast

119899minus 1198614119904lowast

119902


1119904lowast

119899minus 1198612119904lowast

119902

(88)


3119910 + 119861

4119909 and

1198611119910 + 1198612119909 = 1 we get

119904lowast= (119911 minus 1

R minus 1)

119904lowast

119899= 119910(

1198770minus 1

R minus 1)

119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

ℎlowast= ((119911 minus 1) 119877

0

R minus 1)

(89)


119899 119904lowast

119902 119903lowast




119869 (xlowast) =((

(


4(xlowast) + 120572


119902minus 11986140 minus119862

6(xlowast)

12057911198623(xlowast) 120579


11989912057911198625(xlowast) 0 120579

11198626(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) 120579


1199020 12057911198626(xlowast)

0 1198605

1198606

minus1 0

0 minus1198611

minus1198612

0 minus1

))

)

(90)

where 1205791= 1 minus 120579

1and

1198623(xlowast) = 119861

5

119904lowast

119899

ℎlowast+ 1198616

119904lowast

119902

ℎlowast=120572119899119910 (1198770minus 1)

1205791(119911 minus 1)

1198624(xlowast) = 119861

5

119904lowast

ℎlowast=1198615

1198770

1198625(xlowast) = 119861

6

119904lowast

ℎlowast=1198616

1198770

1198626(xlowast) = minus(119861

5


119899

ℎlowast2+ 1198616


119902

ℎlowast2) = minus

120572119899119910 (1198770minus 1)

1205791 (119911 minus 1) 1198770

(91)


1198755(120577 xlowast)

= (120577 + 1)2(1205773+ 1198762(xlowast) 1205772 + 119876

1(xlowast) 120577 + 119876

0(xlowast))

= 0

(92)

where

1198762(xlowast) = 120572

119899+ 120572119902+ 1198623(xlowast) minus 119862

4(xlowast) 120579


1

+ 1

1198761(xlowast) = 120572

119902


1199021198624(xlowast) + 119862

4(xlowast))

+ 11986121198626(xlowast) 120579

1+ 1198623(xlowast) (120572

1199021205791+ 11986131205791+ 11986141205791)

+ 120572119899(120572119902+ 12057911198623(xlowast) minus 119862

5(xlowast) 120579

1+ 1) minus 119862

5(xlowast)

sdot 1205791

1198760(xlowast) = 120572

1199021205791(11986111198626(xlowast) + 119861


4(xlowast))

+ 120572119899(1205791(11986121198626(xlowast) + 119861


5(xlowast)) + 120572

119902)

(93)




119902=

119903lowast




1198616 1198626(xdfe) = 0 and (92) becomes

1198755(120577 xdfe) = (120577 + 1)

3(1205772+ 119876dfe120577 + 119877dfe) = 0 (94)

where


= 120572119899(1 minus 119877

0) + 120572119902+ (1 minus 120579

1) 1198616(

120572119899minus 120572119902

120572119902

)

119877dfe = 120572119902 (120572119899 minus 11986151205791) + 1205721198991198616 (1205791 minus 1)

= 120572119902120572119899(1 minus 119877

0)

(95)


radic1198762







=


119899 119904lowast

119902 119903lowast


and we have

1198755(120577 xlowast) = (120577 + 1)2 (1205773 + 119875x

119890119890

1205772+ 119876x

119890119890

120577 + 119877x119890119890

) = 0 (96)

where


119875x119890119890

=


12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)


119890119890will be the


119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)




119899= 0 and we


1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)





grouping 1198616




1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)





CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References





































119876



119876










119863 or confirmed








119873





6




119889119878

119889119905= Π minus 120582119878 + (1 minus 120579

3) 120573119873119875119873+ (1 minus 120579

2) 120572119873119878119873

+ (1 minus 1205796) 120572119876119878119876+ (1 minus 120579

7) 120573119876119875119876minus 120583119878

(1)





119867119871 (119905) = (119878 + 119878119873 + 119878119876 + 119875119873 + 119875119876 + 119862119873 + 119862119876 + 119868119873

+ 119868119876+ 119877119877) (119905)

(2)

119867(119905) = (119878 + 119878119873+ 119878119876+ 119875119873+ 119875119876+ 119862119873+ 119862119876+ 119868119873+ 119868119876

+ 119877119877+ 119877119863) (119905)

(3)



the following form

120582 =1

119867(1205793120588119873119875119873+ 1205797120588119876119875119876+ 120591119873119862119873+ 120585119873119868119873+ 120591119876119862119876

+ 120585119876119868119876+ 119886119863119877119863)

(4)


120588119876 120591119873 120591119876 120585119873 120585119876 and 119886







2(resp 120579

6) of the




119873and 120572

119876)

a fraction 1205792119887




119889119878119873

119889119905= 1205791120582119878 minus (1 minus 120579

2) 120572119873119878119873minus 1205792119886120572119873119878119873minus 1205792119887120572119873119878119873

minus 120583119878119873

119889119878119876

119889119905= (1 minus 120579

1) 120582119878 minus (1 minus 120579

6) 120572119876119878119876minus 1205796120572119876119878119876minus 120583119878119876

(5)


2= 1205792119886+ 1205792119887 so that 1 minus 120579

2+ 1205792119886+ 1205792119887= 1


6of sus-


119873and 120573

119876) when some


119873and 120572


1 minus 1205793and 1 minus 120579






3119886



119889119875119873

119889119905= 1205792119886120572119873119878119873minus (1 minus 120579

3) 120573119873119875119873minus 1205793119886120573119873119875119873

minus 1205793119887120573119873119875119873minus 120583119875119873

119889119875119876

119889119905= 1205792119887120572119873119878119873+ 1205796120572119876119878119876minus (1 minus 120579

7) 120573119876119875119876minus 1205797120573119876119875119876

minus 120583119875119876

(6)



119864119873for



119873or 120574119876




4escape


119889119862119873

119889119905= 1205793119886120573119873119875119873minus 119903119864119873119862119873minus 1205794120574119873119862119873


119889119862119876


(7)



4and 120579



119871119873and 119903119871119876


119873and 120575





119889119868119873


119889119868119876

119889119905= (1 minus 120579

4) 120574119873119862119873+ 120590119873119868119873+ 120574119876119862119876minus 120575119876119868119876minus 119903119871119876119868119876

minus 120583119868119876

(8)


119873and 120575

119876and the


119889119877119877

119889119905= 119903119864119873119862119873+ 119903119871119873119868119873+ 119903119864119876119862119876+ 119903119871119876119868119876minus 120583119868119873

119889119877119863

119889119905= 120575119873119868119873+ 120575119876119868119876minus 119887119877119863

(9)


119873and keeps



119889119877119873

119889119905= 120583119867119871

119889119863119863

119889119905= 119887119877119863

(10)


119889119878

119889119905= Π minus 120582119878 +

3120573119873119875119873+ 2120572119873119878119873+ 6120572119876119878119876

+ 7120573119876119875119876minus 120583119878

(11)

119889119878119873

119889119905= 1205791120582119878 minus (120572

119873+ 120583) 119878

119873 (12)

119889119878119876

119889119905= 1120582119878 minus (120572

119876+ 120583) 119878

119876 (13)

119889119875119873

119889119905= 1205792119886120572119873119878119873minus (120573119873+ 120583) 119875

119873 (14)

119889119875119876

119889119905= 1205792119887120572119873119878119873+ 1205796120572119876119878119876minus (120573119876+ 120583) 119875

119876 (15)

119889119862119873

119889119905= 1205793119886120573119873119875119873minus (119903119864119873+ 120574119873+ 120583)119862

119873 (16)

119889119862119876

119889119905= 1205793119887120573119873119875119873+ 1205797120573119876119875119876minus (119903119864119876+ 120574119876+ 120583)119862

119876 (17)

119889119868119873

119889119905= 1205794120574119873119862119873minus (119903119871119873+ 120575119873+ 120583) 119868

119873 (18)

119889119868119876

119889119905= 4120574119873119862119873+ 120590119873119868119873+ 120574119876119862119876

minus (119903119871119876+ 120575119876+ 120583) 119868

119876

(19)

119889119877119877

119889119905= 119903119864119873119862119873+ 119903119871119873119868119873+ 119903119864119876119862119876+ 119903119871119876119868119876minus 120583119877119877 (20)

119889119877119863

119889119905= 120575119873119868119873+ 120575119876119868119876minus 119887119877119863 (21)

119889119877119873

119889119905= 120583119867119871

(22)

119889119863119863

119889119905= 119887119877119863 (23)







119878 (0) = 1198780

119868119873(0) = 119868

0

119878119873 (0) = 119878119876 (0) = 119875119873 (0) = 0

119875119876(0) = 119862

119873(0) = 119862

119876(0) = 119868

119876(0) = 119877

119863(0) = 119877

119877(0)

= 119877119873(0) = 119863

119863(0) = 0

(24)





119889119867119871

119889119905= Π minus 120583119867

119871minus 120575119873119868119873minus 120575119876119868119876 (25)

119889119867


119863 (26)



119863 119877119873 and



119871minus 119887119877119863







119873 119878119876



1= 119878 119909

2= 119878119873 119909

11= 119877119863 Let f(x) =

(1198911(x) 1198912(x) 119891





119878119873 and 119891




dx119889119905= f (x) x (0) = x

0 (27)





119868lowast and 119877







10038171003817100381710038171003817f1015840 (z y minus x)10038171003817100381710038171003817infin =

1003817100381710038171003817100381710038171003817100381710038171003817

11

sum

119896=1


1003817100381710038171003817100381710038171003817100381710038171003817infin

le

1003817100381710038171003817100381710038171003817100381710038171003817

11

sum

119896=1

nabla119891119896 (z)1003817100381710038171003817100381710038171003817100381710038171003817infin

1003817100381710038171003817y minus x1003817100381710038171003817infin

(29)


+ It is now





11

sum

119896=1

nabla119891119896 (z)1003817100381710038171003817100381710038171003817100381710038171003817infin



1003817100381710038171003817infinle 119872

1003817100381710038171003817y minus x1003817100381710038171003817infin (31)









1) = 0 1198781015840(119905

1) lt 0


1 119878(119905) gt 0 and 119878

119873(119905) gt 0 119878

119876(119905) gt 0

119875119873(119905) gt 0 119875

119876(119905) gt 0 119862

119873(119905) gt 0 119862

119876(119905) gt 0 119868

119873(119905) gt 0

119868119876(119905) gt 0 119877

119877(119905) gt 0 and 119877

119863(119905) gt 0 So at the point 119905 = 119905

1


119889119878

119889119905

10038161003816100381610038161003816100381610038161003816119905=1199051

= Π minus 120582119878 (1199051) + (1 minus 120579

3) 120573119873119875119873(1199051)

+ (1 minus 1205792) 120572119873119878119873(1199051) + (1 minus 120579

6) 120572119876119878119876(1199051)

+ (1 minus 1205797) 120573119876119875119876(1199051) minus 120583119878 (119905

1)

= Π + (1 minus 1205793) 120573119873119875119873(1199051)

+ (1 minus 1205792) 120572119873119878119873(1199051) + (1 minus 120579

6) 120572119876119878119876(1199051)

+ (1 minus 1205797) 120573119876119875119876(1199051) gt 0

(32)


1) lt



+ the solution will







Ω119867119871

= 119867119871(119905) 0 le 119867

119871(119905) le

Π

120583 sub R

11

+ (33)


119871is bounded



119889119867119871

119889119905= Π minus 120583119867


119871997904rArr

119867119871(119905) le

Π

120583+ (119867119871(0) minus

Π

120583) 119890minus120583119905

(34)


119871(119905)


119871(0) lt Π then119867



119867119871(119905) le lim119905rarrinfin

sup(Π120583+ (119867119871(0) minus

Π

120583) 119890minus120583119905) (35)





119868lowast and 119877


119871




119904 =119878

1198780

119904119899=119878119873

1198780

119873

119904119902=119878119876

1198780

119876

119901119899=119875119873

1198750

119873

119901119902=119875119876

1198750

119876

119888119899=119862119873

1198620

119873

119888119902=119862119876

1198620

119876

ℎ =119867

1198670

119894119899=119868119873

1198680

119873

119894119902=119868119876

1198680

119876

119903119903=119877119877

1198770

119877

119903119889=119877119863

1198770

119863

119903119899=119877119873

1198770

119873

119889119863=119863119863

1198630

119863

119905lowast=119905

1198790

ℎ119897=119867119871

1198670

119871

(36)

where

1198780=Π

120583

1198780

119873= 1198780

119876= 1198780

1198750

119873=12057921198861205721198731198780

119873

120573119873+ 120583

1198750

119876=12057921198871205721198731198780

119873

120573119876+ 120583

1198620

119873=12057931198861205731198731198750

119873

119903119864119873+ 120574119873+ 120583

1198620

119876=12057931198871205731198731198750

119873

119903119864119876+ 120574119876+ 120583

1198680

119873=

12057941205741198731198620

119873

119903119871119873+ 120575119873+ 120583

1198680

119876=(1 minus 120579

4) 1205741198731198620

119873

119903119871119876+ 120575119876+ 120583

1198770

119877= 1199031198641198731198620

1198731198790

1198770

119863=1205751198731198680

N119887

1198770

119873= 12058311987901198670

119871

1198630

119863= 11988711987901198770

119863

1198670= 1198670

119871=Π

120583= 1198780

1198790=1

120583

(37)


120588119899=12057931205881198731198750

119873

1205831198670

120588119902=

12057971205881198761198750

119876

1205831198670

120591119899=1205911198731198620

119873

1205831198670

120591119902=

1205911198761198620

119876

1205831198670

120585119899=1205851198731198680

119873

1205831198670

120585119902=

1205851198761198680

119876

1205831198670

119886119889=1198861198631198770

119863

1205831198670

120572119899= (120572119873+ 120583)119879

0

120572119902= (120572119876+ 120583)119879

0

120573119899= (120573119873+ 120583) 119879

0

120573119902= (120573119876+ 120583)119879

0


120583119889= 1198871198790

1198871=(1 minus 120579

3) 1205731198731198750

119873

1205831198780

1198872=(1 minus 120579

2) 1205721198731198780

119873

1205831198780

1198873=

(1 minus 1205796) 1205721198761198780

119876

1205831198780

1198874=

(1 minus 1205797) 1205731198761198750

119876

1205831198780

1198875=1205751198731198680

119873

1205831198670

1198876=

1205751198761198680

119876

1205831198670

1198878=(119887 minus 120583) 119877

0

119863

1205831198670

1198861=

12057961205721198761198780

119876

12057921198871205721198731198780

119873

1198862=

12057971205731198761198750

119876

12057931198871205731198731198750

119873

1198863=

1205901198731198680

119873

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198864=

1205741198761198620

119876

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198865=1199031198711198731198680

119873

1199031198641198731198620

119873

1198866=

1199031198641198761198620

119876

1199031198641198731198620

119873

1198867=

1199031198711198761198680

119876

1199031198641198731198620

119873

1198868=

1205751198761198680

119876

1205751198731198680

119873

120574119899= (119903119864119873+ 120574119873+ 120583) 119879

0

120574119902= (119903119864119876+ 120574119876+ 120583) 119879

0

120575119902= (119903119871119876+ 120575119876+ 120583) 119879

0

120575119899= (119903119871119873+ 120575119873+ 120583) 119879

0

(38)


120582 = 120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)

(39)


119889119904

119889119905= 1 minus 120582119904 + 119887

1119901119899+ 1198872119904119899+ 1198873119904119902+ 1198874119901119902minus 119904 (40)

119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (41)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (42)

119889119901119899

119889119905= 120573119899(119904119899minus 119901119899) (43)

119889119901119902

119889119905= 120573119902(119904119899+ 1198861119904119902minus 119901119902) (44)

119889119888119899

119889119905= 120574119899(119901119899minus 119888119899) (45)

119889119888119902

119889119905= 120574119902(119901119899+ 1198862119901119902minus 119888119902) (46)

119889119894119899

119889119905= 120575119899(119888119899minus 119894119899) (47)

119889119894119902

119889119905= 120575119902(119888119899+ 1198863119894119899+ 1198864119888119902minus 119894119902) (48)

119889119903119903

119889119905= 119888119899+ 1198865119894119899+ 1198866119888119902+ 1198867119894119902minus 119903119903 (49)

119889119903119889

119889119905= 120583119889(119894119899+ 1198868119894119902minus 119903119889) (50)

119889119903119899

119889119905= ℎ119897 (51)

119889119889119863

119889119905= 119889119863 (52)


119889ℎ119897

119889119905= 1 minus ℎ

119897minus 1198875119894119899minus 1198876119894119902 (53)

119889ℎ

119889119905= 1 minus ℎ minus 119887

8119903119889 (54)




120588119899=12057931205792119886120588119873120572119873

(120573119873+ 120583) 120583

120588119902=12057971205792119887120588119876120572119873

(120573119876+ 120583) 120583

120591119899=

12057931198861205792119886120591119873120572119873120573119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) 120583

120591119902=

12057931198871205792119886120591119876120572119873120573119873

(120573119873+ 120583) (119903

119864119876+ 120574119876+ 120583) 120583

120585119899=

120579311988612057921198861205794120585119873120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 120583

120585119902=

12057931198861205792119886(1 minus 120579

4) 120585119876120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119876+ 120575119876+ 120583) 120583

119886119889=

120579211988612057931198861205794120575119873120574119873120573119873120572119873119886119863

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 119887120583

1198871=(1 minus 120579

3) 1205792119886120573119873120572119873

(120573119873+ 120583) 120583

1198872=(1 minus 120579

2) 120572119873

120583

1198873=(1 minus 120579

6) 120572119876

120583

1198874=(1 minus 120579

7) 1205792119887120573119876120572119873

(120573119876+ 120583) 120583

1198875=

120579412057921198861205793119886120575119873120574119873120573119873120572119873

(119903119871119873+ 120575119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198876=

(1 minus 1205794) 12057921198861205793119886120575119876120574119873120573119873120572119873

(119903119871119876+ 120575119876+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198878=

(119887 minus 120583) 120579412057921198861205793119886120572119873120573119873120574119873120575119873

119887120583 (120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583)

120575119899=119903119871119873+ 120575119873+ 120583

120583

120572119899=120572119873+ 120583

120583

120572119902=120572119876+ 120583

120583

120573119899=120573119873+ 120583

120583

120573119902=120573119876+ 120583

120583

120574119899=119903119864119873+ 120574119873+ 120583

120583

120574119902=119903119864119876+ 120574119876+ 120583

120583

120575119902=119903119871119876+ 120575119876+ 120583

120583

1198861=1205796120572119876

1205792119887120572119873

1198862=12057971205792119887120573119876(120573119873+ 120583)

12057921198861205793119887(120573119876+ 120583) 120573

119873

1198863=

1205794120590119873

(1 minus 1205794) (119903119871119873+ 120575119873+ 120583)

1198864=

1205793119887120574119876(119903119864119873+ 120574119873+ 120583)

(1 minus 1205794) 1205793119886120574119873(119903119864119876+ 120574119876+ 120583)

1198865=

1199031198711198731205794120574119873

119903119864119873(119903119871119873+ 120575119873+ 120583)

120583119889=119887

120583

1198866=1205793119887119903119864119876(119903119864119873+ 120574119873+ 120583)

1205793119886119903119864119873(119903119864119876+ 120574119876+ 120583)

1198867=119903119871119876(1 minus 120579

4) 120574119873

119903119864119873(119903119871119876+ 120575119876+ 120583)

1198868=(1 minus 120579

4) 120575119876(119903119871119873+ 120575119873+ 120583)

1205794120575119873(119903119871119876+ 120575119876+ 120583)

(55)


119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast



119904lowast

119899= 119901lowast

119899= 119888lowast

119899= 119894lowast

119899(56)



119899and

119904lowast


119901lowast

119902(119901lowast

119899 119904lowast

119902) = 119901lowast

119899+ 1198861119904lowast

119902

119888lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

1119901lowast

119899+ 1198602119904lowast

119902

119894lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

3119901lowast

119899+ 1198604119904lowast

119902

119903lowast

119903(119901lowast

119899 119904lowast

119902) = 119860

5119901lowast

119899+ 1198606119904lowast

119902

119903lowast

119889(119901lowast

119899 119904lowast

119902) = 119860

7119901lowast

119899+ 1198608119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902

ℎlowast

119897(119901lowast

119899 119904lowast

119902) = 1 minus (119887

5+ 11988761198603) 119901lowast

119899minus 11988761198604119904lowast

119902

(57)

where

1198601= 1 + 119886

2

1198602= 11988611198862

1198603= 1 + 119886

3+ 11988641198601

1198604= 11988641198602

1198605= 1 + 119886

5+ 11988661198601+ 11988671198603

1198606= 11988661198602+ 11988671198604

1198607= 1 + 119886

81198603

1198608= 11988681198604

1198611= 11988781198607

1198612= 11988781198608


1198614= 120572119902minus 1198873minus 11988741198861

(58)




3and 119861


1198613= 1

+120572119873

120583(1205731198731205792119886(1205793minus 1)

120573119873+ 120583

+1205731198761205792119887(1205797minus 1)

120573119876+ 120583

+ 1205792)

gt 0 since 1205792= 1205792119886+ 1205792119887

1198614=1205721198761205796(1205731198761205797+ 120583) + 120583 (120573

119876+ 120583)

120583 (120573119876+ 120583)

gt 0

(59)


4gt 0





1205791(1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119899119901lowast

119899(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(60)

(1 minus 1205791) (1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119902119904lowast

119902(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(61)

where

1198615= 120588119899+ 120588119902+ 120591119899+ 1205911199021198601+ 120585119899+ 1205851199021198603+ 1198861198891198607

1198616= 1205881199021198861+ 1205911199021198602+ 1205851199021198604+ 1198861198891198608

(62)



(1 minus 1198611119901lowast

119899minus 1198612119904lowast

119902) (120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast

1199021205791) = 0 (63)


119899minus 1198612119904lowast

119902= 0 or (ii)

120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast


system


119899minus 1198614119904lowast

119902= 0


119899minus 1198612119904lowast

119902= 0

(64)



119901lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902 (65)



119904lowast

119902= 0

or 119904lowast119902=1198617

1198618

=(1 minus 120579

1) 120572119899(1198770minus 1)

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612) (R minus 1)

= 119909 (1198770minus 1

R minus 1)

(66)

where

1198617= 120572119902120572119899(1 minus 120579

1) ((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

) minus 1)

= 120572119902120572119899(1 minus 120579

1) (1198770minus 1)

1198618= 120572119902[(12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614)

minus (12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612)] = 120572

119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612)((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (

12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614

12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612

) minus 1) = 120572119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612) (1198770119911 minus 1)

(67)

with

119909 =(1 minus 120579

1) 120572119899

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612)

119910 =

1205791120572119902119909

(1 minus 1205791) 120572119899

119911 =

(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

(68)

1198770=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

R =1198770(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

= 1199111198770

(69)


1198612= 11988781198868119886411988611198862=1205796120572119876

120583

1205797120573119876

(120573119876+ 120583)

((1 minus120583

119887)

sdot120575119876

(119903119871119876+ 120575119876+ 120583)

120574119876

(119903119864119876+ 120574119876+ 120583)) lt

1205796120572119876

120583

sdot1205797120573119876

(120573119876+ 120583)

lt1205796120572119876

120583

(1205797120573119876+ 120583)

(120573119876+ 120583)

+ 1 = 1198614

(70)



3lt 1198611 we will


1120572119902(1minus120579

1)120572119899)(1198611minus1198613)


system as follows




0gt 1 and the


0




119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119894lowast

119899= 119888lowast

119899= 119904lowast

119899= 119901lowast

119899= 119910(

1198770minus 1

R minus 1)

119901lowast

119902= (119910 + 119886

1119909) (1198770minus 1

R minus 1)

119888lowast

119902= (1198601119910 + 119860

2119909) (1198770minus 1

R minus 1)

119894lowast

119902= (1198603119910 + 119860

4119909) (1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

119903lowast

119889= (1198607119910 + 119860

8119909) (1198770minus 1

R minus 1)

119904lowast= 1 minus (119861

3119910 + 1198614119909) (1198770minus 1

R minus 1)


1119910 + 1198612119909) (1198770minus 1

R minus 1)

(71)



119889119891119890=


119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast







119869dfe =

((((((((((((((((((((((

(


1198890

0 minus1205721198990 120579

1120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 0 minus120572119902

1205791120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 120573119899

0 minus120573119899

0 0 0 0 0 0 0 0

0 1205731199021198861120573119902

0 minus120573119902

0 0 0 0 0 0 0

0 0 0 120574119899

0 minus1205741198990 0 0 0 0 0

0 0 0 120574119902

1198862120574119902

0 minus1205741199020 0 0 0 0

0 0 0 0 0 120575119899

0 minus1205751198990 0 0 0

0 0 0 0 0 12057511990211988641205751199021198863120575119902minus1205751199020 0 0

0 0 0 0 0 1 11988661198865

1198867minus1 0 0

0 0 0 0 0 0 0 12058311988912058311988911988680 minus120583

1198890

0 0 0 0 0 0 0 0 0 0 minus1198878minus1

))))))))))))))))))))))

)

(72)

where 1205791= 1 minus 120579



119875 (120577) =1003816100381610038161003816119869dfe minus 120577I

1003816100381610038161003816 = (120577 + 1)31198759(120577) = 0 (73)


1198759 (120577) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus120572119899minus 120577 0 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

0 minus120572119902minus 120577 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

120573119899

0 minus120573119899minus 120577 0 0 0 0 0 0

120573119902

1198861120573119902

0 minus120573119902minus 120577 0 0 0 0 0

0 0 120574119899

0 minus120574119899minus 120577 0 0 0 0

0 0 120574119902

1198862120574119902

0 minus120574119902minus 120577 0 0 0

0 0 0 0 120575119899

0 minus120575119899minus 120577 0 0

0 0 0 0 120575119902

1198864120575119902

1198863120575119902minus120575119902minus 120577 0

0 0 0 0 0 0 120583119889

1205831198891198868minus120583119889minus 120577

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(74)


119875 (120577) = (120577 + 1)3

9

sum

119896=0

119888119894120577119894 (75)

where1198889= 1

1198888= 120572119899+ 120572119902+ 120573119899+ 120573119902+ 120574119899+ 120574119902+ 120575119899+ 120575119902+ 120583119889

1198880= 1205731198991205731199021205741198991205741199021205751198991205751199021205831198891205721198991205721199021 minus

12057911198615

120572119899


1) 1198616

120572119902

= 120573119899120573119902120574119899120574119902120575119899120575119902120583119889120572119899120572119902(1 minus 119877

0)

(76)




0= 1 to





0 1198770




0= 1


0specifically


0which is fixedRmay






0is the


119899 119901119902 119888119899 119888119902 119894119899 119894119902 119903119903 119903119889 and ℎ The




F =

(((((((((((((((((

(

0

1205791120582119904

(1 minus 1205791) 120582119904

0

0

0

0

0

0

0

0

0

)))))))))))))))))

)

V =

((((((((((((((((((((((((((((((((

(


120572119899119904119899

120572119902119904119902

120573119899(minus119904119899+ 119901119899)

120573119902(minus119904119899minus 1198861119904119902+ 119901119902)

120574119899(minus119901119899+ 119888119899)

120574119902(minus119901119899minus 1198862119901119902+ 119888119902)

120575119899(minus119888119899+ 119894119899)



minus120583119889119894119899minus 1205831198891198868119894119902+ 120583119889119903119889

minus1 + ℎ + 1198878119903119889

))))))))))))))))))))))))))))))))

)

(77)


119894119895=

120597F119894120597119909119895and (119881)

119894119895= 120597V



119901119902= 119888119899= 119888119902= 119894119899= 119894119902= 119903119903= 119903119889= 0 The basic


1198770= 984858 (119865119881

minus1) =

1

120572119899120572119902

[1205721199021205791(1

+ (1 + 1198863+ (1 + 119886

2) 1198864) 119886119889

+ 120585119902(11988621198864+ 1198863+ 1198864+ 1) + 119886

2120591119902+ 120585119899+ 120588119899+ 120588119902

+ 120591119899+ 120591119902) minus 120572

119899(1205791minus 1) (119886

1120588119902

+ 1198861[11988621198864(1198868119886119889+ 120585119902) + 1198862120591119902])]

=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

(78)






119899(which








120583


120583 + rate120583

asymp large (79)


119899 1120573119902



119901119899= 119888119899= 119894119899= 119904119899

119901119902= 119904119899+ 1198861119904119902

119888119902= 1198601119904119899+ 1198602119904119902

119894119902= 1198603119904119899+ 1198604119904119902

119903119889= 1198607119904119899+ 1198608119904119902

(80)


119889119905= 1 minus 120582119904 + (120572


119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (82)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (83)

119889119903119903

119889119905= 1198605119904119899+ 1198606119904119902minus 119903119903 (84)

119889ℎ

119889119905= 1 minus ℎ minus 119861

1119904119899minus 1198612119904119902 (85)


120582119904 = (120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)) 119904 = (119861

5(119904119899

ℎ)

+ 1198616(

119904119902

ℎ)) 119904

(86)





119899 119904lowast

119902 119903lowast


119864dfe = (1 0 0 0 1)

119864119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast)

(87)


119904lowast

119902=1198617

1198618

119904lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902

119903lowast

119903= 1198605119904lowast

119899+ 1198606119904lowast

119902

119904lowast= 1 minus 119861

3119904lowast

119899minus 1198614119904lowast

119902


1119904lowast

119899minus 1198612119904lowast

119902

(88)


3119910 + 119861

4119909 and

1198611119910 + 1198612119909 = 1 we get

119904lowast= (119911 minus 1

R minus 1)

119904lowast

119899= 119910(

1198770minus 1

R minus 1)

119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

ℎlowast= ((119911 minus 1) 119877

0

R minus 1)

(89)


119899 119904lowast

119902 119903lowast




119869 (xlowast) =((

(


4(xlowast) + 120572


119902minus 11986140 minus119862

6(xlowast)

12057911198623(xlowast) 120579


11989912057911198625(xlowast) 0 120579

11198626(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) 120579


1199020 12057911198626(xlowast)

0 1198605

1198606

minus1 0

0 minus1198611

minus1198612

0 minus1

))

)

(90)

where 1205791= 1 minus 120579

1and

1198623(xlowast) = 119861

5

119904lowast

119899

ℎlowast+ 1198616

119904lowast

119902

ℎlowast=120572119899119910 (1198770minus 1)

1205791(119911 minus 1)

1198624(xlowast) = 119861

5

119904lowast

ℎlowast=1198615

1198770

1198625(xlowast) = 119861

6

119904lowast

ℎlowast=1198616

1198770

1198626(xlowast) = minus(119861

5


119899

ℎlowast2+ 1198616


119902

ℎlowast2) = minus

120572119899119910 (1198770minus 1)

1205791 (119911 minus 1) 1198770

(91)


1198755(120577 xlowast)

= (120577 + 1)2(1205773+ 1198762(xlowast) 1205772 + 119876

1(xlowast) 120577 + 119876

0(xlowast))

= 0

(92)

where

1198762(xlowast) = 120572

119899+ 120572119902+ 1198623(xlowast) minus 119862

4(xlowast) 120579


1

+ 1

1198761(xlowast) = 120572

119902


1199021198624(xlowast) + 119862

4(xlowast))

+ 11986121198626(xlowast) 120579

1+ 1198623(xlowast) (120572

1199021205791+ 11986131205791+ 11986141205791)

+ 120572119899(120572119902+ 12057911198623(xlowast) minus 119862

5(xlowast) 120579

1+ 1) minus 119862

5(xlowast)

sdot 1205791

1198760(xlowast) = 120572

1199021205791(11986111198626(xlowast) + 119861


4(xlowast))

+ 120572119899(1205791(11986121198626(xlowast) + 119861


5(xlowast)) + 120572

119902)

(93)




119902=

119903lowast




1198616 1198626(xdfe) = 0 and (92) becomes

1198755(120577 xdfe) = (120577 + 1)

3(1205772+ 119876dfe120577 + 119877dfe) = 0 (94)

where


= 120572119899(1 minus 119877

0) + 120572119902+ (1 minus 120579

1) 1198616(

120572119899minus 120572119902

120572119902

)

119877dfe = 120572119902 (120572119899 minus 11986151205791) + 1205721198991198616 (1205791 minus 1)

= 120572119902120572119899(1 minus 119877

0)

(95)


radic1198762







=


119899 119904lowast

119902 119903lowast


and we have

1198755(120577 xlowast) = (120577 + 1)2 (1205773 + 119875x

119890119890

1205772+ 119876x

119890119890

120577 + 119877x119890119890

) = 0 (96)

where


119875x119890119890

=


12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)


119890119890will be the


119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)




119899= 0 and we


1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)





grouping 1198616




1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)





CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References






































119867119871 (119905) = (119878 + 119878119873 + 119878119876 + 119875119873 + 119875119876 + 119862119873 + 119862119876 + 119868119873

+ 119868119876+ 119877119877) (119905)

(2)

119867(119905) = (119878 + 119878119873+ 119878119876+ 119875119873+ 119875119876+ 119862119873+ 119862119876+ 119868119873+ 119868119876

+ 119877119877+ 119877119863) (119905)

(3)



the following form

120582 =1

119867(1205793120588119873119875119873+ 1205797120588119876119875119876+ 120591119873119862119873+ 120585119873119868119873+ 120591119876119862119876

+ 120585119876119868119876+ 119886119863119877119863)

(4)


120588119876 120591119873 120591119876 120585119873 120585119876 and 119886







2(resp 120579

6) of the




119873and 120572

119876)

a fraction 1205792119887




119889119878119873

119889119905= 1205791120582119878 minus (1 minus 120579

2) 120572119873119878119873minus 1205792119886120572119873119878119873minus 1205792119887120572119873119878119873

minus 120583119878119873

119889119878119876

119889119905= (1 minus 120579

1) 120582119878 minus (1 minus 120579

6) 120572119876119878119876minus 1205796120572119876119878119876minus 120583119878119876

(5)


2= 1205792119886+ 1205792119887 so that 1 minus 120579

2+ 1205792119886+ 1205792119887= 1


6of sus-


119873and 120573

119876) when some


119873and 120572


1 minus 1205793and 1 minus 120579






3119886



119889119875119873

119889119905= 1205792119886120572119873119878119873minus (1 minus 120579

3) 120573119873119875119873minus 1205793119886120573119873119875119873

minus 1205793119887120573119873119875119873minus 120583119875119873

119889119875119876

119889119905= 1205792119887120572119873119878119873+ 1205796120572119876119878119876minus (1 minus 120579

7) 120573119876119875119876minus 1205797120573119876119875119876

minus 120583119875119876

(6)



119864119873for



119873or 120574119876




4escape


119889119862119873

119889119905= 1205793119886120573119873119875119873minus 119903119864119873119862119873minus 1205794120574119873119862119873


119889119862119876


(7)



4and 120579



119871119873and 119903119871119876


119873and 120575





119889119868119873


119889119868119876

119889119905= (1 minus 120579

4) 120574119873119862119873+ 120590119873119868119873+ 120574119876119862119876minus 120575119876119868119876minus 119903119871119876119868119876

minus 120583119868119876

(8)


119873and 120575

119876and the


119889119877119877

119889119905= 119903119864119873119862119873+ 119903119871119873119868119873+ 119903119864119876119862119876+ 119903119871119876119868119876minus 120583119868119873

119889119877119863

119889119905= 120575119873119868119873+ 120575119876119868119876minus 119887119877119863

(9)


119873and keeps



119889119877119873

119889119905= 120583119867119871

119889119863119863

119889119905= 119887119877119863

(10)


119889119878

119889119905= Π minus 120582119878 +

3120573119873119875119873+ 2120572119873119878119873+ 6120572119876119878119876

+ 7120573119876119875119876minus 120583119878

(11)

119889119878119873

119889119905= 1205791120582119878 minus (120572

119873+ 120583) 119878

119873 (12)

119889119878119876

119889119905= 1120582119878 minus (120572

119876+ 120583) 119878

119876 (13)

119889119875119873

119889119905= 1205792119886120572119873119878119873minus (120573119873+ 120583) 119875

119873 (14)

119889119875119876

119889119905= 1205792119887120572119873119878119873+ 1205796120572119876119878119876minus (120573119876+ 120583) 119875

119876 (15)

119889119862119873

119889119905= 1205793119886120573119873119875119873minus (119903119864119873+ 120574119873+ 120583)119862

119873 (16)

119889119862119876

119889119905= 1205793119887120573119873119875119873+ 1205797120573119876119875119876minus (119903119864119876+ 120574119876+ 120583)119862

119876 (17)

119889119868119873

119889119905= 1205794120574119873119862119873minus (119903119871119873+ 120575119873+ 120583) 119868

119873 (18)

119889119868119876

119889119905= 4120574119873119862119873+ 120590119873119868119873+ 120574119876119862119876

minus (119903119871119876+ 120575119876+ 120583) 119868

119876

(19)

119889119877119877

119889119905= 119903119864119873119862119873+ 119903119871119873119868119873+ 119903119864119876119862119876+ 119903119871119876119868119876minus 120583119877119877 (20)

119889119877119863

119889119905= 120575119873119868119873+ 120575119876119868119876minus 119887119877119863 (21)

119889119877119873

119889119905= 120583119867119871

(22)

119889119863119863

119889119905= 119887119877119863 (23)







119878 (0) = 1198780

119868119873(0) = 119868

0

119878119873 (0) = 119878119876 (0) = 119875119873 (0) = 0

119875119876(0) = 119862

119873(0) = 119862

119876(0) = 119868

119876(0) = 119877

119863(0) = 119877

119877(0)

= 119877119873(0) = 119863

119863(0) = 0

(24)





119889119867119871

119889119905= Π minus 120583119867

119871minus 120575119873119868119873minus 120575119876119868119876 (25)

119889119867


119863 (26)



119863 119877119873 and



119871minus 119887119877119863







119873 119878119876



1= 119878 119909

2= 119878119873 119909

11= 119877119863 Let f(x) =

(1198911(x) 1198912(x) 119891





119878119873 and 119891




dx119889119905= f (x) x (0) = x

0 (27)





119868lowast and 119877







10038171003817100381710038171003817f1015840 (z y minus x)10038171003817100381710038171003817infin =

1003817100381710038171003817100381710038171003817100381710038171003817

11

sum

119896=1


1003817100381710038171003817100381710038171003817100381710038171003817infin

le

1003817100381710038171003817100381710038171003817100381710038171003817

11

sum

119896=1

nabla119891119896 (z)1003817100381710038171003817100381710038171003817100381710038171003817infin

1003817100381710038171003817y minus x1003817100381710038171003817infin

(29)


+ It is now





11

sum

119896=1

nabla119891119896 (z)1003817100381710038171003817100381710038171003817100381710038171003817infin



1003817100381710038171003817infinle 119872

1003817100381710038171003817y minus x1003817100381710038171003817infin (31)









1) = 0 1198781015840(119905

1) lt 0


1 119878(119905) gt 0 and 119878

119873(119905) gt 0 119878

119876(119905) gt 0

119875119873(119905) gt 0 119875

119876(119905) gt 0 119862

119873(119905) gt 0 119862

119876(119905) gt 0 119868

119873(119905) gt 0

119868119876(119905) gt 0 119877

119877(119905) gt 0 and 119877

119863(119905) gt 0 So at the point 119905 = 119905

1


119889119878

119889119905

10038161003816100381610038161003816100381610038161003816119905=1199051

= Π minus 120582119878 (1199051) + (1 minus 120579

3) 120573119873119875119873(1199051)

+ (1 minus 1205792) 120572119873119878119873(1199051) + (1 minus 120579

6) 120572119876119878119876(1199051)

+ (1 minus 1205797) 120573119876119875119876(1199051) minus 120583119878 (119905

1)

= Π + (1 minus 1205793) 120573119873119875119873(1199051)

+ (1 minus 1205792) 120572119873119878119873(1199051) + (1 minus 120579

6) 120572119876119878119876(1199051)

+ (1 minus 1205797) 120573119876119875119876(1199051) gt 0

(32)


1) lt



+ the solution will







Ω119867119871

= 119867119871(119905) 0 le 119867

119871(119905) le

Π

120583 sub R

11

+ (33)


119871is bounded



119889119867119871

119889119905= Π minus 120583119867


119871997904rArr

119867119871(119905) le

Π

120583+ (119867119871(0) minus

Π

120583) 119890minus120583119905

(34)


119871(119905)


119871(0) lt Π then119867



119867119871(119905) le lim119905rarrinfin

sup(Π120583+ (119867119871(0) minus

Π

120583) 119890minus120583119905) (35)





119868lowast and 119877


119871




119904 =119878

1198780

119904119899=119878119873

1198780

119873

119904119902=119878119876

1198780

119876

119901119899=119875119873

1198750

119873

119901119902=119875119876

1198750

119876

119888119899=119862119873

1198620

119873

119888119902=119862119876

1198620

119876

ℎ =119867

1198670

119894119899=119868119873

1198680

119873

119894119902=119868119876

1198680

119876

119903119903=119877119877

1198770

119877

119903119889=119877119863

1198770

119863

119903119899=119877119873

1198770

119873

119889119863=119863119863

1198630

119863

119905lowast=119905

1198790

ℎ119897=119867119871

1198670

119871

(36)

where

1198780=Π

120583

1198780

119873= 1198780

119876= 1198780

1198750

119873=12057921198861205721198731198780

119873

120573119873+ 120583

1198750

119876=12057921198871205721198731198780

119873

120573119876+ 120583

1198620

119873=12057931198861205731198731198750

119873

119903119864119873+ 120574119873+ 120583

1198620

119876=12057931198871205731198731198750

119873

119903119864119876+ 120574119876+ 120583

1198680

119873=

12057941205741198731198620

119873

119903119871119873+ 120575119873+ 120583

1198680

119876=(1 minus 120579

4) 1205741198731198620

119873

119903119871119876+ 120575119876+ 120583

1198770

119877= 1199031198641198731198620

1198731198790

1198770

119863=1205751198731198680

N119887

1198770

119873= 12058311987901198670

119871

1198630

119863= 11988711987901198770

119863

1198670= 1198670

119871=Π

120583= 1198780

1198790=1

120583

(37)


120588119899=12057931205881198731198750

119873

1205831198670

120588119902=

12057971205881198761198750

119876

1205831198670

120591119899=1205911198731198620

119873

1205831198670

120591119902=

1205911198761198620

119876

1205831198670

120585119899=1205851198731198680

119873

1205831198670

120585119902=

1205851198761198680

119876

1205831198670

119886119889=1198861198631198770

119863

1205831198670

120572119899= (120572119873+ 120583)119879

0

120572119902= (120572119876+ 120583)119879

0

120573119899= (120573119873+ 120583) 119879

0

120573119902= (120573119876+ 120583)119879

0


120583119889= 1198871198790

1198871=(1 minus 120579

3) 1205731198731198750

119873

1205831198780

1198872=(1 minus 120579

2) 1205721198731198780

119873

1205831198780

1198873=

(1 minus 1205796) 1205721198761198780

119876

1205831198780

1198874=

(1 minus 1205797) 1205731198761198750

119876

1205831198780

1198875=1205751198731198680

119873

1205831198670

1198876=

1205751198761198680

119876

1205831198670

1198878=(119887 minus 120583) 119877

0

119863

1205831198670

1198861=

12057961205721198761198780

119876

12057921198871205721198731198780

119873

1198862=

12057971205731198761198750

119876

12057931198871205731198731198750

119873

1198863=

1205901198731198680

119873

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198864=

1205741198761198620

119876

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198865=1199031198711198731198680

119873

1199031198641198731198620

119873

1198866=

1199031198641198761198620

119876

1199031198641198731198620

119873

1198867=

1199031198711198761198680

119876

1199031198641198731198620

119873

1198868=

1205751198761198680

119876

1205751198731198680

119873

120574119899= (119903119864119873+ 120574119873+ 120583) 119879

0

120574119902= (119903119864119876+ 120574119876+ 120583) 119879

0

120575119902= (119903119871119876+ 120575119876+ 120583) 119879

0

120575119899= (119903119871119873+ 120575119873+ 120583) 119879

0

(38)


120582 = 120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)

(39)


119889119904

119889119905= 1 minus 120582119904 + 119887

1119901119899+ 1198872119904119899+ 1198873119904119902+ 1198874119901119902minus 119904 (40)

119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (41)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (42)

119889119901119899

119889119905= 120573119899(119904119899minus 119901119899) (43)

119889119901119902

119889119905= 120573119902(119904119899+ 1198861119904119902minus 119901119902) (44)

119889119888119899

119889119905= 120574119899(119901119899minus 119888119899) (45)

119889119888119902

119889119905= 120574119902(119901119899+ 1198862119901119902minus 119888119902) (46)

119889119894119899

119889119905= 120575119899(119888119899minus 119894119899) (47)

119889119894119902

119889119905= 120575119902(119888119899+ 1198863119894119899+ 1198864119888119902minus 119894119902) (48)

119889119903119903

119889119905= 119888119899+ 1198865119894119899+ 1198866119888119902+ 1198867119894119902minus 119903119903 (49)

119889119903119889

119889119905= 120583119889(119894119899+ 1198868119894119902minus 119903119889) (50)

119889119903119899

119889119905= ℎ119897 (51)

119889119889119863

119889119905= 119889119863 (52)


119889ℎ119897

119889119905= 1 minus ℎ

119897minus 1198875119894119899minus 1198876119894119902 (53)

119889ℎ

119889119905= 1 minus ℎ minus 119887

8119903119889 (54)




120588119899=12057931205792119886120588119873120572119873

(120573119873+ 120583) 120583

120588119902=12057971205792119887120588119876120572119873

(120573119876+ 120583) 120583

120591119899=

12057931198861205792119886120591119873120572119873120573119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) 120583

120591119902=

12057931198871205792119886120591119876120572119873120573119873

(120573119873+ 120583) (119903

119864119876+ 120574119876+ 120583) 120583

120585119899=

120579311988612057921198861205794120585119873120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 120583

120585119902=

12057931198861205792119886(1 minus 120579

4) 120585119876120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119876+ 120575119876+ 120583) 120583

119886119889=

120579211988612057931198861205794120575119873120574119873120573119873120572119873119886119863

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 119887120583

1198871=(1 minus 120579

3) 1205792119886120573119873120572119873

(120573119873+ 120583) 120583

1198872=(1 minus 120579

2) 120572119873

120583

1198873=(1 minus 120579

6) 120572119876

120583

1198874=(1 minus 120579

7) 1205792119887120573119876120572119873

(120573119876+ 120583) 120583

1198875=

120579412057921198861205793119886120575119873120574119873120573119873120572119873

(119903119871119873+ 120575119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198876=

(1 minus 1205794) 12057921198861205793119886120575119876120574119873120573119873120572119873

(119903119871119876+ 120575119876+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198878=

(119887 minus 120583) 120579412057921198861205793119886120572119873120573119873120574119873120575119873

119887120583 (120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583)

120575119899=119903119871119873+ 120575119873+ 120583

120583

120572119899=120572119873+ 120583

120583

120572119902=120572119876+ 120583

120583

120573119899=120573119873+ 120583

120583

120573119902=120573119876+ 120583

120583

120574119899=119903119864119873+ 120574119873+ 120583

120583

120574119902=119903119864119876+ 120574119876+ 120583

120583

120575119902=119903119871119876+ 120575119876+ 120583

120583

1198861=1205796120572119876

1205792119887120572119873

1198862=12057971205792119887120573119876(120573119873+ 120583)

12057921198861205793119887(120573119876+ 120583) 120573

119873

1198863=

1205794120590119873

(1 minus 1205794) (119903119871119873+ 120575119873+ 120583)

1198864=

1205793119887120574119876(119903119864119873+ 120574119873+ 120583)

(1 minus 1205794) 1205793119886120574119873(119903119864119876+ 120574119876+ 120583)

1198865=

1199031198711198731205794120574119873

119903119864119873(119903119871119873+ 120575119873+ 120583)

120583119889=119887

120583

1198866=1205793119887119903119864119876(119903119864119873+ 120574119873+ 120583)

1205793119886119903119864119873(119903119864119876+ 120574119876+ 120583)

1198867=119903119871119876(1 minus 120579

4) 120574119873

119903119864119873(119903119871119876+ 120575119876+ 120583)

1198868=(1 minus 120579

4) 120575119876(119903119871119873+ 120575119873+ 120583)

1205794120575119873(119903119871119876+ 120575119876+ 120583)

(55)


119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast



119904lowast

119899= 119901lowast

119899= 119888lowast

119899= 119894lowast

119899(56)



119899and

119904lowast


119901lowast

119902(119901lowast

119899 119904lowast

119902) = 119901lowast

119899+ 1198861119904lowast

119902

119888lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

1119901lowast

119899+ 1198602119904lowast

119902

119894lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

3119901lowast

119899+ 1198604119904lowast

119902

119903lowast

119903(119901lowast

119899 119904lowast

119902) = 119860

5119901lowast

119899+ 1198606119904lowast

119902

119903lowast

119889(119901lowast

119899 119904lowast

119902) = 119860

7119901lowast

119899+ 1198608119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902

ℎlowast

119897(119901lowast

119899 119904lowast

119902) = 1 minus (119887

5+ 11988761198603) 119901lowast

119899minus 11988761198604119904lowast

119902

(57)

where

1198601= 1 + 119886

2

1198602= 11988611198862

1198603= 1 + 119886

3+ 11988641198601

1198604= 11988641198602

1198605= 1 + 119886

5+ 11988661198601+ 11988671198603

1198606= 11988661198602+ 11988671198604

1198607= 1 + 119886

81198603

1198608= 11988681198604

1198611= 11988781198607

1198612= 11988781198608


1198614= 120572119902minus 1198873minus 11988741198861

(58)




3and 119861


1198613= 1

+120572119873

120583(1205731198731205792119886(1205793minus 1)

120573119873+ 120583

+1205731198761205792119887(1205797minus 1)

120573119876+ 120583

+ 1205792)

gt 0 since 1205792= 1205792119886+ 1205792119887

1198614=1205721198761205796(1205731198761205797+ 120583) + 120583 (120573

119876+ 120583)

120583 (120573119876+ 120583)

gt 0

(59)


4gt 0





1205791(1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119899119901lowast

119899(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(60)

(1 minus 1205791) (1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119902119904lowast

119902(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(61)

where

1198615= 120588119899+ 120588119902+ 120591119899+ 1205911199021198601+ 120585119899+ 1205851199021198603+ 1198861198891198607

1198616= 1205881199021198861+ 1205911199021198602+ 1205851199021198604+ 1198861198891198608

(62)



(1 minus 1198611119901lowast

119899minus 1198612119904lowast

119902) (120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast

1199021205791) = 0 (63)


119899minus 1198612119904lowast

119902= 0 or (ii)

120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast


system


119899minus 1198614119904lowast

119902= 0


119899minus 1198612119904lowast

119902= 0

(64)



119901lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902 (65)



119904lowast

119902= 0

or 119904lowast119902=1198617

1198618

=(1 minus 120579

1) 120572119899(1198770minus 1)

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612) (R minus 1)

= 119909 (1198770minus 1

R minus 1)

(66)

where

1198617= 120572119902120572119899(1 minus 120579

1) ((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

) minus 1)

= 120572119902120572119899(1 minus 120579

1) (1198770minus 1)

1198618= 120572119902[(12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614)

minus (12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612)] = 120572

119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612)((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (

12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614

12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612

) minus 1) = 120572119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612) (1198770119911 minus 1)

(67)

with

119909 =(1 minus 120579

1) 120572119899

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612)

119910 =

1205791120572119902119909

(1 minus 1205791) 120572119899

119911 =

(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

(68)

1198770=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

R =1198770(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

= 1199111198770

(69)


1198612= 11988781198868119886411988611198862=1205796120572119876

120583

1205797120573119876

(120573119876+ 120583)

((1 minus120583

119887)

sdot120575119876

(119903119871119876+ 120575119876+ 120583)

120574119876

(119903119864119876+ 120574119876+ 120583)) lt

1205796120572119876

120583

sdot1205797120573119876

(120573119876+ 120583)

lt1205796120572119876

120583

(1205797120573119876+ 120583)

(120573119876+ 120583)

+ 1 = 1198614

(70)



3lt 1198611 we will


1120572119902(1minus120579

1)120572119899)(1198611minus1198613)


system as follows




0gt 1 and the


0




119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119894lowast

119899= 119888lowast

119899= 119904lowast

119899= 119901lowast

119899= 119910(

1198770minus 1

R minus 1)

119901lowast

119902= (119910 + 119886

1119909) (1198770minus 1

R minus 1)

119888lowast

119902= (1198601119910 + 119860

2119909) (1198770minus 1

R minus 1)

119894lowast

119902= (1198603119910 + 119860

4119909) (1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

119903lowast

119889= (1198607119910 + 119860

8119909) (1198770minus 1

R minus 1)

119904lowast= 1 minus (119861

3119910 + 1198614119909) (1198770minus 1

R minus 1)


1119910 + 1198612119909) (1198770minus 1

R minus 1)

(71)



119889119891119890=


119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast







119869dfe =

((((((((((((((((((((((

(


1198890

0 minus1205721198990 120579

1120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 0 minus120572119902

1205791120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 120573119899

0 minus120573119899

0 0 0 0 0 0 0 0

0 1205731199021198861120573119902

0 minus120573119902

0 0 0 0 0 0 0

0 0 0 120574119899

0 minus1205741198990 0 0 0 0 0

0 0 0 120574119902

1198862120574119902

0 minus1205741199020 0 0 0 0

0 0 0 0 0 120575119899

0 minus1205751198990 0 0 0

0 0 0 0 0 12057511990211988641205751199021198863120575119902minus1205751199020 0 0

0 0 0 0 0 1 11988661198865

1198867minus1 0 0

0 0 0 0 0 0 0 12058311988912058311988911988680 minus120583

1198890

0 0 0 0 0 0 0 0 0 0 minus1198878minus1

))))))))))))))))))))))

)

(72)

where 1205791= 1 minus 120579



119875 (120577) =1003816100381610038161003816119869dfe minus 120577I

1003816100381610038161003816 = (120577 + 1)31198759(120577) = 0 (73)


1198759 (120577) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus120572119899minus 120577 0 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

0 minus120572119902minus 120577 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

120573119899

0 minus120573119899minus 120577 0 0 0 0 0 0

120573119902

1198861120573119902

0 minus120573119902minus 120577 0 0 0 0 0

0 0 120574119899

0 minus120574119899minus 120577 0 0 0 0

0 0 120574119902

1198862120574119902

0 minus120574119902minus 120577 0 0 0

0 0 0 0 120575119899

0 minus120575119899minus 120577 0 0

0 0 0 0 120575119902

1198864120575119902

1198863120575119902minus120575119902minus 120577 0

0 0 0 0 0 0 120583119889

1205831198891198868minus120583119889minus 120577

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(74)


119875 (120577) = (120577 + 1)3

9

sum

119896=0

119888119894120577119894 (75)

where1198889= 1

1198888= 120572119899+ 120572119902+ 120573119899+ 120573119902+ 120574119899+ 120574119902+ 120575119899+ 120575119902+ 120583119889

1198880= 1205731198991205731199021205741198991205741199021205751198991205751199021205831198891205721198991205721199021 minus

12057911198615

120572119899


1) 1198616

120572119902

= 120573119899120573119902120574119899120574119902120575119899120575119902120583119889120572119899120572119902(1 minus 119877

0)

(76)




0= 1 to





0 1198770




0= 1


0specifically


0which is fixedRmay






0is the


119899 119901119902 119888119899 119888119902 119894119899 119894119902 119903119903 119903119889 and ℎ The




F =

(((((((((((((((((

(

0

1205791120582119904

(1 minus 1205791) 120582119904

0

0

0

0

0

0

0

0

0

)))))))))))))))))

)

V =

((((((((((((((((((((((((((((((((

(


120572119899119904119899

120572119902119904119902

120573119899(minus119904119899+ 119901119899)

120573119902(minus119904119899minus 1198861119904119902+ 119901119902)

120574119899(minus119901119899+ 119888119899)

120574119902(minus119901119899minus 1198862119901119902+ 119888119902)

120575119899(minus119888119899+ 119894119899)



minus120583119889119894119899minus 1205831198891198868119894119902+ 120583119889119903119889

minus1 + ℎ + 1198878119903119889

))))))))))))))))))))))))))))))))

)

(77)


119894119895=

120597F119894120597119909119895and (119881)

119894119895= 120597V



119901119902= 119888119899= 119888119902= 119894119899= 119894119902= 119903119903= 119903119889= 0 The basic


1198770= 984858 (119865119881

minus1) =

1

120572119899120572119902

[1205721199021205791(1

+ (1 + 1198863+ (1 + 119886

2) 1198864) 119886119889

+ 120585119902(11988621198864+ 1198863+ 1198864+ 1) + 119886

2120591119902+ 120585119899+ 120588119899+ 120588119902

+ 120591119899+ 120591119902) minus 120572

119899(1205791minus 1) (119886

1120588119902

+ 1198861[11988621198864(1198868119886119889+ 120585119902) + 1198862120591119902])]

=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

(78)






119899(which








120583


120583 + rate120583

asymp large (79)


119899 1120573119902



119901119899= 119888119899= 119894119899= 119904119899

119901119902= 119904119899+ 1198861119904119902

119888119902= 1198601119904119899+ 1198602119904119902

119894119902= 1198603119904119899+ 1198604119904119902

119903119889= 1198607119904119899+ 1198608119904119902

(80)


119889119905= 1 minus 120582119904 + (120572


119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (82)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (83)

119889119903119903

119889119905= 1198605119904119899+ 1198606119904119902minus 119903119903 (84)

119889ℎ

119889119905= 1 minus ℎ minus 119861

1119904119899minus 1198612119904119902 (85)


120582119904 = (120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)) 119904 = (119861

5(119904119899

ℎ)

+ 1198616(

119904119902

ℎ)) 119904

(86)





119899 119904lowast

119902 119903lowast


119864dfe = (1 0 0 0 1)

119864119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast)

(87)


119904lowast

119902=1198617

1198618

119904lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902

119903lowast

119903= 1198605119904lowast

119899+ 1198606119904lowast

119902

119904lowast= 1 minus 119861

3119904lowast

119899minus 1198614119904lowast

119902


1119904lowast

119899minus 1198612119904lowast

119902

(88)


3119910 + 119861

4119909 and

1198611119910 + 1198612119909 = 1 we get

119904lowast= (119911 minus 1

R minus 1)

119904lowast

119899= 119910(

1198770minus 1

R minus 1)

119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

ℎlowast= ((119911 minus 1) 119877

0

R minus 1)

(89)


119899 119904lowast

119902 119903lowast




119869 (xlowast) =((

(


4(xlowast) + 120572


119902minus 11986140 minus119862

6(xlowast)

12057911198623(xlowast) 120579


11989912057911198625(xlowast) 0 120579

11198626(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) 120579


1199020 12057911198626(xlowast)

0 1198605

1198606

minus1 0

0 minus1198611

minus1198612

0 minus1

))

)

(90)

where 1205791= 1 minus 120579

1and

1198623(xlowast) = 119861

5

119904lowast

119899

ℎlowast+ 1198616

119904lowast

119902

ℎlowast=120572119899119910 (1198770minus 1)

1205791(119911 minus 1)

1198624(xlowast) = 119861

5

119904lowast

ℎlowast=1198615

1198770

1198625(xlowast) = 119861

6

119904lowast

ℎlowast=1198616

1198770

1198626(xlowast) = minus(119861

5


119899

ℎlowast2+ 1198616


119902

ℎlowast2) = minus

120572119899119910 (1198770minus 1)

1205791 (119911 minus 1) 1198770

(91)


1198755(120577 xlowast)

= (120577 + 1)2(1205773+ 1198762(xlowast) 1205772 + 119876

1(xlowast) 120577 + 119876

0(xlowast))

= 0

(92)

where

1198762(xlowast) = 120572

119899+ 120572119902+ 1198623(xlowast) minus 119862

4(xlowast) 120579


1

+ 1

1198761(xlowast) = 120572

119902


1199021198624(xlowast) + 119862

4(xlowast))

+ 11986121198626(xlowast) 120579

1+ 1198623(xlowast) (120572

1199021205791+ 11986131205791+ 11986141205791)

+ 120572119899(120572119902+ 12057911198623(xlowast) minus 119862

5(xlowast) 120579

1+ 1) minus 119862

5(xlowast)

sdot 1205791

1198760(xlowast) = 120572

1199021205791(11986111198626(xlowast) + 119861


4(xlowast))

+ 120572119899(1205791(11986121198626(xlowast) + 119861


5(xlowast)) + 120572

119902)

(93)




119902=

119903lowast




1198616 1198626(xdfe) = 0 and (92) becomes

1198755(120577 xdfe) = (120577 + 1)

3(1205772+ 119876dfe120577 + 119877dfe) = 0 (94)

where


= 120572119899(1 minus 119877

0) + 120572119902+ (1 minus 120579

1) 1198616(

120572119899minus 120572119902

120572119902

)

119877dfe = 120572119902 (120572119899 minus 11986151205791) + 1205721198991198616 (1205791 minus 1)

= 120572119902120572119899(1 minus 119877

0)

(95)


radic1198762







=


119899 119904lowast

119902 119903lowast


and we have

1198755(120577 xlowast) = (120577 + 1)2 (1205773 + 119875x

119890119890

1205772+ 119876x

119890119890

120577 + 119877x119890119890

) = 0 (96)

where


119875x119890119890

=


12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)


119890119890will be the


119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)




119899= 0 and we


1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)





grouping 1198616




1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)





CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References





































4and 120579



119871119873and 119903119871119876


119873and 120575





119889119868119873


119889119868119876

119889119905= (1 minus 120579

4) 120574119873119862119873+ 120590119873119868119873+ 120574119876119862119876minus 120575119876119868119876minus 119903119871119876119868119876

minus 120583119868119876

(8)


119873and 120575

119876and the


119889119877119877

119889119905= 119903119864119873119862119873+ 119903119871119873119868119873+ 119903119864119876119862119876+ 119903119871119876119868119876minus 120583119868119873

119889119877119863

119889119905= 120575119873119868119873+ 120575119876119868119876minus 119887119877119863

(9)


119873and keeps



119889119877119873

119889119905= 120583119867119871

119889119863119863

119889119905= 119887119877119863

(10)


119889119878

119889119905= Π minus 120582119878 +

3120573119873119875119873+ 2120572119873119878119873+ 6120572119876119878119876

+ 7120573119876119875119876minus 120583119878

(11)

119889119878119873

119889119905= 1205791120582119878 minus (120572

119873+ 120583) 119878

119873 (12)

119889119878119876

119889119905= 1120582119878 minus (120572

119876+ 120583) 119878

119876 (13)

119889119875119873

119889119905= 1205792119886120572119873119878119873minus (120573119873+ 120583) 119875

119873 (14)

119889119875119876

119889119905= 1205792119887120572119873119878119873+ 1205796120572119876119878119876minus (120573119876+ 120583) 119875

119876 (15)

119889119862119873

119889119905= 1205793119886120573119873119875119873minus (119903119864119873+ 120574119873+ 120583)119862

119873 (16)

119889119862119876

119889119905= 1205793119887120573119873119875119873+ 1205797120573119876119875119876minus (119903119864119876+ 120574119876+ 120583)119862

119876 (17)

119889119868119873

119889119905= 1205794120574119873119862119873minus (119903119871119873+ 120575119873+ 120583) 119868

119873 (18)

119889119868119876

119889119905= 4120574119873119862119873+ 120590119873119868119873+ 120574119876119862119876

minus (119903119871119876+ 120575119876+ 120583) 119868

119876

(19)

119889119877119877

119889119905= 119903119864119873119862119873+ 119903119871119873119868119873+ 119903119864119876119862119876+ 119903119871119876119868119876minus 120583119877119877 (20)

119889119877119863

119889119905= 120575119873119868119873+ 120575119876119868119876minus 119887119877119863 (21)

119889119877119873

119889119905= 120583119867119871

(22)

119889119863119863

119889119905= 119887119877119863 (23)







119878 (0) = 1198780

119868119873(0) = 119868

0

119878119873 (0) = 119878119876 (0) = 119875119873 (0) = 0

119875119876(0) = 119862

119873(0) = 119862

119876(0) = 119868

119876(0) = 119877

119863(0) = 119877

119877(0)

= 119877119873(0) = 119863

119863(0) = 0

(24)





119889119867119871

119889119905= Π minus 120583119867

119871minus 120575119873119868119873minus 120575119876119868119876 (25)

119889119867


119863 (26)



119863 119877119873 and



119871minus 119887119877119863







119873 119878119876



1= 119878 119909

2= 119878119873 119909

11= 119877119863 Let f(x) =

(1198911(x) 1198912(x) 119891





119878119873 and 119891




dx119889119905= f (x) x (0) = x

0 (27)





119868lowast and 119877







10038171003817100381710038171003817f1015840 (z y minus x)10038171003817100381710038171003817infin =

1003817100381710038171003817100381710038171003817100381710038171003817

11

sum

119896=1


1003817100381710038171003817100381710038171003817100381710038171003817infin

le

1003817100381710038171003817100381710038171003817100381710038171003817

11

sum

119896=1

nabla119891119896 (z)1003817100381710038171003817100381710038171003817100381710038171003817infin

1003817100381710038171003817y minus x1003817100381710038171003817infin

(29)


+ It is now





11

sum

119896=1

nabla119891119896 (z)1003817100381710038171003817100381710038171003817100381710038171003817infin



1003817100381710038171003817infinle 119872

1003817100381710038171003817y minus x1003817100381710038171003817infin (31)









1) = 0 1198781015840(119905

1) lt 0


1 119878(119905) gt 0 and 119878

119873(119905) gt 0 119878

119876(119905) gt 0

119875119873(119905) gt 0 119875

119876(119905) gt 0 119862

119873(119905) gt 0 119862

119876(119905) gt 0 119868

119873(119905) gt 0

119868119876(119905) gt 0 119877

119877(119905) gt 0 and 119877

119863(119905) gt 0 So at the point 119905 = 119905

1


119889119878

119889119905

10038161003816100381610038161003816100381610038161003816119905=1199051

= Π minus 120582119878 (1199051) + (1 minus 120579

3) 120573119873119875119873(1199051)

+ (1 minus 1205792) 120572119873119878119873(1199051) + (1 minus 120579

6) 120572119876119878119876(1199051)

+ (1 minus 1205797) 120573119876119875119876(1199051) minus 120583119878 (119905

1)

= Π + (1 minus 1205793) 120573119873119875119873(1199051)

+ (1 minus 1205792) 120572119873119878119873(1199051) + (1 minus 120579

6) 120572119876119878119876(1199051)

+ (1 minus 1205797) 120573119876119875119876(1199051) gt 0

(32)


1) lt



+ the solution will







Ω119867119871

= 119867119871(119905) 0 le 119867

119871(119905) le

Π

120583 sub R

11

+ (33)


119871is bounded



119889119867119871

119889119905= Π minus 120583119867


119871997904rArr

119867119871(119905) le

Π

120583+ (119867119871(0) minus

Π

120583) 119890minus120583119905

(34)


119871(119905)


119871(0) lt Π then119867



119867119871(119905) le lim119905rarrinfin

sup(Π120583+ (119867119871(0) minus

Π

120583) 119890minus120583119905) (35)





119868lowast and 119877


119871




119904 =119878

1198780

119904119899=119878119873

1198780

119873

119904119902=119878119876

1198780

119876

119901119899=119875119873

1198750

119873

119901119902=119875119876

1198750

119876

119888119899=119862119873

1198620

119873

119888119902=119862119876

1198620

119876

ℎ =119867

1198670

119894119899=119868119873

1198680

119873

119894119902=119868119876

1198680

119876

119903119903=119877119877

1198770

119877

119903119889=119877119863

1198770

119863

119903119899=119877119873

1198770

119873

119889119863=119863119863

1198630

119863

119905lowast=119905

1198790

ℎ119897=119867119871

1198670

119871

(36)

where

1198780=Π

120583

1198780

119873= 1198780

119876= 1198780

1198750

119873=12057921198861205721198731198780

119873

120573119873+ 120583

1198750

119876=12057921198871205721198731198780

119873

120573119876+ 120583

1198620

119873=12057931198861205731198731198750

119873

119903119864119873+ 120574119873+ 120583

1198620

119876=12057931198871205731198731198750

119873

119903119864119876+ 120574119876+ 120583

1198680

119873=

12057941205741198731198620

119873

119903119871119873+ 120575119873+ 120583

1198680

119876=(1 minus 120579

4) 1205741198731198620

119873

119903119871119876+ 120575119876+ 120583

1198770

119877= 1199031198641198731198620

1198731198790

1198770

119863=1205751198731198680

N119887

1198770

119873= 12058311987901198670

119871

1198630

119863= 11988711987901198770

119863

1198670= 1198670

119871=Π

120583= 1198780

1198790=1

120583

(37)


120588119899=12057931205881198731198750

119873

1205831198670

120588119902=

12057971205881198761198750

119876

1205831198670

120591119899=1205911198731198620

119873

1205831198670

120591119902=

1205911198761198620

119876

1205831198670

120585119899=1205851198731198680

119873

1205831198670

120585119902=

1205851198761198680

119876

1205831198670

119886119889=1198861198631198770

119863

1205831198670

120572119899= (120572119873+ 120583)119879

0

120572119902= (120572119876+ 120583)119879

0

120573119899= (120573119873+ 120583) 119879

0

120573119902= (120573119876+ 120583)119879

0


120583119889= 1198871198790

1198871=(1 minus 120579

3) 1205731198731198750

119873

1205831198780

1198872=(1 minus 120579

2) 1205721198731198780

119873

1205831198780

1198873=

(1 minus 1205796) 1205721198761198780

119876

1205831198780

1198874=

(1 minus 1205797) 1205731198761198750

119876

1205831198780

1198875=1205751198731198680

119873

1205831198670

1198876=

1205751198761198680

119876

1205831198670

1198878=(119887 minus 120583) 119877

0

119863

1205831198670

1198861=

12057961205721198761198780

119876

12057921198871205721198731198780

119873

1198862=

12057971205731198761198750

119876

12057931198871205731198731198750

119873

1198863=

1205901198731198680

119873

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198864=

1205741198761198620

119876

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198865=1199031198711198731198680

119873

1199031198641198731198620

119873

1198866=

1199031198641198761198620

119876

1199031198641198731198620

119873

1198867=

1199031198711198761198680

119876

1199031198641198731198620

119873

1198868=

1205751198761198680

119876

1205751198731198680

119873

120574119899= (119903119864119873+ 120574119873+ 120583) 119879

0

120574119902= (119903119864119876+ 120574119876+ 120583) 119879

0

120575119902= (119903119871119876+ 120575119876+ 120583) 119879

0

120575119899= (119903119871119873+ 120575119873+ 120583) 119879

0

(38)


120582 = 120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)

(39)


119889119904

119889119905= 1 minus 120582119904 + 119887

1119901119899+ 1198872119904119899+ 1198873119904119902+ 1198874119901119902minus 119904 (40)

119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (41)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (42)

119889119901119899

119889119905= 120573119899(119904119899minus 119901119899) (43)

119889119901119902

119889119905= 120573119902(119904119899+ 1198861119904119902minus 119901119902) (44)

119889119888119899

119889119905= 120574119899(119901119899minus 119888119899) (45)

119889119888119902

119889119905= 120574119902(119901119899+ 1198862119901119902minus 119888119902) (46)

119889119894119899

119889119905= 120575119899(119888119899minus 119894119899) (47)

119889119894119902

119889119905= 120575119902(119888119899+ 1198863119894119899+ 1198864119888119902minus 119894119902) (48)

119889119903119903

119889119905= 119888119899+ 1198865119894119899+ 1198866119888119902+ 1198867119894119902minus 119903119903 (49)

119889119903119889

119889119905= 120583119889(119894119899+ 1198868119894119902minus 119903119889) (50)

119889119903119899

119889119905= ℎ119897 (51)

119889119889119863

119889119905= 119889119863 (52)


119889ℎ119897

119889119905= 1 minus ℎ

119897minus 1198875119894119899minus 1198876119894119902 (53)

119889ℎ

119889119905= 1 minus ℎ minus 119887

8119903119889 (54)




120588119899=12057931205792119886120588119873120572119873

(120573119873+ 120583) 120583

120588119902=12057971205792119887120588119876120572119873

(120573119876+ 120583) 120583

120591119899=

12057931198861205792119886120591119873120572119873120573119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) 120583

120591119902=

12057931198871205792119886120591119876120572119873120573119873

(120573119873+ 120583) (119903

119864119876+ 120574119876+ 120583) 120583

120585119899=

120579311988612057921198861205794120585119873120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 120583

120585119902=

12057931198861205792119886(1 minus 120579

4) 120585119876120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119876+ 120575119876+ 120583) 120583

119886119889=

120579211988612057931198861205794120575119873120574119873120573119873120572119873119886119863

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 119887120583

1198871=(1 minus 120579

3) 1205792119886120573119873120572119873

(120573119873+ 120583) 120583

1198872=(1 minus 120579

2) 120572119873

120583

1198873=(1 minus 120579

6) 120572119876

120583

1198874=(1 minus 120579

7) 1205792119887120573119876120572119873

(120573119876+ 120583) 120583

1198875=

120579412057921198861205793119886120575119873120574119873120573119873120572119873

(119903119871119873+ 120575119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198876=

(1 minus 1205794) 12057921198861205793119886120575119876120574119873120573119873120572119873

(119903119871119876+ 120575119876+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198878=

(119887 minus 120583) 120579412057921198861205793119886120572119873120573119873120574119873120575119873

119887120583 (120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583)

120575119899=119903119871119873+ 120575119873+ 120583

120583

120572119899=120572119873+ 120583

120583

120572119902=120572119876+ 120583

120583

120573119899=120573119873+ 120583

120583

120573119902=120573119876+ 120583

120583

120574119899=119903119864119873+ 120574119873+ 120583

120583

120574119902=119903119864119876+ 120574119876+ 120583

120583

120575119902=119903119871119876+ 120575119876+ 120583

120583

1198861=1205796120572119876

1205792119887120572119873

1198862=12057971205792119887120573119876(120573119873+ 120583)

12057921198861205793119887(120573119876+ 120583) 120573

119873

1198863=

1205794120590119873

(1 minus 1205794) (119903119871119873+ 120575119873+ 120583)

1198864=

1205793119887120574119876(119903119864119873+ 120574119873+ 120583)

(1 minus 1205794) 1205793119886120574119873(119903119864119876+ 120574119876+ 120583)

1198865=

1199031198711198731205794120574119873

119903119864119873(119903119871119873+ 120575119873+ 120583)

120583119889=119887

120583

1198866=1205793119887119903119864119876(119903119864119873+ 120574119873+ 120583)

1205793119886119903119864119873(119903119864119876+ 120574119876+ 120583)

1198867=119903119871119876(1 minus 120579

4) 120574119873

119903119864119873(119903119871119876+ 120575119876+ 120583)

1198868=(1 minus 120579

4) 120575119876(119903119871119873+ 120575119873+ 120583)

1205794120575119873(119903119871119876+ 120575119876+ 120583)

(55)


119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast



119904lowast

119899= 119901lowast

119899= 119888lowast

119899= 119894lowast

119899(56)



119899and

119904lowast


119901lowast

119902(119901lowast

119899 119904lowast

119902) = 119901lowast

119899+ 1198861119904lowast

119902

119888lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

1119901lowast

119899+ 1198602119904lowast

119902

119894lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

3119901lowast

119899+ 1198604119904lowast

119902

119903lowast

119903(119901lowast

119899 119904lowast

119902) = 119860

5119901lowast

119899+ 1198606119904lowast

119902

119903lowast

119889(119901lowast

119899 119904lowast

119902) = 119860

7119901lowast

119899+ 1198608119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902

ℎlowast

119897(119901lowast

119899 119904lowast

119902) = 1 minus (119887

5+ 11988761198603) 119901lowast

119899minus 11988761198604119904lowast

119902

(57)

where

1198601= 1 + 119886

2

1198602= 11988611198862

1198603= 1 + 119886

3+ 11988641198601

1198604= 11988641198602

1198605= 1 + 119886

5+ 11988661198601+ 11988671198603

1198606= 11988661198602+ 11988671198604

1198607= 1 + 119886

81198603

1198608= 11988681198604

1198611= 11988781198607

1198612= 11988781198608


1198614= 120572119902minus 1198873minus 11988741198861

(58)




3and 119861


1198613= 1

+120572119873

120583(1205731198731205792119886(1205793minus 1)

120573119873+ 120583

+1205731198761205792119887(1205797minus 1)

120573119876+ 120583

+ 1205792)

gt 0 since 1205792= 1205792119886+ 1205792119887

1198614=1205721198761205796(1205731198761205797+ 120583) + 120583 (120573

119876+ 120583)

120583 (120573119876+ 120583)

gt 0

(59)


4gt 0





1205791(1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119899119901lowast

119899(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(60)

(1 minus 1205791) (1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119902119904lowast

119902(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(61)

where

1198615= 120588119899+ 120588119902+ 120591119899+ 1205911199021198601+ 120585119899+ 1205851199021198603+ 1198861198891198607

1198616= 1205881199021198861+ 1205911199021198602+ 1205851199021198604+ 1198861198891198608

(62)



(1 minus 1198611119901lowast

119899minus 1198612119904lowast

119902) (120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast

1199021205791) = 0 (63)


119899minus 1198612119904lowast

119902= 0 or (ii)

120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast


system


119899minus 1198614119904lowast

119902= 0


119899minus 1198612119904lowast

119902= 0

(64)



119901lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902 (65)



119904lowast

119902= 0

or 119904lowast119902=1198617

1198618

=(1 minus 120579

1) 120572119899(1198770minus 1)

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612) (R minus 1)

= 119909 (1198770minus 1

R minus 1)

(66)

where

1198617= 120572119902120572119899(1 minus 120579

1) ((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

) minus 1)

= 120572119902120572119899(1 minus 120579

1) (1198770minus 1)

1198618= 120572119902[(12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614)

minus (12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612)] = 120572

119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612)((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (

12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614

12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612

) minus 1) = 120572119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612) (1198770119911 minus 1)

(67)

with

119909 =(1 minus 120579

1) 120572119899

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612)

119910 =

1205791120572119902119909

(1 minus 1205791) 120572119899

119911 =

(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

(68)

1198770=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

R =1198770(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

= 1199111198770

(69)


1198612= 11988781198868119886411988611198862=1205796120572119876

120583

1205797120573119876

(120573119876+ 120583)

((1 minus120583

119887)

sdot120575119876

(119903119871119876+ 120575119876+ 120583)

120574119876

(119903119864119876+ 120574119876+ 120583)) lt

1205796120572119876

120583

sdot1205797120573119876

(120573119876+ 120583)

lt1205796120572119876

120583

(1205797120573119876+ 120583)

(120573119876+ 120583)

+ 1 = 1198614

(70)



3lt 1198611 we will


1120572119902(1minus120579

1)120572119899)(1198611minus1198613)


system as follows




0gt 1 and the


0




119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119894lowast

119899= 119888lowast

119899= 119904lowast

119899= 119901lowast

119899= 119910(

1198770minus 1

R minus 1)

119901lowast

119902= (119910 + 119886

1119909) (1198770minus 1

R minus 1)

119888lowast

119902= (1198601119910 + 119860

2119909) (1198770minus 1

R minus 1)

119894lowast

119902= (1198603119910 + 119860

4119909) (1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

119903lowast

119889= (1198607119910 + 119860

8119909) (1198770minus 1

R minus 1)

119904lowast= 1 minus (119861

3119910 + 1198614119909) (1198770minus 1

R minus 1)


1119910 + 1198612119909) (1198770minus 1

R minus 1)

(71)



119889119891119890=


119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast







119869dfe =

((((((((((((((((((((((

(


1198890

0 minus1205721198990 120579

1120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 0 minus120572119902

1205791120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 120573119899

0 minus120573119899

0 0 0 0 0 0 0 0

0 1205731199021198861120573119902

0 minus120573119902

0 0 0 0 0 0 0

0 0 0 120574119899

0 minus1205741198990 0 0 0 0 0

0 0 0 120574119902

1198862120574119902

0 minus1205741199020 0 0 0 0

0 0 0 0 0 120575119899

0 minus1205751198990 0 0 0

0 0 0 0 0 12057511990211988641205751199021198863120575119902minus1205751199020 0 0

0 0 0 0 0 1 11988661198865

1198867minus1 0 0

0 0 0 0 0 0 0 12058311988912058311988911988680 minus120583

1198890

0 0 0 0 0 0 0 0 0 0 minus1198878minus1

))))))))))))))))))))))

)

(72)

where 1205791= 1 minus 120579



119875 (120577) =1003816100381610038161003816119869dfe minus 120577I

1003816100381610038161003816 = (120577 + 1)31198759(120577) = 0 (73)


1198759 (120577) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus120572119899minus 120577 0 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

0 minus120572119902minus 120577 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

120573119899

0 minus120573119899minus 120577 0 0 0 0 0 0

120573119902

1198861120573119902

0 minus120573119902minus 120577 0 0 0 0 0

0 0 120574119899

0 minus120574119899minus 120577 0 0 0 0

0 0 120574119902

1198862120574119902

0 minus120574119902minus 120577 0 0 0

0 0 0 0 120575119899

0 minus120575119899minus 120577 0 0

0 0 0 0 120575119902

1198864120575119902

1198863120575119902minus120575119902minus 120577 0

0 0 0 0 0 0 120583119889

1205831198891198868minus120583119889minus 120577

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(74)


119875 (120577) = (120577 + 1)3

9

sum

119896=0

119888119894120577119894 (75)

where1198889= 1

1198888= 120572119899+ 120572119902+ 120573119899+ 120573119902+ 120574119899+ 120574119902+ 120575119899+ 120575119902+ 120583119889

1198880= 1205731198991205731199021205741198991205741199021205751198991205751199021205831198891205721198991205721199021 minus

12057911198615

120572119899


1) 1198616

120572119902

= 120573119899120573119902120574119899120574119902120575119899120575119902120583119889120572119899120572119902(1 minus 119877

0)

(76)




0= 1 to





0 1198770




0= 1


0specifically


0which is fixedRmay






0is the


119899 119901119902 119888119899 119888119902 119894119899 119894119902 119903119903 119903119889 and ℎ The




F =

(((((((((((((((((

(

0

1205791120582119904

(1 minus 1205791) 120582119904

0

0

0

0

0

0

0

0

0

)))))))))))))))))

)

V =

((((((((((((((((((((((((((((((((

(


120572119899119904119899

120572119902119904119902

120573119899(minus119904119899+ 119901119899)

120573119902(minus119904119899minus 1198861119904119902+ 119901119902)

120574119899(minus119901119899+ 119888119899)

120574119902(minus119901119899minus 1198862119901119902+ 119888119902)

120575119899(minus119888119899+ 119894119899)



minus120583119889119894119899minus 1205831198891198868119894119902+ 120583119889119903119889

minus1 + ℎ + 1198878119903119889

))))))))))))))))))))))))))))))))

)

(77)


119894119895=

120597F119894120597119909119895and (119881)

119894119895= 120597V



119901119902= 119888119899= 119888119902= 119894119899= 119894119902= 119903119903= 119903119889= 0 The basic


1198770= 984858 (119865119881

minus1) =

1

120572119899120572119902

[1205721199021205791(1

+ (1 + 1198863+ (1 + 119886

2) 1198864) 119886119889

+ 120585119902(11988621198864+ 1198863+ 1198864+ 1) + 119886

2120591119902+ 120585119899+ 120588119899+ 120588119902

+ 120591119899+ 120591119902) minus 120572

119899(1205791minus 1) (119886

1120588119902

+ 1198861[11988621198864(1198868119886119889+ 120585119902) + 1198862120591119902])]

=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

(78)






119899(which








120583


120583 + rate120583

asymp large (79)


119899 1120573119902



119901119899= 119888119899= 119894119899= 119904119899

119901119902= 119904119899+ 1198861119904119902

119888119902= 1198601119904119899+ 1198602119904119902

119894119902= 1198603119904119899+ 1198604119904119902

119903119889= 1198607119904119899+ 1198608119904119902

(80)


119889119905= 1 minus 120582119904 + (120572


119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (82)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (83)

119889119903119903

119889119905= 1198605119904119899+ 1198606119904119902minus 119903119903 (84)

119889ℎ

119889119905= 1 minus ℎ minus 119861

1119904119899minus 1198612119904119902 (85)


120582119904 = (120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)) 119904 = (119861

5(119904119899

ℎ)

+ 1198616(

119904119902

ℎ)) 119904

(86)





119899 119904lowast

119902 119903lowast


119864dfe = (1 0 0 0 1)

119864119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast)

(87)


119904lowast

119902=1198617

1198618

119904lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902

119903lowast

119903= 1198605119904lowast

119899+ 1198606119904lowast

119902

119904lowast= 1 minus 119861

3119904lowast

119899minus 1198614119904lowast

119902


1119904lowast

119899minus 1198612119904lowast

119902

(88)


3119910 + 119861

4119909 and

1198611119910 + 1198612119909 = 1 we get

119904lowast= (119911 minus 1

R minus 1)

119904lowast

119899= 119910(

1198770minus 1

R minus 1)

119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

ℎlowast= ((119911 minus 1) 119877

0

R minus 1)

(89)


119899 119904lowast

119902 119903lowast




119869 (xlowast) =((

(


4(xlowast) + 120572


119902minus 11986140 minus119862

6(xlowast)

12057911198623(xlowast) 120579


11989912057911198625(xlowast) 0 120579

11198626(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) 120579


1199020 12057911198626(xlowast)

0 1198605

1198606

minus1 0

0 minus1198611

minus1198612

0 minus1

))

)

(90)

where 1205791= 1 minus 120579

1and

1198623(xlowast) = 119861

5

119904lowast

119899

ℎlowast+ 1198616

119904lowast

119902

ℎlowast=120572119899119910 (1198770minus 1)

1205791(119911 minus 1)

1198624(xlowast) = 119861

5

119904lowast

ℎlowast=1198615

1198770

1198625(xlowast) = 119861

6

119904lowast

ℎlowast=1198616

1198770

1198626(xlowast) = minus(119861

5


119899

ℎlowast2+ 1198616


119902

ℎlowast2) = minus

120572119899119910 (1198770minus 1)

1205791 (119911 minus 1) 1198770

(91)


1198755(120577 xlowast)

= (120577 + 1)2(1205773+ 1198762(xlowast) 1205772 + 119876

1(xlowast) 120577 + 119876

0(xlowast))

= 0

(92)

where

1198762(xlowast) = 120572

119899+ 120572119902+ 1198623(xlowast) minus 119862

4(xlowast) 120579


1

+ 1

1198761(xlowast) = 120572

119902


1199021198624(xlowast) + 119862

4(xlowast))

+ 11986121198626(xlowast) 120579

1+ 1198623(xlowast) (120572

1199021205791+ 11986131205791+ 11986141205791)

+ 120572119899(120572119902+ 12057911198623(xlowast) minus 119862

5(xlowast) 120579

1+ 1) minus 119862

5(xlowast)

sdot 1205791

1198760(xlowast) = 120572

1199021205791(11986111198626(xlowast) + 119861


4(xlowast))

+ 120572119899(1205791(11986121198626(xlowast) + 119861


5(xlowast)) + 120572

119902)

(93)




119902=

119903lowast




1198616 1198626(xdfe) = 0 and (92) becomes

1198755(120577 xdfe) = (120577 + 1)

3(1205772+ 119876dfe120577 + 119877dfe) = 0 (94)

where


= 120572119899(1 minus 119877

0) + 120572119902+ (1 minus 120579

1) 1198616(

120572119899minus 120572119902

120572119902

)

119877dfe = 120572119902 (120572119899 minus 11986151205791) + 1205721198991198616 (1205791 minus 1)

= 120572119902120572119899(1 minus 119877

0)

(95)


radic1198762







=


119899 119904lowast

119902 119903lowast


and we have

1198755(120577 xlowast) = (120577 + 1)2 (1205773 + 119875x

119890119890

1205772+ 119876x

119890119890

120577 + 119877x119890119890

) = 0 (96)

where


119875x119890119890

=


12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)


119890119890will be the


119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)




119899= 0 and we


1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)





grouping 1198616




1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)





CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References





































119878 (0) = 1198780

119868119873(0) = 119868

0

119878119873 (0) = 119878119876 (0) = 119875119873 (0) = 0

119875119876(0) = 119862

119873(0) = 119862

119876(0) = 119868

119876(0) = 119877

119863(0) = 119877

119877(0)

= 119877119873(0) = 119863

119863(0) = 0

(24)





119889119867119871

119889119905= Π minus 120583119867

119871minus 120575119873119868119873minus 120575119876119868119876 (25)

119889119867


119863 (26)



119863 119877119873 and



119871minus 119887119877119863







119873 119878119876



1= 119878 119909

2= 119878119873 119909

11= 119877119863 Let f(x) =

(1198911(x) 1198912(x) 119891





119878119873 and 119891




dx119889119905= f (x) x (0) = x

0 (27)





119868lowast and 119877







10038171003817100381710038171003817f1015840 (z y minus x)10038171003817100381710038171003817infin =

1003817100381710038171003817100381710038171003817100381710038171003817

11

sum

119896=1


1003817100381710038171003817100381710038171003817100381710038171003817infin

le

1003817100381710038171003817100381710038171003817100381710038171003817

11

sum

119896=1

nabla119891119896 (z)1003817100381710038171003817100381710038171003817100381710038171003817infin

1003817100381710038171003817y minus x1003817100381710038171003817infin

(29)


+ It is now





11

sum

119896=1

nabla119891119896 (z)1003817100381710038171003817100381710038171003817100381710038171003817infin



1003817100381710038171003817infinle 119872

1003817100381710038171003817y minus x1003817100381710038171003817infin (31)









1) = 0 1198781015840(119905

1) lt 0


1 119878(119905) gt 0 and 119878

119873(119905) gt 0 119878

119876(119905) gt 0

119875119873(119905) gt 0 119875

119876(119905) gt 0 119862

119873(119905) gt 0 119862

119876(119905) gt 0 119868

119873(119905) gt 0

119868119876(119905) gt 0 119877

119877(119905) gt 0 and 119877

119863(119905) gt 0 So at the point 119905 = 119905

1


119889119878

119889119905

10038161003816100381610038161003816100381610038161003816119905=1199051

= Π minus 120582119878 (1199051) + (1 minus 120579

3) 120573119873119875119873(1199051)

+ (1 minus 1205792) 120572119873119878119873(1199051) + (1 minus 120579

6) 120572119876119878119876(1199051)

+ (1 minus 1205797) 120573119876119875119876(1199051) minus 120583119878 (119905

1)

= Π + (1 minus 1205793) 120573119873119875119873(1199051)

+ (1 minus 1205792) 120572119873119878119873(1199051) + (1 minus 120579

6) 120572119876119878119876(1199051)

+ (1 minus 1205797) 120573119876119875119876(1199051) gt 0

(32)


1) lt



+ the solution will







Ω119867119871

= 119867119871(119905) 0 le 119867

119871(119905) le

Π

120583 sub R

11

+ (33)


119871is bounded



119889119867119871

119889119905= Π minus 120583119867


119871997904rArr

119867119871(119905) le

Π

120583+ (119867119871(0) minus

Π

120583) 119890minus120583119905

(34)


119871(119905)


119871(0) lt Π then119867



119867119871(119905) le lim119905rarrinfin

sup(Π120583+ (119867119871(0) minus

Π

120583) 119890minus120583119905) (35)





119868lowast and 119877


119871




119904 =119878

1198780

119904119899=119878119873

1198780

119873

119904119902=119878119876

1198780

119876

119901119899=119875119873

1198750

119873

119901119902=119875119876

1198750

119876

119888119899=119862119873

1198620

119873

119888119902=119862119876

1198620

119876

ℎ =119867

1198670

119894119899=119868119873

1198680

119873

119894119902=119868119876

1198680

119876

119903119903=119877119877

1198770

119877

119903119889=119877119863

1198770

119863

119903119899=119877119873

1198770

119873

119889119863=119863119863

1198630

119863

119905lowast=119905

1198790

ℎ119897=119867119871

1198670

119871

(36)

where

1198780=Π

120583

1198780

119873= 1198780

119876= 1198780

1198750

119873=12057921198861205721198731198780

119873

120573119873+ 120583

1198750

119876=12057921198871205721198731198780

119873

120573119876+ 120583

1198620

119873=12057931198861205731198731198750

119873

119903119864119873+ 120574119873+ 120583

1198620

119876=12057931198871205731198731198750

119873

119903119864119876+ 120574119876+ 120583

1198680

119873=

12057941205741198731198620

119873

119903119871119873+ 120575119873+ 120583

1198680

119876=(1 minus 120579

4) 1205741198731198620

119873

119903119871119876+ 120575119876+ 120583

1198770

119877= 1199031198641198731198620

1198731198790

1198770

119863=1205751198731198680

N119887

1198770

119873= 12058311987901198670

119871

1198630

119863= 11988711987901198770

119863

1198670= 1198670

119871=Π

120583= 1198780

1198790=1

120583

(37)


120588119899=12057931205881198731198750

119873

1205831198670

120588119902=

12057971205881198761198750

119876

1205831198670

120591119899=1205911198731198620

119873

1205831198670

120591119902=

1205911198761198620

119876

1205831198670

120585119899=1205851198731198680

119873

1205831198670

120585119902=

1205851198761198680

119876

1205831198670

119886119889=1198861198631198770

119863

1205831198670

120572119899= (120572119873+ 120583)119879

0

120572119902= (120572119876+ 120583)119879

0

120573119899= (120573119873+ 120583) 119879

0

120573119902= (120573119876+ 120583)119879

0


120583119889= 1198871198790

1198871=(1 minus 120579

3) 1205731198731198750

119873

1205831198780

1198872=(1 minus 120579

2) 1205721198731198780

119873

1205831198780

1198873=

(1 minus 1205796) 1205721198761198780

119876

1205831198780

1198874=

(1 minus 1205797) 1205731198761198750

119876

1205831198780

1198875=1205751198731198680

119873

1205831198670

1198876=

1205751198761198680

119876

1205831198670

1198878=(119887 minus 120583) 119877

0

119863

1205831198670

1198861=

12057961205721198761198780

119876

12057921198871205721198731198780

119873

1198862=

12057971205731198761198750

119876

12057931198871205731198731198750

119873

1198863=

1205901198731198680

119873

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198864=

1205741198761198620

119876

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198865=1199031198711198731198680

119873

1199031198641198731198620

119873

1198866=

1199031198641198761198620

119876

1199031198641198731198620

119873

1198867=

1199031198711198761198680

119876

1199031198641198731198620

119873

1198868=

1205751198761198680

119876

1205751198731198680

119873

120574119899= (119903119864119873+ 120574119873+ 120583) 119879

0

120574119902= (119903119864119876+ 120574119876+ 120583) 119879

0

120575119902= (119903119871119876+ 120575119876+ 120583) 119879

0

120575119899= (119903119871119873+ 120575119873+ 120583) 119879

0

(38)


120582 = 120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)

(39)


119889119904

119889119905= 1 minus 120582119904 + 119887

1119901119899+ 1198872119904119899+ 1198873119904119902+ 1198874119901119902minus 119904 (40)

119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (41)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (42)

119889119901119899

119889119905= 120573119899(119904119899minus 119901119899) (43)

119889119901119902

119889119905= 120573119902(119904119899+ 1198861119904119902minus 119901119902) (44)

119889119888119899

119889119905= 120574119899(119901119899minus 119888119899) (45)

119889119888119902

119889119905= 120574119902(119901119899+ 1198862119901119902minus 119888119902) (46)

119889119894119899

119889119905= 120575119899(119888119899minus 119894119899) (47)

119889119894119902

119889119905= 120575119902(119888119899+ 1198863119894119899+ 1198864119888119902minus 119894119902) (48)

119889119903119903

119889119905= 119888119899+ 1198865119894119899+ 1198866119888119902+ 1198867119894119902minus 119903119903 (49)

119889119903119889

119889119905= 120583119889(119894119899+ 1198868119894119902minus 119903119889) (50)

119889119903119899

119889119905= ℎ119897 (51)

119889119889119863

119889119905= 119889119863 (52)


119889ℎ119897

119889119905= 1 minus ℎ

119897minus 1198875119894119899minus 1198876119894119902 (53)

119889ℎ

119889119905= 1 minus ℎ minus 119887

8119903119889 (54)




120588119899=12057931205792119886120588119873120572119873

(120573119873+ 120583) 120583

120588119902=12057971205792119887120588119876120572119873

(120573119876+ 120583) 120583

120591119899=

12057931198861205792119886120591119873120572119873120573119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) 120583

120591119902=

12057931198871205792119886120591119876120572119873120573119873

(120573119873+ 120583) (119903

119864119876+ 120574119876+ 120583) 120583

120585119899=

120579311988612057921198861205794120585119873120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 120583

120585119902=

12057931198861205792119886(1 minus 120579

4) 120585119876120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119876+ 120575119876+ 120583) 120583

119886119889=

120579211988612057931198861205794120575119873120574119873120573119873120572119873119886119863

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 119887120583

1198871=(1 minus 120579

3) 1205792119886120573119873120572119873

(120573119873+ 120583) 120583

1198872=(1 minus 120579

2) 120572119873

120583

1198873=(1 minus 120579

6) 120572119876

120583

1198874=(1 minus 120579

7) 1205792119887120573119876120572119873

(120573119876+ 120583) 120583

1198875=

120579412057921198861205793119886120575119873120574119873120573119873120572119873

(119903119871119873+ 120575119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198876=

(1 minus 1205794) 12057921198861205793119886120575119876120574119873120573119873120572119873

(119903119871119876+ 120575119876+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198878=

(119887 minus 120583) 120579412057921198861205793119886120572119873120573119873120574119873120575119873

119887120583 (120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583)

120575119899=119903119871119873+ 120575119873+ 120583

120583

120572119899=120572119873+ 120583

120583

120572119902=120572119876+ 120583

120583

120573119899=120573119873+ 120583

120583

120573119902=120573119876+ 120583

120583

120574119899=119903119864119873+ 120574119873+ 120583

120583

120574119902=119903119864119876+ 120574119876+ 120583

120583

120575119902=119903119871119876+ 120575119876+ 120583

120583

1198861=1205796120572119876

1205792119887120572119873

1198862=12057971205792119887120573119876(120573119873+ 120583)

12057921198861205793119887(120573119876+ 120583) 120573

119873

1198863=

1205794120590119873

(1 minus 1205794) (119903119871119873+ 120575119873+ 120583)

1198864=

1205793119887120574119876(119903119864119873+ 120574119873+ 120583)

(1 minus 1205794) 1205793119886120574119873(119903119864119876+ 120574119876+ 120583)

1198865=

1199031198711198731205794120574119873

119903119864119873(119903119871119873+ 120575119873+ 120583)

120583119889=119887

120583

1198866=1205793119887119903119864119876(119903119864119873+ 120574119873+ 120583)

1205793119886119903119864119873(119903119864119876+ 120574119876+ 120583)

1198867=119903119871119876(1 minus 120579

4) 120574119873

119903119864119873(119903119871119876+ 120575119876+ 120583)

1198868=(1 minus 120579

4) 120575119876(119903119871119873+ 120575119873+ 120583)

1205794120575119873(119903119871119876+ 120575119876+ 120583)

(55)


119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast



119904lowast

119899= 119901lowast

119899= 119888lowast

119899= 119894lowast

119899(56)



119899and

119904lowast


119901lowast

119902(119901lowast

119899 119904lowast

119902) = 119901lowast

119899+ 1198861119904lowast

119902

119888lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

1119901lowast

119899+ 1198602119904lowast

119902

119894lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

3119901lowast

119899+ 1198604119904lowast

119902

119903lowast

119903(119901lowast

119899 119904lowast

119902) = 119860

5119901lowast

119899+ 1198606119904lowast

119902

119903lowast

119889(119901lowast

119899 119904lowast

119902) = 119860

7119901lowast

119899+ 1198608119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902

ℎlowast

119897(119901lowast

119899 119904lowast

119902) = 1 minus (119887

5+ 11988761198603) 119901lowast

119899minus 11988761198604119904lowast

119902

(57)

where

1198601= 1 + 119886

2

1198602= 11988611198862

1198603= 1 + 119886

3+ 11988641198601

1198604= 11988641198602

1198605= 1 + 119886

5+ 11988661198601+ 11988671198603

1198606= 11988661198602+ 11988671198604

1198607= 1 + 119886

81198603

1198608= 11988681198604

1198611= 11988781198607

1198612= 11988781198608


1198614= 120572119902minus 1198873minus 11988741198861

(58)




3and 119861


1198613= 1

+120572119873

120583(1205731198731205792119886(1205793minus 1)

120573119873+ 120583

+1205731198761205792119887(1205797minus 1)

120573119876+ 120583

+ 1205792)

gt 0 since 1205792= 1205792119886+ 1205792119887

1198614=1205721198761205796(1205731198761205797+ 120583) + 120583 (120573

119876+ 120583)

120583 (120573119876+ 120583)

gt 0

(59)


4gt 0





1205791(1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119899119901lowast

119899(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(60)

(1 minus 1205791) (1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119902119904lowast

119902(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(61)

where

1198615= 120588119899+ 120588119902+ 120591119899+ 1205911199021198601+ 120585119899+ 1205851199021198603+ 1198861198891198607

1198616= 1205881199021198861+ 1205911199021198602+ 1205851199021198604+ 1198861198891198608

(62)



(1 minus 1198611119901lowast

119899minus 1198612119904lowast

119902) (120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast

1199021205791) = 0 (63)


119899minus 1198612119904lowast

119902= 0 or (ii)

120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast


system


119899minus 1198614119904lowast

119902= 0


119899minus 1198612119904lowast

119902= 0

(64)



119901lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902 (65)



119904lowast

119902= 0

or 119904lowast119902=1198617

1198618

=(1 minus 120579

1) 120572119899(1198770minus 1)

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612) (R minus 1)

= 119909 (1198770minus 1

R minus 1)

(66)

where

1198617= 120572119902120572119899(1 minus 120579

1) ((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

) minus 1)

= 120572119902120572119899(1 minus 120579

1) (1198770minus 1)

1198618= 120572119902[(12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614)

minus (12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612)] = 120572

119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612)((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (

12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614

12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612

) minus 1) = 120572119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612) (1198770119911 minus 1)

(67)

with

119909 =(1 minus 120579

1) 120572119899

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612)

119910 =

1205791120572119902119909

(1 minus 1205791) 120572119899

119911 =

(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

(68)

1198770=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

R =1198770(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

= 1199111198770

(69)


1198612= 11988781198868119886411988611198862=1205796120572119876

120583

1205797120573119876

(120573119876+ 120583)

((1 minus120583

119887)

sdot120575119876

(119903119871119876+ 120575119876+ 120583)

120574119876

(119903119864119876+ 120574119876+ 120583)) lt

1205796120572119876

120583

sdot1205797120573119876

(120573119876+ 120583)

lt1205796120572119876

120583

(1205797120573119876+ 120583)

(120573119876+ 120583)

+ 1 = 1198614

(70)



3lt 1198611 we will


1120572119902(1minus120579

1)120572119899)(1198611minus1198613)


system as follows




0gt 1 and the


0




119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119894lowast

119899= 119888lowast

119899= 119904lowast

119899= 119901lowast

119899= 119910(

1198770minus 1

R minus 1)

119901lowast

119902= (119910 + 119886

1119909) (1198770minus 1

R minus 1)

119888lowast

119902= (1198601119910 + 119860

2119909) (1198770minus 1

R minus 1)

119894lowast

119902= (1198603119910 + 119860

4119909) (1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

119903lowast

119889= (1198607119910 + 119860

8119909) (1198770minus 1

R minus 1)

119904lowast= 1 minus (119861

3119910 + 1198614119909) (1198770minus 1

R minus 1)


1119910 + 1198612119909) (1198770minus 1

R minus 1)

(71)



119889119891119890=


119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast







119869dfe =

((((((((((((((((((((((

(


1198890

0 minus1205721198990 120579

1120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 0 minus120572119902

1205791120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 120573119899

0 minus120573119899

0 0 0 0 0 0 0 0

0 1205731199021198861120573119902

0 minus120573119902

0 0 0 0 0 0 0

0 0 0 120574119899

0 minus1205741198990 0 0 0 0 0

0 0 0 120574119902

1198862120574119902

0 minus1205741199020 0 0 0 0

0 0 0 0 0 120575119899

0 minus1205751198990 0 0 0

0 0 0 0 0 12057511990211988641205751199021198863120575119902minus1205751199020 0 0

0 0 0 0 0 1 11988661198865

1198867minus1 0 0

0 0 0 0 0 0 0 12058311988912058311988911988680 minus120583

1198890

0 0 0 0 0 0 0 0 0 0 minus1198878minus1

))))))))))))))))))))))

)

(72)

where 1205791= 1 minus 120579



119875 (120577) =1003816100381610038161003816119869dfe minus 120577I

1003816100381610038161003816 = (120577 + 1)31198759(120577) = 0 (73)


1198759 (120577) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus120572119899minus 120577 0 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

0 minus120572119902minus 120577 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

120573119899

0 minus120573119899minus 120577 0 0 0 0 0 0

120573119902

1198861120573119902

0 minus120573119902minus 120577 0 0 0 0 0

0 0 120574119899

0 minus120574119899minus 120577 0 0 0 0

0 0 120574119902

1198862120574119902

0 minus120574119902minus 120577 0 0 0

0 0 0 0 120575119899

0 minus120575119899minus 120577 0 0

0 0 0 0 120575119902

1198864120575119902

1198863120575119902minus120575119902minus 120577 0

0 0 0 0 0 0 120583119889

1205831198891198868minus120583119889minus 120577

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(74)


119875 (120577) = (120577 + 1)3

9

sum

119896=0

119888119894120577119894 (75)

where1198889= 1

1198888= 120572119899+ 120572119902+ 120573119899+ 120573119902+ 120574119899+ 120574119902+ 120575119899+ 120575119902+ 120583119889

1198880= 1205731198991205731199021205741198991205741199021205751198991205751199021205831198891205721198991205721199021 minus

12057911198615

120572119899


1) 1198616

120572119902

= 120573119899120573119902120574119899120574119902120575119899120575119902120583119889120572119899120572119902(1 minus 119877

0)

(76)




0= 1 to





0 1198770




0= 1


0specifically


0which is fixedRmay






0is the


119899 119901119902 119888119899 119888119902 119894119899 119894119902 119903119903 119903119889 and ℎ The




F =

(((((((((((((((((

(

0

1205791120582119904

(1 minus 1205791) 120582119904

0

0

0

0

0

0

0

0

0

)))))))))))))))))

)

V =

((((((((((((((((((((((((((((((((

(


120572119899119904119899

120572119902119904119902

120573119899(minus119904119899+ 119901119899)

120573119902(minus119904119899minus 1198861119904119902+ 119901119902)

120574119899(minus119901119899+ 119888119899)

120574119902(minus119901119899minus 1198862119901119902+ 119888119902)

120575119899(minus119888119899+ 119894119899)



minus120583119889119894119899minus 1205831198891198868119894119902+ 120583119889119903119889

minus1 + ℎ + 1198878119903119889

))))))))))))))))))))))))))))))))

)

(77)


119894119895=

120597F119894120597119909119895and (119881)

119894119895= 120597V



119901119902= 119888119899= 119888119902= 119894119899= 119894119902= 119903119903= 119903119889= 0 The basic


1198770= 984858 (119865119881

minus1) =

1

120572119899120572119902

[1205721199021205791(1

+ (1 + 1198863+ (1 + 119886

2) 1198864) 119886119889

+ 120585119902(11988621198864+ 1198863+ 1198864+ 1) + 119886

2120591119902+ 120585119899+ 120588119899+ 120588119902

+ 120591119899+ 120591119902) minus 120572

119899(1205791minus 1) (119886

1120588119902

+ 1198861[11988621198864(1198868119886119889+ 120585119902) + 1198862120591119902])]

=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

(78)






119899(which








120583


120583 + rate120583

asymp large (79)


119899 1120573119902



119901119899= 119888119899= 119894119899= 119904119899

119901119902= 119904119899+ 1198861119904119902

119888119902= 1198601119904119899+ 1198602119904119902

119894119902= 1198603119904119899+ 1198604119904119902

119903119889= 1198607119904119899+ 1198608119904119902

(80)


119889119905= 1 minus 120582119904 + (120572


119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (82)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (83)

119889119903119903

119889119905= 1198605119904119899+ 1198606119904119902minus 119903119903 (84)

119889ℎ

119889119905= 1 minus ℎ minus 119861

1119904119899minus 1198612119904119902 (85)


120582119904 = (120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)) 119904 = (119861

5(119904119899

ℎ)

+ 1198616(

119904119902

ℎ)) 119904

(86)





119899 119904lowast

119902 119903lowast


119864dfe = (1 0 0 0 1)

119864119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast)

(87)


119904lowast

119902=1198617

1198618

119904lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902

119903lowast

119903= 1198605119904lowast

119899+ 1198606119904lowast

119902

119904lowast= 1 minus 119861

3119904lowast

119899minus 1198614119904lowast

119902


1119904lowast

119899minus 1198612119904lowast

119902

(88)


3119910 + 119861

4119909 and

1198611119910 + 1198612119909 = 1 we get

119904lowast= (119911 minus 1

R minus 1)

119904lowast

119899= 119910(

1198770minus 1

R minus 1)

119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

ℎlowast= ((119911 minus 1) 119877

0

R minus 1)

(89)


119899 119904lowast

119902 119903lowast




119869 (xlowast) =((

(


4(xlowast) + 120572


119902minus 11986140 minus119862

6(xlowast)

12057911198623(xlowast) 120579


11989912057911198625(xlowast) 0 120579

11198626(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) 120579


1199020 12057911198626(xlowast)

0 1198605

1198606

minus1 0

0 minus1198611

minus1198612

0 minus1

))

)

(90)

where 1205791= 1 minus 120579

1and

1198623(xlowast) = 119861

5

119904lowast

119899

ℎlowast+ 1198616

119904lowast

119902

ℎlowast=120572119899119910 (1198770minus 1)

1205791(119911 minus 1)

1198624(xlowast) = 119861

5

119904lowast

ℎlowast=1198615

1198770

1198625(xlowast) = 119861

6

119904lowast

ℎlowast=1198616

1198770

1198626(xlowast) = minus(119861

5


119899

ℎlowast2+ 1198616


119902

ℎlowast2) = minus

120572119899119910 (1198770minus 1)

1205791 (119911 minus 1) 1198770

(91)


1198755(120577 xlowast)

= (120577 + 1)2(1205773+ 1198762(xlowast) 1205772 + 119876

1(xlowast) 120577 + 119876

0(xlowast))

= 0

(92)

where

1198762(xlowast) = 120572

119899+ 120572119902+ 1198623(xlowast) minus 119862

4(xlowast) 120579


1

+ 1

1198761(xlowast) = 120572

119902


1199021198624(xlowast) + 119862

4(xlowast))

+ 11986121198626(xlowast) 120579

1+ 1198623(xlowast) (120572

1199021205791+ 11986131205791+ 11986141205791)

+ 120572119899(120572119902+ 12057911198623(xlowast) minus 119862

5(xlowast) 120579

1+ 1) minus 119862

5(xlowast)

sdot 1205791

1198760(xlowast) = 120572

1199021205791(11986111198626(xlowast) + 119861


4(xlowast))

+ 120572119899(1205791(11986121198626(xlowast) + 119861


5(xlowast)) + 120572

119902)

(93)




119902=

119903lowast




1198616 1198626(xdfe) = 0 and (92) becomes

1198755(120577 xdfe) = (120577 + 1)

3(1205772+ 119876dfe120577 + 119877dfe) = 0 (94)

where


= 120572119899(1 minus 119877

0) + 120572119902+ (1 minus 120579

1) 1198616(

120572119899minus 120572119902

120572119902

)

119877dfe = 120572119902 (120572119899 minus 11986151205791) + 1205721198991198616 (1205791 minus 1)

= 120572119902120572119899(1 minus 119877

0)

(95)


radic1198762







=


119899 119904lowast

119902 119903lowast


and we have

1198755(120577 xlowast) = (120577 + 1)2 (1205773 + 119875x

119890119890

1205772+ 119876x

119890119890

120577 + 119877x119890119890

) = 0 (96)

where


119875x119890119890

=


12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)


119890119890will be the


119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)




119899= 0 and we


1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)





grouping 1198616




1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)





CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References





































10038171003817100381710038171003817f1015840 (z y minus x)10038171003817100381710038171003817infin =

1003817100381710038171003817100381710038171003817100381710038171003817

11

sum

119896=1


1003817100381710038171003817100381710038171003817100381710038171003817infin

le

1003817100381710038171003817100381710038171003817100381710038171003817

11

sum

119896=1

nabla119891119896 (z)1003817100381710038171003817100381710038171003817100381710038171003817infin

1003817100381710038171003817y minus x1003817100381710038171003817infin

(29)


+ It is now





11

sum

119896=1

nabla119891119896 (z)1003817100381710038171003817100381710038171003817100381710038171003817infin



1003817100381710038171003817infinle 119872

1003817100381710038171003817y minus x1003817100381710038171003817infin (31)









1) = 0 1198781015840(119905

1) lt 0


1 119878(119905) gt 0 and 119878

119873(119905) gt 0 119878

119876(119905) gt 0

119875119873(119905) gt 0 119875

119876(119905) gt 0 119862

119873(119905) gt 0 119862

119876(119905) gt 0 119868

119873(119905) gt 0

119868119876(119905) gt 0 119877

119877(119905) gt 0 and 119877

119863(119905) gt 0 So at the point 119905 = 119905

1


119889119878

119889119905

10038161003816100381610038161003816100381610038161003816119905=1199051

= Π minus 120582119878 (1199051) + (1 minus 120579

3) 120573119873119875119873(1199051)

+ (1 minus 1205792) 120572119873119878119873(1199051) + (1 minus 120579

6) 120572119876119878119876(1199051)

+ (1 minus 1205797) 120573119876119875119876(1199051) minus 120583119878 (119905

1)

= Π + (1 minus 1205793) 120573119873119875119873(1199051)

+ (1 minus 1205792) 120572119873119878119873(1199051) + (1 minus 120579

6) 120572119876119878119876(1199051)

+ (1 minus 1205797) 120573119876119875119876(1199051) gt 0

(32)


1) lt



+ the solution will







Ω119867119871

= 119867119871(119905) 0 le 119867

119871(119905) le

Π

120583 sub R

11

+ (33)


119871is bounded



119889119867119871

119889119905= Π minus 120583119867


119871997904rArr

119867119871(119905) le

Π

120583+ (119867119871(0) minus

Π

120583) 119890minus120583119905

(34)


119871(119905)


119871(0) lt Π then119867



119867119871(119905) le lim119905rarrinfin

sup(Π120583+ (119867119871(0) minus

Π

120583) 119890minus120583119905) (35)





119868lowast and 119877


119871




119904 =119878

1198780

119904119899=119878119873

1198780

119873

119904119902=119878119876

1198780

119876

119901119899=119875119873

1198750

119873

119901119902=119875119876

1198750

119876

119888119899=119862119873

1198620

119873

119888119902=119862119876

1198620

119876

ℎ =119867

1198670

119894119899=119868119873

1198680

119873

119894119902=119868119876

1198680

119876

119903119903=119877119877

1198770

119877

119903119889=119877119863

1198770

119863

119903119899=119877119873

1198770

119873

119889119863=119863119863

1198630

119863

119905lowast=119905

1198790

ℎ119897=119867119871

1198670

119871

(36)

where

1198780=Π

120583

1198780

119873= 1198780

119876= 1198780

1198750

119873=12057921198861205721198731198780

119873

120573119873+ 120583

1198750

119876=12057921198871205721198731198780

119873

120573119876+ 120583

1198620

119873=12057931198861205731198731198750

119873

119903119864119873+ 120574119873+ 120583

1198620

119876=12057931198871205731198731198750

119873

119903119864119876+ 120574119876+ 120583

1198680

119873=

12057941205741198731198620

119873

119903119871119873+ 120575119873+ 120583

1198680

119876=(1 minus 120579

4) 1205741198731198620

119873

119903119871119876+ 120575119876+ 120583

1198770

119877= 1199031198641198731198620

1198731198790

1198770

119863=1205751198731198680

N119887

1198770

119873= 12058311987901198670

119871

1198630

119863= 11988711987901198770

119863

1198670= 1198670

119871=Π

120583= 1198780

1198790=1

120583

(37)


120588119899=12057931205881198731198750

119873

1205831198670

120588119902=

12057971205881198761198750

119876

1205831198670

120591119899=1205911198731198620

119873

1205831198670

120591119902=

1205911198761198620

119876

1205831198670

120585119899=1205851198731198680

119873

1205831198670

120585119902=

1205851198761198680

119876

1205831198670

119886119889=1198861198631198770

119863

1205831198670

120572119899= (120572119873+ 120583)119879

0

120572119902= (120572119876+ 120583)119879

0

120573119899= (120573119873+ 120583) 119879

0

120573119902= (120573119876+ 120583)119879

0


120583119889= 1198871198790

1198871=(1 minus 120579

3) 1205731198731198750

119873

1205831198780

1198872=(1 minus 120579

2) 1205721198731198780

119873

1205831198780

1198873=

(1 minus 1205796) 1205721198761198780

119876

1205831198780

1198874=

(1 minus 1205797) 1205731198761198750

119876

1205831198780

1198875=1205751198731198680

119873

1205831198670

1198876=

1205751198761198680

119876

1205831198670

1198878=(119887 minus 120583) 119877

0

119863

1205831198670

1198861=

12057961205721198761198780

119876

12057921198871205721198731198780

119873

1198862=

12057971205731198761198750

119876

12057931198871205731198731198750

119873

1198863=

1205901198731198680

119873

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198864=

1205741198761198620

119876

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198865=1199031198711198731198680

119873

1199031198641198731198620

119873

1198866=

1199031198641198761198620

119876

1199031198641198731198620

119873

1198867=

1199031198711198761198680

119876

1199031198641198731198620

119873

1198868=

1205751198761198680

119876

1205751198731198680

119873

120574119899= (119903119864119873+ 120574119873+ 120583) 119879

0

120574119902= (119903119864119876+ 120574119876+ 120583) 119879

0

120575119902= (119903119871119876+ 120575119876+ 120583) 119879

0

120575119899= (119903119871119873+ 120575119873+ 120583) 119879

0

(38)


120582 = 120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)

(39)


119889119904

119889119905= 1 minus 120582119904 + 119887

1119901119899+ 1198872119904119899+ 1198873119904119902+ 1198874119901119902minus 119904 (40)

119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (41)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (42)

119889119901119899

119889119905= 120573119899(119904119899minus 119901119899) (43)

119889119901119902

119889119905= 120573119902(119904119899+ 1198861119904119902minus 119901119902) (44)

119889119888119899

119889119905= 120574119899(119901119899minus 119888119899) (45)

119889119888119902

119889119905= 120574119902(119901119899+ 1198862119901119902minus 119888119902) (46)

119889119894119899

119889119905= 120575119899(119888119899minus 119894119899) (47)

119889119894119902

119889119905= 120575119902(119888119899+ 1198863119894119899+ 1198864119888119902minus 119894119902) (48)

119889119903119903

119889119905= 119888119899+ 1198865119894119899+ 1198866119888119902+ 1198867119894119902minus 119903119903 (49)

119889119903119889

119889119905= 120583119889(119894119899+ 1198868119894119902minus 119903119889) (50)

119889119903119899

119889119905= ℎ119897 (51)

119889119889119863

119889119905= 119889119863 (52)


119889ℎ119897

119889119905= 1 minus ℎ

119897minus 1198875119894119899minus 1198876119894119902 (53)

119889ℎ

119889119905= 1 minus ℎ minus 119887

8119903119889 (54)




120588119899=12057931205792119886120588119873120572119873

(120573119873+ 120583) 120583

120588119902=12057971205792119887120588119876120572119873

(120573119876+ 120583) 120583

120591119899=

12057931198861205792119886120591119873120572119873120573119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) 120583

120591119902=

12057931198871205792119886120591119876120572119873120573119873

(120573119873+ 120583) (119903

119864119876+ 120574119876+ 120583) 120583

120585119899=

120579311988612057921198861205794120585119873120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 120583

120585119902=

12057931198861205792119886(1 minus 120579

4) 120585119876120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119876+ 120575119876+ 120583) 120583

119886119889=

120579211988612057931198861205794120575119873120574119873120573119873120572119873119886119863

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 119887120583

1198871=(1 minus 120579

3) 1205792119886120573119873120572119873

(120573119873+ 120583) 120583

1198872=(1 minus 120579

2) 120572119873

120583

1198873=(1 minus 120579

6) 120572119876

120583

1198874=(1 minus 120579

7) 1205792119887120573119876120572119873

(120573119876+ 120583) 120583

1198875=

120579412057921198861205793119886120575119873120574119873120573119873120572119873

(119903119871119873+ 120575119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198876=

(1 minus 1205794) 12057921198861205793119886120575119876120574119873120573119873120572119873

(119903119871119876+ 120575119876+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198878=

(119887 minus 120583) 120579412057921198861205793119886120572119873120573119873120574119873120575119873

119887120583 (120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583)

120575119899=119903119871119873+ 120575119873+ 120583

120583

120572119899=120572119873+ 120583

120583

120572119902=120572119876+ 120583

120583

120573119899=120573119873+ 120583

120583

120573119902=120573119876+ 120583

120583

120574119899=119903119864119873+ 120574119873+ 120583

120583

120574119902=119903119864119876+ 120574119876+ 120583

120583

120575119902=119903119871119876+ 120575119876+ 120583

120583

1198861=1205796120572119876

1205792119887120572119873

1198862=12057971205792119887120573119876(120573119873+ 120583)

12057921198861205793119887(120573119876+ 120583) 120573

119873

1198863=

1205794120590119873

(1 minus 1205794) (119903119871119873+ 120575119873+ 120583)

1198864=

1205793119887120574119876(119903119864119873+ 120574119873+ 120583)

(1 minus 1205794) 1205793119886120574119873(119903119864119876+ 120574119876+ 120583)

1198865=

1199031198711198731205794120574119873

119903119864119873(119903119871119873+ 120575119873+ 120583)

120583119889=119887

120583

1198866=1205793119887119903119864119876(119903119864119873+ 120574119873+ 120583)

1205793119886119903119864119873(119903119864119876+ 120574119876+ 120583)

1198867=119903119871119876(1 minus 120579

4) 120574119873

119903119864119873(119903119871119876+ 120575119876+ 120583)

1198868=(1 minus 120579

4) 120575119876(119903119871119873+ 120575119873+ 120583)

1205794120575119873(119903119871119876+ 120575119876+ 120583)

(55)


119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast



119904lowast

119899= 119901lowast

119899= 119888lowast

119899= 119894lowast

119899(56)



119899and

119904lowast


119901lowast

119902(119901lowast

119899 119904lowast

119902) = 119901lowast

119899+ 1198861119904lowast

119902

119888lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

1119901lowast

119899+ 1198602119904lowast

119902

119894lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

3119901lowast

119899+ 1198604119904lowast

119902

119903lowast

119903(119901lowast

119899 119904lowast

119902) = 119860

5119901lowast

119899+ 1198606119904lowast

119902

119903lowast

119889(119901lowast

119899 119904lowast

119902) = 119860

7119901lowast

119899+ 1198608119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902

ℎlowast

119897(119901lowast

119899 119904lowast

119902) = 1 minus (119887

5+ 11988761198603) 119901lowast

119899minus 11988761198604119904lowast

119902

(57)

where

1198601= 1 + 119886

2

1198602= 11988611198862

1198603= 1 + 119886

3+ 11988641198601

1198604= 11988641198602

1198605= 1 + 119886

5+ 11988661198601+ 11988671198603

1198606= 11988661198602+ 11988671198604

1198607= 1 + 119886

81198603

1198608= 11988681198604

1198611= 11988781198607

1198612= 11988781198608


1198614= 120572119902minus 1198873minus 11988741198861

(58)




3and 119861


1198613= 1

+120572119873

120583(1205731198731205792119886(1205793minus 1)

120573119873+ 120583

+1205731198761205792119887(1205797minus 1)

120573119876+ 120583

+ 1205792)

gt 0 since 1205792= 1205792119886+ 1205792119887

1198614=1205721198761205796(1205731198761205797+ 120583) + 120583 (120573

119876+ 120583)

120583 (120573119876+ 120583)

gt 0

(59)


4gt 0





1205791(1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119899119901lowast

119899(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(60)

(1 minus 1205791) (1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119902119904lowast

119902(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(61)

where

1198615= 120588119899+ 120588119902+ 120591119899+ 1205911199021198601+ 120585119899+ 1205851199021198603+ 1198861198891198607

1198616= 1205881199021198861+ 1205911199021198602+ 1205851199021198604+ 1198861198891198608

(62)



(1 minus 1198611119901lowast

119899minus 1198612119904lowast

119902) (120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast

1199021205791) = 0 (63)


119899minus 1198612119904lowast

119902= 0 or (ii)

120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast


system


119899minus 1198614119904lowast

119902= 0


119899minus 1198612119904lowast

119902= 0

(64)



119901lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902 (65)



119904lowast

119902= 0

or 119904lowast119902=1198617

1198618

=(1 minus 120579

1) 120572119899(1198770minus 1)

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612) (R minus 1)

= 119909 (1198770minus 1

R minus 1)

(66)

where

1198617= 120572119902120572119899(1 minus 120579

1) ((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

) minus 1)

= 120572119902120572119899(1 minus 120579

1) (1198770minus 1)

1198618= 120572119902[(12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614)

minus (12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612)] = 120572

119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612)((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (

12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614

12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612

) minus 1) = 120572119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612) (1198770119911 minus 1)

(67)

with

119909 =(1 minus 120579

1) 120572119899

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612)

119910 =

1205791120572119902119909

(1 minus 1205791) 120572119899

119911 =

(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

(68)

1198770=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

R =1198770(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

= 1199111198770

(69)


1198612= 11988781198868119886411988611198862=1205796120572119876

120583

1205797120573119876

(120573119876+ 120583)

((1 minus120583

119887)

sdot120575119876

(119903119871119876+ 120575119876+ 120583)

120574119876

(119903119864119876+ 120574119876+ 120583)) lt

1205796120572119876

120583

sdot1205797120573119876

(120573119876+ 120583)

lt1205796120572119876

120583

(1205797120573119876+ 120583)

(120573119876+ 120583)

+ 1 = 1198614

(70)



3lt 1198611 we will


1120572119902(1minus120579

1)120572119899)(1198611minus1198613)


system as follows




0gt 1 and the


0




119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119894lowast

119899= 119888lowast

119899= 119904lowast

119899= 119901lowast

119899= 119910(

1198770minus 1

R minus 1)

119901lowast

119902= (119910 + 119886

1119909) (1198770minus 1

R minus 1)

119888lowast

119902= (1198601119910 + 119860

2119909) (1198770minus 1

R minus 1)

119894lowast

119902= (1198603119910 + 119860

4119909) (1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

119903lowast

119889= (1198607119910 + 119860

8119909) (1198770minus 1

R minus 1)

119904lowast= 1 minus (119861

3119910 + 1198614119909) (1198770minus 1

R minus 1)


1119910 + 1198612119909) (1198770minus 1

R minus 1)

(71)



119889119891119890=


119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast







119869dfe =

((((((((((((((((((((((

(


1198890

0 minus1205721198990 120579

1120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 0 minus120572119902

1205791120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 120573119899

0 minus120573119899

0 0 0 0 0 0 0 0

0 1205731199021198861120573119902

0 minus120573119902

0 0 0 0 0 0 0

0 0 0 120574119899

0 minus1205741198990 0 0 0 0 0

0 0 0 120574119902

1198862120574119902

0 minus1205741199020 0 0 0 0

0 0 0 0 0 120575119899

0 minus1205751198990 0 0 0

0 0 0 0 0 12057511990211988641205751199021198863120575119902minus1205751199020 0 0

0 0 0 0 0 1 11988661198865

1198867minus1 0 0

0 0 0 0 0 0 0 12058311988912058311988911988680 minus120583

1198890

0 0 0 0 0 0 0 0 0 0 minus1198878minus1

))))))))))))))))))))))

)

(72)

where 1205791= 1 minus 120579



119875 (120577) =1003816100381610038161003816119869dfe minus 120577I

1003816100381610038161003816 = (120577 + 1)31198759(120577) = 0 (73)


1198759 (120577) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus120572119899minus 120577 0 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

0 minus120572119902minus 120577 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

120573119899

0 minus120573119899minus 120577 0 0 0 0 0 0

120573119902

1198861120573119902

0 minus120573119902minus 120577 0 0 0 0 0

0 0 120574119899

0 minus120574119899minus 120577 0 0 0 0

0 0 120574119902

1198862120574119902

0 minus120574119902minus 120577 0 0 0

0 0 0 0 120575119899

0 minus120575119899minus 120577 0 0

0 0 0 0 120575119902

1198864120575119902

1198863120575119902minus120575119902minus 120577 0

0 0 0 0 0 0 120583119889

1205831198891198868minus120583119889minus 120577

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(74)


119875 (120577) = (120577 + 1)3

9

sum

119896=0

119888119894120577119894 (75)

where1198889= 1

1198888= 120572119899+ 120572119902+ 120573119899+ 120573119902+ 120574119899+ 120574119902+ 120575119899+ 120575119902+ 120583119889

1198880= 1205731198991205731199021205741198991205741199021205751198991205751199021205831198891205721198991205721199021 minus

12057911198615

120572119899


1) 1198616

120572119902

= 120573119899120573119902120574119899120574119902120575119899120575119902120583119889120572119899120572119902(1 minus 119877

0)

(76)




0= 1 to





0 1198770




0= 1


0specifically


0which is fixedRmay






0is the


119899 119901119902 119888119899 119888119902 119894119899 119894119902 119903119903 119903119889 and ℎ The




F =

(((((((((((((((((

(

0

1205791120582119904

(1 minus 1205791) 120582119904

0

0

0

0

0

0

0

0

0

)))))))))))))))))

)

V =

((((((((((((((((((((((((((((((((

(


120572119899119904119899

120572119902119904119902

120573119899(minus119904119899+ 119901119899)

120573119902(minus119904119899minus 1198861119904119902+ 119901119902)

120574119899(minus119901119899+ 119888119899)

120574119902(minus119901119899minus 1198862119901119902+ 119888119902)

120575119899(minus119888119899+ 119894119899)



minus120583119889119894119899minus 1205831198891198868119894119902+ 120583119889119903119889

minus1 + ℎ + 1198878119903119889

))))))))))))))))))))))))))))))))

)

(77)


119894119895=

120597F119894120597119909119895and (119881)

119894119895= 120597V



119901119902= 119888119899= 119888119902= 119894119899= 119894119902= 119903119903= 119903119889= 0 The basic


1198770= 984858 (119865119881

minus1) =

1

120572119899120572119902

[1205721199021205791(1

+ (1 + 1198863+ (1 + 119886

2) 1198864) 119886119889

+ 120585119902(11988621198864+ 1198863+ 1198864+ 1) + 119886

2120591119902+ 120585119899+ 120588119899+ 120588119902

+ 120591119899+ 120591119902) minus 120572

119899(1205791minus 1) (119886

1120588119902

+ 1198861[11988621198864(1198868119886119889+ 120585119902) + 1198862120591119902])]

=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

(78)






119899(which








120583


120583 + rate120583

asymp large (79)


119899 1120573119902



119901119899= 119888119899= 119894119899= 119904119899

119901119902= 119904119899+ 1198861119904119902

119888119902= 1198601119904119899+ 1198602119904119902

119894119902= 1198603119904119899+ 1198604119904119902

119903119889= 1198607119904119899+ 1198608119904119902

(80)


119889119905= 1 minus 120582119904 + (120572


119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (82)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (83)

119889119903119903

119889119905= 1198605119904119899+ 1198606119904119902minus 119903119903 (84)

119889ℎ

119889119905= 1 minus ℎ minus 119861

1119904119899minus 1198612119904119902 (85)


120582119904 = (120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)) 119904 = (119861

5(119904119899

ℎ)

+ 1198616(

119904119902

ℎ)) 119904

(86)





119899 119904lowast

119902 119903lowast


119864dfe = (1 0 0 0 1)

119864119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast)

(87)


119904lowast

119902=1198617

1198618

119904lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902

119903lowast

119903= 1198605119904lowast

119899+ 1198606119904lowast

119902

119904lowast= 1 minus 119861

3119904lowast

119899minus 1198614119904lowast

119902


1119904lowast

119899minus 1198612119904lowast

119902

(88)


3119910 + 119861

4119909 and

1198611119910 + 1198612119909 = 1 we get

119904lowast= (119911 minus 1

R minus 1)

119904lowast

119899= 119910(

1198770minus 1

R minus 1)

119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

ℎlowast= ((119911 minus 1) 119877

0

R minus 1)

(89)


119899 119904lowast

119902 119903lowast




119869 (xlowast) =((

(


4(xlowast) + 120572


119902minus 11986140 minus119862

6(xlowast)

12057911198623(xlowast) 120579


11989912057911198625(xlowast) 0 120579

11198626(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) 120579


1199020 12057911198626(xlowast)

0 1198605

1198606

minus1 0

0 minus1198611

minus1198612

0 minus1

))

)

(90)

where 1205791= 1 minus 120579

1and

1198623(xlowast) = 119861

5

119904lowast

119899

ℎlowast+ 1198616

119904lowast

119902

ℎlowast=120572119899119910 (1198770minus 1)

1205791(119911 minus 1)

1198624(xlowast) = 119861

5

119904lowast

ℎlowast=1198615

1198770

1198625(xlowast) = 119861

6

119904lowast

ℎlowast=1198616

1198770

1198626(xlowast) = minus(119861

5


119899

ℎlowast2+ 1198616


119902

ℎlowast2) = minus

120572119899119910 (1198770minus 1)

1205791 (119911 minus 1) 1198770

(91)


1198755(120577 xlowast)

= (120577 + 1)2(1205773+ 1198762(xlowast) 1205772 + 119876

1(xlowast) 120577 + 119876

0(xlowast))

= 0

(92)

where

1198762(xlowast) = 120572

119899+ 120572119902+ 1198623(xlowast) minus 119862

4(xlowast) 120579


1

+ 1

1198761(xlowast) = 120572

119902


1199021198624(xlowast) + 119862

4(xlowast))

+ 11986121198626(xlowast) 120579

1+ 1198623(xlowast) (120572

1199021205791+ 11986131205791+ 11986141205791)

+ 120572119899(120572119902+ 12057911198623(xlowast) minus 119862

5(xlowast) 120579

1+ 1) minus 119862

5(xlowast)

sdot 1205791

1198760(xlowast) = 120572

1199021205791(11986111198626(xlowast) + 119861


4(xlowast))

+ 120572119899(1205791(11986121198626(xlowast) + 119861


5(xlowast)) + 120572

119902)

(93)




119902=

119903lowast




1198616 1198626(xdfe) = 0 and (92) becomes

1198755(120577 xdfe) = (120577 + 1)

3(1205772+ 119876dfe120577 + 119877dfe) = 0 (94)

where


= 120572119899(1 minus 119877

0) + 120572119902+ (1 minus 120579

1) 1198616(

120572119899minus 120572119902

120572119902

)

119877dfe = 120572119902 (120572119899 minus 11986151205791) + 1205721198991198616 (1205791 minus 1)

= 120572119902120572119899(1 minus 119877

0)

(95)


radic1198762







=


119899 119904lowast

119902 119903lowast


and we have

1198755(120577 xlowast) = (120577 + 1)2 (1205773 + 119875x

119890119890

1205772+ 119876x

119890119890

120577 + 119877x119890119890

) = 0 (96)

where


119875x119890119890

=


12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)


119890119890will be the


119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)




119899= 0 and we


1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)





grouping 1198616




1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)





CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References





































119904 =119878

1198780

119904119899=119878119873

1198780

119873

119904119902=119878119876

1198780

119876

119901119899=119875119873

1198750

119873

119901119902=119875119876

1198750

119876

119888119899=119862119873

1198620

119873

119888119902=119862119876

1198620

119876

ℎ =119867

1198670

119894119899=119868119873

1198680

119873

119894119902=119868119876

1198680

119876

119903119903=119877119877

1198770

119877

119903119889=119877119863

1198770

119863

119903119899=119877119873

1198770

119873

119889119863=119863119863

1198630

119863

119905lowast=119905

1198790

ℎ119897=119867119871

1198670

119871

(36)

where

1198780=Π

120583

1198780

119873= 1198780

119876= 1198780

1198750

119873=12057921198861205721198731198780

119873

120573119873+ 120583

1198750

119876=12057921198871205721198731198780

119873

120573119876+ 120583

1198620

119873=12057931198861205731198731198750

119873

119903119864119873+ 120574119873+ 120583

1198620

119876=12057931198871205731198731198750

119873

119903119864119876+ 120574119876+ 120583

1198680

119873=

12057941205741198731198620

119873

119903119871119873+ 120575119873+ 120583

1198680

119876=(1 minus 120579

4) 1205741198731198620

119873

119903119871119876+ 120575119876+ 120583

1198770

119877= 1199031198641198731198620

1198731198790

1198770

119863=1205751198731198680

N119887

1198770

119873= 12058311987901198670

119871

1198630

119863= 11988711987901198770

119863

1198670= 1198670

119871=Π

120583= 1198780

1198790=1

120583

(37)


120588119899=12057931205881198731198750

119873

1205831198670

120588119902=

12057971205881198761198750

119876

1205831198670

120591119899=1205911198731198620

119873

1205831198670

120591119902=

1205911198761198620

119876

1205831198670

120585119899=1205851198731198680

119873

1205831198670

120585119902=

1205851198761198680

119876

1205831198670

119886119889=1198861198631198770

119863

1205831198670

120572119899= (120572119873+ 120583)119879

0

120572119902= (120572119876+ 120583)119879

0

120573119899= (120573119873+ 120583) 119879

0

120573119902= (120573119876+ 120583)119879

0


120583119889= 1198871198790

1198871=(1 minus 120579

3) 1205731198731198750

119873

1205831198780

1198872=(1 minus 120579

2) 1205721198731198780

119873

1205831198780

1198873=

(1 minus 1205796) 1205721198761198780

119876

1205831198780

1198874=

(1 minus 1205797) 1205731198761198750

119876

1205831198780

1198875=1205751198731198680

119873

1205831198670

1198876=

1205751198761198680

119876

1205831198670

1198878=(119887 minus 120583) 119877

0

119863

1205831198670

1198861=

12057961205721198761198780

119876

12057921198871205721198731198780

119873

1198862=

12057971205731198761198750

119876

12057931198871205731198731198750

119873

1198863=

1205901198731198680

119873

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198864=

1205741198761198620

119876

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198865=1199031198711198731198680

119873

1199031198641198731198620

119873

1198866=

1199031198641198761198620

119876

1199031198641198731198620

119873

1198867=

1199031198711198761198680

119876

1199031198641198731198620

119873

1198868=

1205751198761198680

119876

1205751198731198680

119873

120574119899= (119903119864119873+ 120574119873+ 120583) 119879

0

120574119902= (119903119864119876+ 120574119876+ 120583) 119879

0

120575119902= (119903119871119876+ 120575119876+ 120583) 119879

0

120575119899= (119903119871119873+ 120575119873+ 120583) 119879

0

(38)


120582 = 120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)

(39)


119889119904

119889119905= 1 minus 120582119904 + 119887

1119901119899+ 1198872119904119899+ 1198873119904119902+ 1198874119901119902minus 119904 (40)

119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (41)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (42)

119889119901119899

119889119905= 120573119899(119904119899minus 119901119899) (43)

119889119901119902

119889119905= 120573119902(119904119899+ 1198861119904119902minus 119901119902) (44)

119889119888119899

119889119905= 120574119899(119901119899minus 119888119899) (45)

119889119888119902

119889119905= 120574119902(119901119899+ 1198862119901119902minus 119888119902) (46)

119889119894119899

119889119905= 120575119899(119888119899minus 119894119899) (47)

119889119894119902

119889119905= 120575119902(119888119899+ 1198863119894119899+ 1198864119888119902minus 119894119902) (48)

119889119903119903

119889119905= 119888119899+ 1198865119894119899+ 1198866119888119902+ 1198867119894119902minus 119903119903 (49)

119889119903119889

119889119905= 120583119889(119894119899+ 1198868119894119902minus 119903119889) (50)

119889119903119899

119889119905= ℎ119897 (51)

119889119889119863

119889119905= 119889119863 (52)


119889ℎ119897

119889119905= 1 minus ℎ

119897minus 1198875119894119899minus 1198876119894119902 (53)

119889ℎ

119889119905= 1 minus ℎ minus 119887

8119903119889 (54)




120588119899=12057931205792119886120588119873120572119873

(120573119873+ 120583) 120583

120588119902=12057971205792119887120588119876120572119873

(120573119876+ 120583) 120583

120591119899=

12057931198861205792119886120591119873120572119873120573119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) 120583

120591119902=

12057931198871205792119886120591119876120572119873120573119873

(120573119873+ 120583) (119903

119864119876+ 120574119876+ 120583) 120583

120585119899=

120579311988612057921198861205794120585119873120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 120583

120585119902=

12057931198861205792119886(1 minus 120579

4) 120585119876120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119876+ 120575119876+ 120583) 120583

119886119889=

120579211988612057931198861205794120575119873120574119873120573119873120572119873119886119863

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 119887120583

1198871=(1 minus 120579

3) 1205792119886120573119873120572119873

(120573119873+ 120583) 120583

1198872=(1 minus 120579

2) 120572119873

120583

1198873=(1 minus 120579

6) 120572119876

120583

1198874=(1 minus 120579

7) 1205792119887120573119876120572119873

(120573119876+ 120583) 120583

1198875=

120579412057921198861205793119886120575119873120574119873120573119873120572119873

(119903119871119873+ 120575119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198876=

(1 minus 1205794) 12057921198861205793119886120575119876120574119873120573119873120572119873

(119903119871119876+ 120575119876+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198878=

(119887 minus 120583) 120579412057921198861205793119886120572119873120573119873120574119873120575119873

119887120583 (120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583)

120575119899=119903119871119873+ 120575119873+ 120583

120583

120572119899=120572119873+ 120583

120583

120572119902=120572119876+ 120583

120583

120573119899=120573119873+ 120583

120583

120573119902=120573119876+ 120583

120583

120574119899=119903119864119873+ 120574119873+ 120583

120583

120574119902=119903119864119876+ 120574119876+ 120583

120583

120575119902=119903119871119876+ 120575119876+ 120583

120583

1198861=1205796120572119876

1205792119887120572119873

1198862=12057971205792119887120573119876(120573119873+ 120583)

12057921198861205793119887(120573119876+ 120583) 120573

119873

1198863=

1205794120590119873

(1 minus 1205794) (119903119871119873+ 120575119873+ 120583)

1198864=

1205793119887120574119876(119903119864119873+ 120574119873+ 120583)

(1 minus 1205794) 1205793119886120574119873(119903119864119876+ 120574119876+ 120583)

1198865=

1199031198711198731205794120574119873

119903119864119873(119903119871119873+ 120575119873+ 120583)

120583119889=119887

120583

1198866=1205793119887119903119864119876(119903119864119873+ 120574119873+ 120583)

1205793119886119903119864119873(119903119864119876+ 120574119876+ 120583)

1198867=119903119871119876(1 minus 120579

4) 120574119873

119903119864119873(119903119871119876+ 120575119876+ 120583)

1198868=(1 minus 120579

4) 120575119876(119903119871119873+ 120575119873+ 120583)

1205794120575119873(119903119871119876+ 120575119876+ 120583)

(55)


119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast



119904lowast

119899= 119901lowast

119899= 119888lowast

119899= 119894lowast

119899(56)



119899and

119904lowast


119901lowast

119902(119901lowast

119899 119904lowast

119902) = 119901lowast

119899+ 1198861119904lowast

119902

119888lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

1119901lowast

119899+ 1198602119904lowast

119902

119894lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

3119901lowast

119899+ 1198604119904lowast

119902

119903lowast

119903(119901lowast

119899 119904lowast

119902) = 119860

5119901lowast

119899+ 1198606119904lowast

119902

119903lowast

119889(119901lowast

119899 119904lowast

119902) = 119860

7119901lowast

119899+ 1198608119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902

ℎlowast

119897(119901lowast

119899 119904lowast

119902) = 1 minus (119887

5+ 11988761198603) 119901lowast

119899minus 11988761198604119904lowast

119902

(57)

where

1198601= 1 + 119886

2

1198602= 11988611198862

1198603= 1 + 119886

3+ 11988641198601

1198604= 11988641198602

1198605= 1 + 119886

5+ 11988661198601+ 11988671198603

1198606= 11988661198602+ 11988671198604

1198607= 1 + 119886

81198603

1198608= 11988681198604

1198611= 11988781198607

1198612= 11988781198608


1198614= 120572119902minus 1198873minus 11988741198861

(58)




3and 119861


1198613= 1

+120572119873

120583(1205731198731205792119886(1205793minus 1)

120573119873+ 120583

+1205731198761205792119887(1205797minus 1)

120573119876+ 120583

+ 1205792)

gt 0 since 1205792= 1205792119886+ 1205792119887

1198614=1205721198761205796(1205731198761205797+ 120583) + 120583 (120573

119876+ 120583)

120583 (120573119876+ 120583)

gt 0

(59)


4gt 0





1205791(1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119899119901lowast

119899(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(60)

(1 minus 1205791) (1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119902119904lowast

119902(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(61)

where

1198615= 120588119899+ 120588119902+ 120591119899+ 1205911199021198601+ 120585119899+ 1205851199021198603+ 1198861198891198607

1198616= 1205881199021198861+ 1205911199021198602+ 1205851199021198604+ 1198861198891198608

(62)



(1 minus 1198611119901lowast

119899minus 1198612119904lowast

119902) (120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast

1199021205791) = 0 (63)


119899minus 1198612119904lowast

119902= 0 or (ii)

120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast


system


119899minus 1198614119904lowast

119902= 0


119899minus 1198612119904lowast

119902= 0

(64)



119901lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902 (65)



119904lowast

119902= 0

or 119904lowast119902=1198617

1198618

=(1 minus 120579

1) 120572119899(1198770minus 1)

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612) (R minus 1)

= 119909 (1198770minus 1

R minus 1)

(66)

where

1198617= 120572119902120572119899(1 minus 120579

1) ((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

) minus 1)

= 120572119902120572119899(1 minus 120579

1) (1198770minus 1)

1198618= 120572119902[(12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614)

minus (12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612)] = 120572

119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612)((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (

12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614

12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612

) minus 1) = 120572119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612) (1198770119911 minus 1)

(67)

with

119909 =(1 minus 120579

1) 120572119899

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612)

119910 =

1205791120572119902119909

(1 minus 1205791) 120572119899

119911 =

(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

(68)

1198770=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

R =1198770(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

= 1199111198770

(69)


1198612= 11988781198868119886411988611198862=1205796120572119876

120583

1205797120573119876

(120573119876+ 120583)

((1 minus120583

119887)

sdot120575119876

(119903119871119876+ 120575119876+ 120583)

120574119876

(119903119864119876+ 120574119876+ 120583)) lt

1205796120572119876

120583

sdot1205797120573119876

(120573119876+ 120583)

lt1205796120572119876

120583

(1205797120573119876+ 120583)

(120573119876+ 120583)

+ 1 = 1198614

(70)



3lt 1198611 we will


1120572119902(1minus120579

1)120572119899)(1198611minus1198613)


system as follows




0gt 1 and the


0




119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119894lowast

119899= 119888lowast

119899= 119904lowast

119899= 119901lowast

119899= 119910(

1198770minus 1

R minus 1)

119901lowast

119902= (119910 + 119886

1119909) (1198770minus 1

R minus 1)

119888lowast

119902= (1198601119910 + 119860

2119909) (1198770minus 1

R minus 1)

119894lowast

119902= (1198603119910 + 119860

4119909) (1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

119903lowast

119889= (1198607119910 + 119860

8119909) (1198770minus 1

R minus 1)

119904lowast= 1 minus (119861

3119910 + 1198614119909) (1198770minus 1

R minus 1)


1119910 + 1198612119909) (1198770minus 1

R minus 1)

(71)



119889119891119890=


119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast







119869dfe =

((((((((((((((((((((((

(


1198890

0 minus1205721198990 120579

1120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 0 minus120572119902

1205791120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 120573119899

0 minus120573119899

0 0 0 0 0 0 0 0

0 1205731199021198861120573119902

0 minus120573119902

0 0 0 0 0 0 0

0 0 0 120574119899

0 minus1205741198990 0 0 0 0 0

0 0 0 120574119902

1198862120574119902

0 minus1205741199020 0 0 0 0

0 0 0 0 0 120575119899

0 minus1205751198990 0 0 0

0 0 0 0 0 12057511990211988641205751199021198863120575119902minus1205751199020 0 0

0 0 0 0 0 1 11988661198865

1198867minus1 0 0

0 0 0 0 0 0 0 12058311988912058311988911988680 minus120583

1198890

0 0 0 0 0 0 0 0 0 0 minus1198878minus1

))))))))))))))))))))))

)

(72)

where 1205791= 1 minus 120579



119875 (120577) =1003816100381610038161003816119869dfe minus 120577I

1003816100381610038161003816 = (120577 + 1)31198759(120577) = 0 (73)


1198759 (120577) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus120572119899minus 120577 0 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

0 minus120572119902minus 120577 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

120573119899

0 minus120573119899minus 120577 0 0 0 0 0 0

120573119902

1198861120573119902

0 minus120573119902minus 120577 0 0 0 0 0

0 0 120574119899

0 minus120574119899minus 120577 0 0 0 0

0 0 120574119902

1198862120574119902

0 minus120574119902minus 120577 0 0 0

0 0 0 0 120575119899

0 minus120575119899minus 120577 0 0

0 0 0 0 120575119902

1198864120575119902

1198863120575119902minus120575119902minus 120577 0

0 0 0 0 0 0 120583119889

1205831198891198868minus120583119889minus 120577

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(74)


119875 (120577) = (120577 + 1)3

9

sum

119896=0

119888119894120577119894 (75)

where1198889= 1

1198888= 120572119899+ 120572119902+ 120573119899+ 120573119902+ 120574119899+ 120574119902+ 120575119899+ 120575119902+ 120583119889

1198880= 1205731198991205731199021205741198991205741199021205751198991205751199021205831198891205721198991205721199021 minus

12057911198615

120572119899


1) 1198616

120572119902

= 120573119899120573119902120574119899120574119902120575119899120575119902120583119889120572119899120572119902(1 minus 119877

0)

(76)




0= 1 to





0 1198770




0= 1


0specifically


0which is fixedRmay






0is the


119899 119901119902 119888119899 119888119902 119894119899 119894119902 119903119903 119903119889 and ℎ The




F =

(((((((((((((((((

(

0

1205791120582119904

(1 minus 1205791) 120582119904

0

0

0

0

0

0

0

0

0

)))))))))))))))))

)

V =

((((((((((((((((((((((((((((((((

(


120572119899119904119899

120572119902119904119902

120573119899(minus119904119899+ 119901119899)

120573119902(minus119904119899minus 1198861119904119902+ 119901119902)

120574119899(minus119901119899+ 119888119899)

120574119902(minus119901119899minus 1198862119901119902+ 119888119902)

120575119899(minus119888119899+ 119894119899)



minus120583119889119894119899minus 1205831198891198868119894119902+ 120583119889119903119889

minus1 + ℎ + 1198878119903119889

))))))))))))))))))))))))))))))))

)

(77)


119894119895=

120597F119894120597119909119895and (119881)

119894119895= 120597V



119901119902= 119888119899= 119888119902= 119894119899= 119894119902= 119903119903= 119903119889= 0 The basic


1198770= 984858 (119865119881

minus1) =

1

120572119899120572119902

[1205721199021205791(1

+ (1 + 1198863+ (1 + 119886

2) 1198864) 119886119889

+ 120585119902(11988621198864+ 1198863+ 1198864+ 1) + 119886

2120591119902+ 120585119899+ 120588119899+ 120588119902

+ 120591119899+ 120591119902) minus 120572

119899(1205791minus 1) (119886

1120588119902

+ 1198861[11988621198864(1198868119886119889+ 120585119902) + 1198862120591119902])]

=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

(78)






119899(which








120583


120583 + rate120583

asymp large (79)


119899 1120573119902



119901119899= 119888119899= 119894119899= 119904119899

119901119902= 119904119899+ 1198861119904119902

119888119902= 1198601119904119899+ 1198602119904119902

119894119902= 1198603119904119899+ 1198604119904119902

119903119889= 1198607119904119899+ 1198608119904119902

(80)


119889119905= 1 minus 120582119904 + (120572


119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (82)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (83)

119889119903119903

119889119905= 1198605119904119899+ 1198606119904119902minus 119903119903 (84)

119889ℎ

119889119905= 1 minus ℎ minus 119861

1119904119899minus 1198612119904119902 (85)


120582119904 = (120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)) 119904 = (119861

5(119904119899

ℎ)

+ 1198616(

119904119902

ℎ)) 119904

(86)





119899 119904lowast

119902 119903lowast


119864dfe = (1 0 0 0 1)

119864119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast)

(87)


119904lowast

119902=1198617

1198618

119904lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902

119903lowast

119903= 1198605119904lowast

119899+ 1198606119904lowast

119902

119904lowast= 1 minus 119861

3119904lowast

119899minus 1198614119904lowast

119902


1119904lowast

119899minus 1198612119904lowast

119902

(88)


3119910 + 119861

4119909 and

1198611119910 + 1198612119909 = 1 we get

119904lowast= (119911 minus 1

R minus 1)

119904lowast

119899= 119910(

1198770minus 1

R minus 1)

119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

ℎlowast= ((119911 minus 1) 119877

0

R minus 1)

(89)


119899 119904lowast

119902 119903lowast




119869 (xlowast) =((

(


4(xlowast) + 120572


119902minus 11986140 minus119862

6(xlowast)

12057911198623(xlowast) 120579


11989912057911198625(xlowast) 0 120579

11198626(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) 120579


1199020 12057911198626(xlowast)

0 1198605

1198606

minus1 0

0 minus1198611

minus1198612

0 minus1

))

)

(90)

where 1205791= 1 minus 120579

1and

1198623(xlowast) = 119861

5

119904lowast

119899

ℎlowast+ 1198616

119904lowast

119902

ℎlowast=120572119899119910 (1198770minus 1)

1205791(119911 minus 1)

1198624(xlowast) = 119861

5

119904lowast

ℎlowast=1198615

1198770

1198625(xlowast) = 119861

6

119904lowast

ℎlowast=1198616

1198770

1198626(xlowast) = minus(119861

5


119899

ℎlowast2+ 1198616


119902

ℎlowast2) = minus

120572119899119910 (1198770minus 1)

1205791 (119911 minus 1) 1198770

(91)


1198755(120577 xlowast)

= (120577 + 1)2(1205773+ 1198762(xlowast) 1205772 + 119876

1(xlowast) 120577 + 119876

0(xlowast))

= 0

(92)

where

1198762(xlowast) = 120572

119899+ 120572119902+ 1198623(xlowast) minus 119862

4(xlowast) 120579


1

+ 1

1198761(xlowast) = 120572

119902


1199021198624(xlowast) + 119862

4(xlowast))

+ 11986121198626(xlowast) 120579

1+ 1198623(xlowast) (120572

1199021205791+ 11986131205791+ 11986141205791)

+ 120572119899(120572119902+ 12057911198623(xlowast) minus 119862

5(xlowast) 120579

1+ 1) minus 119862

5(xlowast)

sdot 1205791

1198760(xlowast) = 120572

1199021205791(11986111198626(xlowast) + 119861


4(xlowast))

+ 120572119899(1205791(11986121198626(xlowast) + 119861


5(xlowast)) + 120572

119902)

(93)




119902=

119903lowast




1198616 1198626(xdfe) = 0 and (92) becomes

1198755(120577 xdfe) = (120577 + 1)

3(1205772+ 119876dfe120577 + 119877dfe) = 0 (94)

where


= 120572119899(1 minus 119877

0) + 120572119902+ (1 minus 120579

1) 1198616(

120572119899minus 120572119902

120572119902

)

119877dfe = 120572119902 (120572119899 minus 11986151205791) + 1205721198991198616 (1205791 minus 1)

= 120572119902120572119899(1 minus 119877

0)

(95)


radic1198762







=


119899 119904lowast

119902 119903lowast


and we have

1198755(120577 xlowast) = (120577 + 1)2 (1205773 + 119875x

119890119890

1205772+ 119876x

119890119890

120577 + 119877x119890119890

) = 0 (96)

where


119875x119890119890

=


12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)


119890119890will be the


119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)




119899= 0 and we


1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)





grouping 1198616




1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)





CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References




































120583119889= 1198871198790

1198871=(1 minus 120579

3) 1205731198731198750

119873

1205831198780

1198872=(1 minus 120579

2) 1205721198731198780

119873

1205831198780

1198873=

(1 minus 1205796) 1205721198761198780

119876

1205831198780

1198874=

(1 minus 1205797) 1205731198761198750

119876

1205831198780

1198875=1205751198731198680

119873

1205831198670

1198876=

1205751198761198680

119876

1205831198670

1198878=(119887 minus 120583) 119877

0

119863

1205831198670

1198861=

12057961205721198761198780

119876

12057921198871205721198731198780

119873

1198862=

12057971205731198761198750

119876

12057931198871205731198731198750

119873

1198863=

1205901198731198680

119873

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198864=

1205741198761198620

119876

(119903119871119876+ 120575119876+ 120583) 119868

0

119876

1198865=1199031198711198731198680

119873

1199031198641198731198620

119873

1198866=

1199031198641198761198620

119876

1199031198641198731198620

119873

1198867=

1199031198711198761198680

119876

1199031198641198731198620

119873

1198868=

1205751198761198680

119876

1205751198731198680

119873

120574119899= (119903119864119873+ 120574119873+ 120583) 119879

0

120574119902= (119903119864119876+ 120574119876+ 120583) 119879

0

120575119902= (119903119871119876+ 120575119876+ 120583) 119879

0

120575119899= (119903119871119873+ 120575119873+ 120583) 119879

0

(38)


120582 = 120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)

(39)


119889119904

119889119905= 1 minus 120582119904 + 119887

1119901119899+ 1198872119904119899+ 1198873119904119902+ 1198874119901119902minus 119904 (40)

119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (41)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (42)

119889119901119899

119889119905= 120573119899(119904119899minus 119901119899) (43)

119889119901119902

119889119905= 120573119902(119904119899+ 1198861119904119902minus 119901119902) (44)

119889119888119899

119889119905= 120574119899(119901119899minus 119888119899) (45)

119889119888119902

119889119905= 120574119902(119901119899+ 1198862119901119902minus 119888119902) (46)

119889119894119899

119889119905= 120575119899(119888119899minus 119894119899) (47)

119889119894119902

119889119905= 120575119902(119888119899+ 1198863119894119899+ 1198864119888119902minus 119894119902) (48)

119889119903119903

119889119905= 119888119899+ 1198865119894119899+ 1198866119888119902+ 1198867119894119902minus 119903119903 (49)

119889119903119889

119889119905= 120583119889(119894119899+ 1198868119894119902minus 119903119889) (50)

119889119903119899

119889119905= ℎ119897 (51)

119889119889119863

119889119905= 119889119863 (52)


119889ℎ119897

119889119905= 1 minus ℎ

119897minus 1198875119894119899minus 1198876119894119902 (53)

119889ℎ

119889119905= 1 minus ℎ minus 119887

8119903119889 (54)




120588119899=12057931205792119886120588119873120572119873

(120573119873+ 120583) 120583

120588119902=12057971205792119887120588119876120572119873

(120573119876+ 120583) 120583

120591119899=

12057931198861205792119886120591119873120572119873120573119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) 120583

120591119902=

12057931198871205792119886120591119876120572119873120573119873

(120573119873+ 120583) (119903

119864119876+ 120574119876+ 120583) 120583

120585119899=

120579311988612057921198861205794120585119873120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 120583

120585119902=

12057931198861205792119886(1 minus 120579

4) 120585119876120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119876+ 120575119876+ 120583) 120583

119886119889=

120579211988612057931198861205794120575119873120574119873120573119873120572119873119886119863

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 119887120583

1198871=(1 minus 120579

3) 1205792119886120573119873120572119873

(120573119873+ 120583) 120583

1198872=(1 minus 120579

2) 120572119873

120583

1198873=(1 minus 120579

6) 120572119876

120583

1198874=(1 minus 120579

7) 1205792119887120573119876120572119873

(120573119876+ 120583) 120583

1198875=

120579412057921198861205793119886120575119873120574119873120573119873120572119873

(119903119871119873+ 120575119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198876=

(1 minus 1205794) 12057921198861205793119886120575119876120574119873120573119873120572119873

(119903119871119876+ 120575119876+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198878=

(119887 minus 120583) 120579412057921198861205793119886120572119873120573119873120574119873120575119873

119887120583 (120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583)

120575119899=119903119871119873+ 120575119873+ 120583

120583

120572119899=120572119873+ 120583

120583

120572119902=120572119876+ 120583

120583

120573119899=120573119873+ 120583

120583

120573119902=120573119876+ 120583

120583

120574119899=119903119864119873+ 120574119873+ 120583

120583

120574119902=119903119864119876+ 120574119876+ 120583

120583

120575119902=119903119871119876+ 120575119876+ 120583

120583

1198861=1205796120572119876

1205792119887120572119873

1198862=12057971205792119887120573119876(120573119873+ 120583)

12057921198861205793119887(120573119876+ 120583) 120573

119873

1198863=

1205794120590119873

(1 minus 1205794) (119903119871119873+ 120575119873+ 120583)

1198864=

1205793119887120574119876(119903119864119873+ 120574119873+ 120583)

(1 minus 1205794) 1205793119886120574119873(119903119864119876+ 120574119876+ 120583)

1198865=

1199031198711198731205794120574119873

119903119864119873(119903119871119873+ 120575119873+ 120583)

120583119889=119887

120583

1198866=1205793119887119903119864119876(119903119864119873+ 120574119873+ 120583)

1205793119886119903119864119873(119903119864119876+ 120574119876+ 120583)

1198867=119903119871119876(1 minus 120579

4) 120574119873

119903119864119873(119903119871119876+ 120575119876+ 120583)

1198868=(1 minus 120579

4) 120575119876(119903119871119873+ 120575119873+ 120583)

1205794120575119873(119903119871119876+ 120575119876+ 120583)

(55)


119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast



119904lowast

119899= 119901lowast

119899= 119888lowast

119899= 119894lowast

119899(56)



119899and

119904lowast


119901lowast

119902(119901lowast

119899 119904lowast

119902) = 119901lowast

119899+ 1198861119904lowast

119902

119888lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

1119901lowast

119899+ 1198602119904lowast

119902

119894lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

3119901lowast

119899+ 1198604119904lowast

119902

119903lowast

119903(119901lowast

119899 119904lowast

119902) = 119860

5119901lowast

119899+ 1198606119904lowast

119902

119903lowast

119889(119901lowast

119899 119904lowast

119902) = 119860

7119901lowast

119899+ 1198608119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902

ℎlowast

119897(119901lowast

119899 119904lowast

119902) = 1 minus (119887

5+ 11988761198603) 119901lowast

119899minus 11988761198604119904lowast

119902

(57)

where

1198601= 1 + 119886

2

1198602= 11988611198862

1198603= 1 + 119886

3+ 11988641198601

1198604= 11988641198602

1198605= 1 + 119886

5+ 11988661198601+ 11988671198603

1198606= 11988661198602+ 11988671198604

1198607= 1 + 119886

81198603

1198608= 11988681198604

1198611= 11988781198607

1198612= 11988781198608


1198614= 120572119902minus 1198873minus 11988741198861

(58)




3and 119861


1198613= 1

+120572119873

120583(1205731198731205792119886(1205793minus 1)

120573119873+ 120583

+1205731198761205792119887(1205797minus 1)

120573119876+ 120583

+ 1205792)

gt 0 since 1205792= 1205792119886+ 1205792119887

1198614=1205721198761205796(1205731198761205797+ 120583) + 120583 (120573

119876+ 120583)

120583 (120573119876+ 120583)

gt 0

(59)


4gt 0





1205791(1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119899119901lowast

119899(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(60)

(1 minus 1205791) (1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119902119904lowast

119902(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(61)

where

1198615= 120588119899+ 120588119902+ 120591119899+ 1205911199021198601+ 120585119899+ 1205851199021198603+ 1198861198891198607

1198616= 1205881199021198861+ 1205911199021198602+ 1205851199021198604+ 1198861198891198608

(62)



(1 minus 1198611119901lowast

119899minus 1198612119904lowast

119902) (120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast

1199021205791) = 0 (63)


119899minus 1198612119904lowast

119902= 0 or (ii)

120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast


system


119899minus 1198614119904lowast

119902= 0


119899minus 1198612119904lowast

119902= 0

(64)



119901lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902 (65)



119904lowast

119902= 0

or 119904lowast119902=1198617

1198618

=(1 minus 120579

1) 120572119899(1198770minus 1)

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612) (R minus 1)

= 119909 (1198770minus 1

R minus 1)

(66)

where

1198617= 120572119902120572119899(1 minus 120579

1) ((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

) minus 1)

= 120572119902120572119899(1 minus 120579

1) (1198770minus 1)

1198618= 120572119902[(12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614)

minus (12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612)] = 120572

119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612)((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (

12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614

12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612

) minus 1) = 120572119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612) (1198770119911 minus 1)

(67)

with

119909 =(1 minus 120579

1) 120572119899

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612)

119910 =

1205791120572119902119909

(1 minus 1205791) 120572119899

119911 =

(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

(68)

1198770=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

R =1198770(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

= 1199111198770

(69)


1198612= 11988781198868119886411988611198862=1205796120572119876

120583

1205797120573119876

(120573119876+ 120583)

((1 minus120583

119887)

sdot120575119876

(119903119871119876+ 120575119876+ 120583)

120574119876

(119903119864119876+ 120574119876+ 120583)) lt

1205796120572119876

120583

sdot1205797120573119876

(120573119876+ 120583)

lt1205796120572119876

120583

(1205797120573119876+ 120583)

(120573119876+ 120583)

+ 1 = 1198614

(70)



3lt 1198611 we will


1120572119902(1minus120579

1)120572119899)(1198611minus1198613)


system as follows




0gt 1 and the


0




119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119894lowast

119899= 119888lowast

119899= 119904lowast

119899= 119901lowast

119899= 119910(

1198770minus 1

R minus 1)

119901lowast

119902= (119910 + 119886

1119909) (1198770minus 1

R minus 1)

119888lowast

119902= (1198601119910 + 119860

2119909) (1198770minus 1

R minus 1)

119894lowast

119902= (1198603119910 + 119860

4119909) (1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

119903lowast

119889= (1198607119910 + 119860

8119909) (1198770minus 1

R minus 1)

119904lowast= 1 minus (119861

3119910 + 1198614119909) (1198770minus 1

R minus 1)


1119910 + 1198612119909) (1198770minus 1

R minus 1)

(71)



119889119891119890=


119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast







119869dfe =

((((((((((((((((((((((

(


1198890

0 minus1205721198990 120579

1120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 0 minus120572119902

1205791120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 120573119899

0 minus120573119899

0 0 0 0 0 0 0 0

0 1205731199021198861120573119902

0 minus120573119902

0 0 0 0 0 0 0

0 0 0 120574119899

0 minus1205741198990 0 0 0 0 0

0 0 0 120574119902

1198862120574119902

0 minus1205741199020 0 0 0 0

0 0 0 0 0 120575119899

0 minus1205751198990 0 0 0

0 0 0 0 0 12057511990211988641205751199021198863120575119902minus1205751199020 0 0

0 0 0 0 0 1 11988661198865

1198867minus1 0 0

0 0 0 0 0 0 0 12058311988912058311988911988680 minus120583

1198890

0 0 0 0 0 0 0 0 0 0 minus1198878minus1

))))))))))))))))))))))

)

(72)

where 1205791= 1 minus 120579



119875 (120577) =1003816100381610038161003816119869dfe minus 120577I

1003816100381610038161003816 = (120577 + 1)31198759(120577) = 0 (73)


1198759 (120577) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus120572119899minus 120577 0 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

0 minus120572119902minus 120577 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

120573119899

0 minus120573119899minus 120577 0 0 0 0 0 0

120573119902

1198861120573119902

0 minus120573119902minus 120577 0 0 0 0 0

0 0 120574119899

0 minus120574119899minus 120577 0 0 0 0

0 0 120574119902

1198862120574119902

0 minus120574119902minus 120577 0 0 0

0 0 0 0 120575119899

0 minus120575119899minus 120577 0 0

0 0 0 0 120575119902

1198864120575119902

1198863120575119902minus120575119902minus 120577 0

0 0 0 0 0 0 120583119889

1205831198891198868minus120583119889minus 120577

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(74)


119875 (120577) = (120577 + 1)3

9

sum

119896=0

119888119894120577119894 (75)

where1198889= 1

1198888= 120572119899+ 120572119902+ 120573119899+ 120573119902+ 120574119899+ 120574119902+ 120575119899+ 120575119902+ 120583119889

1198880= 1205731198991205731199021205741198991205741199021205751198991205751199021205831198891205721198991205721199021 minus

12057911198615

120572119899


1) 1198616

120572119902

= 120573119899120573119902120574119899120574119902120575119899120575119902120583119889120572119899120572119902(1 minus 119877

0)

(76)




0= 1 to





0 1198770




0= 1


0specifically


0which is fixedRmay






0is the


119899 119901119902 119888119899 119888119902 119894119899 119894119902 119903119903 119903119889 and ℎ The




F =

(((((((((((((((((

(

0

1205791120582119904

(1 minus 1205791) 120582119904

0

0

0

0

0

0

0

0

0

)))))))))))))))))

)

V =

((((((((((((((((((((((((((((((((

(


120572119899119904119899

120572119902119904119902

120573119899(minus119904119899+ 119901119899)

120573119902(minus119904119899minus 1198861119904119902+ 119901119902)

120574119899(minus119901119899+ 119888119899)

120574119902(minus119901119899minus 1198862119901119902+ 119888119902)

120575119899(minus119888119899+ 119894119899)



minus120583119889119894119899minus 1205831198891198868119894119902+ 120583119889119903119889

minus1 + ℎ + 1198878119903119889

))))))))))))))))))))))))))))))))

)

(77)


119894119895=

120597F119894120597119909119895and (119881)

119894119895= 120597V



119901119902= 119888119899= 119888119902= 119894119899= 119894119902= 119903119903= 119903119889= 0 The basic


1198770= 984858 (119865119881

minus1) =

1

120572119899120572119902

[1205721199021205791(1

+ (1 + 1198863+ (1 + 119886

2) 1198864) 119886119889

+ 120585119902(11988621198864+ 1198863+ 1198864+ 1) + 119886

2120591119902+ 120585119899+ 120588119899+ 120588119902

+ 120591119899+ 120591119902) minus 120572

119899(1205791minus 1) (119886

1120588119902

+ 1198861[11988621198864(1198868119886119889+ 120585119902) + 1198862120591119902])]

=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

(78)






119899(which








120583


120583 + rate120583

asymp large (79)


119899 1120573119902



119901119899= 119888119899= 119894119899= 119904119899

119901119902= 119904119899+ 1198861119904119902

119888119902= 1198601119904119899+ 1198602119904119902

119894119902= 1198603119904119899+ 1198604119904119902

119903119889= 1198607119904119899+ 1198608119904119902

(80)


119889119905= 1 minus 120582119904 + (120572


119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (82)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (83)

119889119903119903

119889119905= 1198605119904119899+ 1198606119904119902minus 119903119903 (84)

119889ℎ

119889119905= 1 minus ℎ minus 119861

1119904119899minus 1198612119904119902 (85)


120582119904 = (120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)) 119904 = (119861

5(119904119899

ℎ)

+ 1198616(

119904119902

ℎ)) 119904

(86)





119899 119904lowast

119902 119903lowast


119864dfe = (1 0 0 0 1)

119864119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast)

(87)


119904lowast

119902=1198617

1198618

119904lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902

119903lowast

119903= 1198605119904lowast

119899+ 1198606119904lowast

119902

119904lowast= 1 minus 119861

3119904lowast

119899minus 1198614119904lowast

119902


1119904lowast

119899minus 1198612119904lowast

119902

(88)


3119910 + 119861

4119909 and

1198611119910 + 1198612119909 = 1 we get

119904lowast= (119911 minus 1

R minus 1)

119904lowast

119899= 119910(

1198770minus 1

R minus 1)

119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

ℎlowast= ((119911 minus 1) 119877

0

R minus 1)

(89)


119899 119904lowast

119902 119903lowast




119869 (xlowast) =((

(


4(xlowast) + 120572


119902minus 11986140 minus119862

6(xlowast)

12057911198623(xlowast) 120579


11989912057911198625(xlowast) 0 120579

11198626(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) 120579


1199020 12057911198626(xlowast)

0 1198605

1198606

minus1 0

0 minus1198611

minus1198612

0 minus1

))

)

(90)

where 1205791= 1 minus 120579

1and

1198623(xlowast) = 119861

5

119904lowast

119899

ℎlowast+ 1198616

119904lowast

119902

ℎlowast=120572119899119910 (1198770minus 1)

1205791(119911 minus 1)

1198624(xlowast) = 119861

5

119904lowast

ℎlowast=1198615

1198770

1198625(xlowast) = 119861

6

119904lowast

ℎlowast=1198616

1198770

1198626(xlowast) = minus(119861

5


119899

ℎlowast2+ 1198616


119902

ℎlowast2) = minus

120572119899119910 (1198770minus 1)

1205791 (119911 minus 1) 1198770

(91)


1198755(120577 xlowast)

= (120577 + 1)2(1205773+ 1198762(xlowast) 1205772 + 119876

1(xlowast) 120577 + 119876

0(xlowast))

= 0

(92)

where

1198762(xlowast) = 120572

119899+ 120572119902+ 1198623(xlowast) minus 119862

4(xlowast) 120579


1

+ 1

1198761(xlowast) = 120572

119902


1199021198624(xlowast) + 119862

4(xlowast))

+ 11986121198626(xlowast) 120579

1+ 1198623(xlowast) (120572

1199021205791+ 11986131205791+ 11986141205791)

+ 120572119899(120572119902+ 12057911198623(xlowast) minus 119862

5(xlowast) 120579

1+ 1) minus 119862

5(xlowast)

sdot 1205791

1198760(xlowast) = 120572

1199021205791(11986111198626(xlowast) + 119861


4(xlowast))

+ 120572119899(1205791(11986121198626(xlowast) + 119861


5(xlowast)) + 120572

119902)

(93)




119902=

119903lowast




1198616 1198626(xdfe) = 0 and (92) becomes

1198755(120577 xdfe) = (120577 + 1)

3(1205772+ 119876dfe120577 + 119877dfe) = 0 (94)

where


= 120572119899(1 minus 119877

0) + 120572119902+ (1 minus 120579

1) 1198616(

120572119899minus 120572119902

120572119902

)

119877dfe = 120572119902 (120572119899 minus 11986151205791) + 1205721198991198616 (1205791 minus 1)

= 120572119902120572119899(1 minus 119877

0)

(95)


radic1198762







=


119899 119904lowast

119902 119903lowast


and we have

1198755(120577 xlowast) = (120577 + 1)2 (1205773 + 119875x

119890119890

1205772+ 119876x

119890119890

120577 + 119877x119890119890

) = 0 (96)

where


119875x119890119890

=


12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)


119890119890will be the


119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)




119899= 0 and we


1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)





grouping 1198616




1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)





CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References






































120588119899=12057931205792119886120588119873120572119873

(120573119873+ 120583) 120583

120588119902=12057971205792119887120588119876120572119873

(120573119876+ 120583) 120583

120591119899=

12057931198861205792119886120591119873120572119873120573119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) 120583

120591119902=

12057931198871205792119886120591119876120572119873120573119873

(120573119873+ 120583) (119903

119864119876+ 120574119876+ 120583) 120583

120585119899=

120579311988612057921198861205794120585119873120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 120583

120585119902=

12057931198861205792119886(1 minus 120579

4) 120585119876120574119873120573119873120572119873

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119876+ 120575119876+ 120583) 120583

119886119889=

120579211988612057931198861205794120575119873120574119873120573119873120572119873119886119863

(120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583) 119887120583

1198871=(1 minus 120579

3) 1205792119886120573119873120572119873

(120573119873+ 120583) 120583

1198872=(1 minus 120579

2) 120572119873

120583

1198873=(1 minus 120579

6) 120572119876

120583

1198874=(1 minus 120579

7) 1205792119887120573119876120572119873

(120573119876+ 120583) 120583

1198875=

120579412057921198861205793119886120575119873120574119873120573119873120572119873

(119903119871119873+ 120575119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198876=

(1 minus 1205794) 12057921198861205793119886120575119876120574119873120573119873120572119873

(119903119871119876+ 120575119876+ 120583) (119903

119864119873+ 120574119873+ 120583) (120573

119873+ 120583) 120583

1198878=

(119887 minus 120583) 120579412057921198861205793119886120572119873120573119873120574119873120575119873

119887120583 (120573119873+ 120583) (119903

119864119873+ 120574119873+ 120583) (119903

119871119873+ 120575119873+ 120583)

120575119899=119903119871119873+ 120575119873+ 120583

120583

120572119899=120572119873+ 120583

120583

120572119902=120572119876+ 120583

120583

120573119899=120573119873+ 120583

120583

120573119902=120573119876+ 120583

120583

120574119899=119903119864119873+ 120574119873+ 120583

120583

120574119902=119903119864119876+ 120574119876+ 120583

120583

120575119902=119903119871119876+ 120575119876+ 120583

120583

1198861=1205796120572119876

1205792119887120572119873

1198862=12057971205792119887120573119876(120573119873+ 120583)

12057921198861205793119887(120573119876+ 120583) 120573

119873

1198863=

1205794120590119873

(1 minus 1205794) (119903119871119873+ 120575119873+ 120583)

1198864=

1205793119887120574119876(119903119864119873+ 120574119873+ 120583)

(1 minus 1205794) 1205793119886120574119873(119903119864119876+ 120574119876+ 120583)

1198865=

1199031198711198731205794120574119873

119903119864119873(119903119871119873+ 120575119873+ 120583)

120583119889=119887

120583

1198866=1205793119887119903119864119876(119903119864119873+ 120574119873+ 120583)

1205793119886119903119864119873(119903119864119876+ 120574119876+ 120583)

1198867=119903119871119876(1 minus 120579

4) 120574119873

119903119864119873(119903119871119876+ 120575119876+ 120583)

1198868=(1 minus 120579

4) 120575119876(119903119871119873+ 120575119873+ 120583)

1205794120575119873(119903119871119876+ 120575119876+ 120583)

(55)


119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast



119904lowast

119899= 119901lowast

119899= 119888lowast

119899= 119894lowast

119899(56)



119899and

119904lowast


119901lowast

119902(119901lowast

119899 119904lowast

119902) = 119901lowast

119899+ 1198861119904lowast

119902

119888lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

1119901lowast

119899+ 1198602119904lowast

119902

119894lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

3119901lowast

119899+ 1198604119904lowast

119902

119903lowast

119903(119901lowast

119899 119904lowast

119902) = 119860

5119901lowast

119899+ 1198606119904lowast

119902

119903lowast

119889(119901lowast

119899 119904lowast

119902) = 119860

7119901lowast

119899+ 1198608119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902

ℎlowast

119897(119901lowast

119899 119904lowast

119902) = 1 minus (119887

5+ 11988761198603) 119901lowast

119899minus 11988761198604119904lowast

119902

(57)

where

1198601= 1 + 119886

2

1198602= 11988611198862

1198603= 1 + 119886

3+ 11988641198601

1198604= 11988641198602

1198605= 1 + 119886

5+ 11988661198601+ 11988671198603

1198606= 11988661198602+ 11988671198604

1198607= 1 + 119886

81198603

1198608= 11988681198604

1198611= 11988781198607

1198612= 11988781198608


1198614= 120572119902minus 1198873minus 11988741198861

(58)




3and 119861


1198613= 1

+120572119873

120583(1205731198731205792119886(1205793minus 1)

120573119873+ 120583

+1205731198761205792119887(1205797minus 1)

120573119876+ 120583

+ 1205792)

gt 0 since 1205792= 1205792119886+ 1205792119887

1198614=1205721198761205796(1205731198761205797+ 120583) + 120583 (120573

119876+ 120583)

120583 (120573119876+ 120583)

gt 0

(59)


4gt 0





1205791(1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119899119901lowast

119899(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(60)

(1 minus 1205791) (1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119902119904lowast

119902(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(61)

where

1198615= 120588119899+ 120588119902+ 120591119899+ 1205911199021198601+ 120585119899+ 1205851199021198603+ 1198861198891198607

1198616= 1205881199021198861+ 1205911199021198602+ 1205851199021198604+ 1198861198891198608

(62)



(1 minus 1198611119901lowast

119899minus 1198612119904lowast

119902) (120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast

1199021205791) = 0 (63)


119899minus 1198612119904lowast

119902= 0 or (ii)

120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast


system


119899minus 1198614119904lowast

119902= 0


119899minus 1198612119904lowast

119902= 0

(64)



119901lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902 (65)



119904lowast

119902= 0

or 119904lowast119902=1198617

1198618

=(1 minus 120579

1) 120572119899(1198770minus 1)

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612) (R minus 1)

= 119909 (1198770minus 1

R minus 1)

(66)

where

1198617= 120572119902120572119899(1 minus 120579

1) ((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

) minus 1)

= 120572119902120572119899(1 minus 120579

1) (1198770minus 1)

1198618= 120572119902[(12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614)

minus (12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612)] = 120572

119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612)((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (

12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614

12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612

) minus 1) = 120572119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612) (1198770119911 minus 1)

(67)

with

119909 =(1 minus 120579

1) 120572119899

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612)

119910 =

1205791120572119902119909

(1 minus 1205791) 120572119899

119911 =

(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

(68)

1198770=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

R =1198770(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

= 1199111198770

(69)


1198612= 11988781198868119886411988611198862=1205796120572119876

120583

1205797120573119876

(120573119876+ 120583)

((1 minus120583

119887)

sdot120575119876

(119903119871119876+ 120575119876+ 120583)

120574119876

(119903119864119876+ 120574119876+ 120583)) lt

1205796120572119876

120583

sdot1205797120573119876

(120573119876+ 120583)

lt1205796120572119876

120583

(1205797120573119876+ 120583)

(120573119876+ 120583)

+ 1 = 1198614

(70)



3lt 1198611 we will


1120572119902(1minus120579

1)120572119899)(1198611minus1198613)


system as follows




0gt 1 and the


0




119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119894lowast

119899= 119888lowast

119899= 119904lowast

119899= 119901lowast

119899= 119910(

1198770minus 1

R minus 1)

119901lowast

119902= (119910 + 119886

1119909) (1198770minus 1

R minus 1)

119888lowast

119902= (1198601119910 + 119860

2119909) (1198770minus 1

R minus 1)

119894lowast

119902= (1198603119910 + 119860

4119909) (1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

119903lowast

119889= (1198607119910 + 119860

8119909) (1198770minus 1

R minus 1)

119904lowast= 1 minus (119861

3119910 + 1198614119909) (1198770minus 1

R minus 1)


1119910 + 1198612119909) (1198770minus 1

R minus 1)

(71)



119889119891119890=


119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast







119869dfe =

((((((((((((((((((((((

(


1198890

0 minus1205721198990 120579

1120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 0 minus120572119902

1205791120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 120573119899

0 minus120573119899

0 0 0 0 0 0 0 0

0 1205731199021198861120573119902

0 minus120573119902

0 0 0 0 0 0 0

0 0 0 120574119899

0 minus1205741198990 0 0 0 0 0

0 0 0 120574119902

1198862120574119902

0 minus1205741199020 0 0 0 0

0 0 0 0 0 120575119899

0 minus1205751198990 0 0 0

0 0 0 0 0 12057511990211988641205751199021198863120575119902minus1205751199020 0 0

0 0 0 0 0 1 11988661198865

1198867minus1 0 0

0 0 0 0 0 0 0 12058311988912058311988911988680 minus120583

1198890

0 0 0 0 0 0 0 0 0 0 minus1198878minus1

))))))))))))))))))))))

)

(72)

where 1205791= 1 minus 120579



119875 (120577) =1003816100381610038161003816119869dfe minus 120577I

1003816100381610038161003816 = (120577 + 1)31198759(120577) = 0 (73)


1198759 (120577) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus120572119899minus 120577 0 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

0 minus120572119902minus 120577 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

120573119899

0 minus120573119899minus 120577 0 0 0 0 0 0

120573119902

1198861120573119902

0 minus120573119902minus 120577 0 0 0 0 0

0 0 120574119899

0 minus120574119899minus 120577 0 0 0 0

0 0 120574119902

1198862120574119902

0 minus120574119902minus 120577 0 0 0

0 0 0 0 120575119899

0 minus120575119899minus 120577 0 0

0 0 0 0 120575119902

1198864120575119902

1198863120575119902minus120575119902minus 120577 0

0 0 0 0 0 0 120583119889

1205831198891198868minus120583119889minus 120577

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(74)


119875 (120577) = (120577 + 1)3

9

sum

119896=0

119888119894120577119894 (75)

where1198889= 1

1198888= 120572119899+ 120572119902+ 120573119899+ 120573119902+ 120574119899+ 120574119902+ 120575119899+ 120575119902+ 120583119889

1198880= 1205731198991205731199021205741198991205741199021205751198991205751199021205831198891205721198991205721199021 minus

12057911198615

120572119899


1) 1198616

120572119902

= 120573119899120573119902120574119899120574119902120575119899120575119902120583119889120572119899120572119902(1 minus 119877

0)

(76)




0= 1 to





0 1198770




0= 1


0specifically


0which is fixedRmay






0is the


119899 119901119902 119888119899 119888119902 119894119899 119894119902 119903119903 119903119889 and ℎ The




F =

(((((((((((((((((

(

0

1205791120582119904

(1 minus 1205791) 120582119904

0

0

0

0

0

0

0

0

0

)))))))))))))))))

)

V =

((((((((((((((((((((((((((((((((

(


120572119899119904119899

120572119902119904119902

120573119899(minus119904119899+ 119901119899)

120573119902(minus119904119899minus 1198861119904119902+ 119901119902)

120574119899(minus119901119899+ 119888119899)

120574119902(minus119901119899minus 1198862119901119902+ 119888119902)

120575119899(minus119888119899+ 119894119899)



minus120583119889119894119899minus 1205831198891198868119894119902+ 120583119889119903119889

minus1 + ℎ + 1198878119903119889

))))))))))))))))))))))))))))))))

)

(77)


119894119895=

120597F119894120597119909119895and (119881)

119894119895= 120597V



119901119902= 119888119899= 119888119902= 119894119899= 119894119902= 119903119903= 119903119889= 0 The basic


1198770= 984858 (119865119881

minus1) =

1

120572119899120572119902

[1205721199021205791(1

+ (1 + 1198863+ (1 + 119886

2) 1198864) 119886119889

+ 120585119902(11988621198864+ 1198863+ 1198864+ 1) + 119886

2120591119902+ 120585119899+ 120588119899+ 120588119902

+ 120591119899+ 120591119902) minus 120572

119899(1205791minus 1) (119886

1120588119902

+ 1198861[11988621198864(1198868119886119889+ 120585119902) + 1198862120591119902])]

=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

(78)






119899(which








120583


120583 + rate120583

asymp large (79)


119899 1120573119902



119901119899= 119888119899= 119894119899= 119904119899

119901119902= 119904119899+ 1198861119904119902

119888119902= 1198601119904119899+ 1198602119904119902

119894119902= 1198603119904119899+ 1198604119904119902

119903119889= 1198607119904119899+ 1198608119904119902

(80)


119889119905= 1 minus 120582119904 + (120572


119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (82)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (83)

119889119903119903

119889119905= 1198605119904119899+ 1198606119904119902minus 119903119903 (84)

119889ℎ

119889119905= 1 minus ℎ minus 119861

1119904119899minus 1198612119904119902 (85)


120582119904 = (120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)) 119904 = (119861

5(119904119899

ℎ)

+ 1198616(

119904119902

ℎ)) 119904

(86)





119899 119904lowast

119902 119903lowast


119864dfe = (1 0 0 0 1)

119864119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast)

(87)


119904lowast

119902=1198617

1198618

119904lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902

119903lowast

119903= 1198605119904lowast

119899+ 1198606119904lowast

119902

119904lowast= 1 minus 119861

3119904lowast

119899minus 1198614119904lowast

119902


1119904lowast

119899minus 1198612119904lowast

119902

(88)


3119910 + 119861

4119909 and

1198611119910 + 1198612119909 = 1 we get

119904lowast= (119911 minus 1

R minus 1)

119904lowast

119899= 119910(

1198770minus 1

R minus 1)

119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

ℎlowast= ((119911 minus 1) 119877

0

R minus 1)

(89)


119899 119904lowast

119902 119903lowast




119869 (xlowast) =((

(


4(xlowast) + 120572


119902minus 11986140 minus119862

6(xlowast)

12057911198623(xlowast) 120579


11989912057911198625(xlowast) 0 120579

11198626(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) 120579


1199020 12057911198626(xlowast)

0 1198605

1198606

minus1 0

0 minus1198611

minus1198612

0 minus1

))

)

(90)

where 1205791= 1 minus 120579

1and

1198623(xlowast) = 119861

5

119904lowast

119899

ℎlowast+ 1198616

119904lowast

119902

ℎlowast=120572119899119910 (1198770minus 1)

1205791(119911 minus 1)

1198624(xlowast) = 119861

5

119904lowast

ℎlowast=1198615

1198770

1198625(xlowast) = 119861

6

119904lowast

ℎlowast=1198616

1198770

1198626(xlowast) = minus(119861

5


119899

ℎlowast2+ 1198616


119902

ℎlowast2) = minus

120572119899119910 (1198770minus 1)

1205791 (119911 minus 1) 1198770

(91)


1198755(120577 xlowast)

= (120577 + 1)2(1205773+ 1198762(xlowast) 1205772 + 119876

1(xlowast) 120577 + 119876

0(xlowast))

= 0

(92)

where

1198762(xlowast) = 120572

119899+ 120572119902+ 1198623(xlowast) minus 119862

4(xlowast) 120579


1

+ 1

1198761(xlowast) = 120572

119902


1199021198624(xlowast) + 119862

4(xlowast))

+ 11986121198626(xlowast) 120579

1+ 1198623(xlowast) (120572

1199021205791+ 11986131205791+ 11986141205791)

+ 120572119899(120572119902+ 12057911198623(xlowast) minus 119862

5(xlowast) 120579

1+ 1) minus 119862

5(xlowast)

sdot 1205791

1198760(xlowast) = 120572

1199021205791(11986111198626(xlowast) + 119861


4(xlowast))

+ 120572119899(1205791(11986121198626(xlowast) + 119861


5(xlowast)) + 120572

119902)

(93)




119902=

119903lowast




1198616 1198626(xdfe) = 0 and (92) becomes

1198755(120577 xdfe) = (120577 + 1)

3(1205772+ 119876dfe120577 + 119877dfe) = 0 (94)

where


= 120572119899(1 minus 119877

0) + 120572119902+ (1 minus 120579

1) 1198616(

120572119899minus 120572119902

120572119902

)

119877dfe = 120572119902 (120572119899 minus 11986151205791) + 1205721198991198616 (1205791 minus 1)

= 120572119902120572119899(1 minus 119877

0)

(95)


radic1198762







=


119899 119904lowast

119902 119903lowast


and we have

1198755(120577 xlowast) = (120577 + 1)2 (1205773 + 119875x

119890119890

1205772+ 119876x

119890119890

120577 + 119877x119890119890

) = 0 (96)

where


119875x119890119890

=


12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)


119890119890will be the


119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)




119899= 0 and we


1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)





grouping 1198616




1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)





CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References





































119899and

119904lowast


119901lowast

119902(119901lowast

119899 119904lowast

119902) = 119901lowast

119899+ 1198861119904lowast

119902

119888lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

1119901lowast

119899+ 1198602119904lowast

119902

119894lowast

119902(119901lowast

119899 119904lowast

119902) = 119860

3119901lowast

119899+ 1198604119904lowast

119902

119903lowast

119903(119901lowast

119899 119904lowast

119902) = 119860

5119901lowast

119899+ 1198606119904lowast

119902

119903lowast

119889(119901lowast

119899 119904lowast

119902) = 119860

7119901lowast

119899+ 1198608119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902


119899 119904lowast

119902) = 1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902

ℎlowast

119897(119901lowast

119899 119904lowast

119902) = 1 minus (119887

5+ 11988761198603) 119901lowast

119899minus 11988761198604119904lowast

119902

(57)

where

1198601= 1 + 119886

2

1198602= 11988611198862

1198603= 1 + 119886

3+ 11988641198601

1198604= 11988641198602

1198605= 1 + 119886

5+ 11988661198601+ 11988671198603

1198606= 11988661198602+ 11988671198604

1198607= 1 + 119886

81198603

1198608= 11988681198604

1198611= 11988781198607

1198612= 11988781198608


1198614= 120572119902minus 1198873minus 11988741198861

(58)




3and 119861


1198613= 1

+120572119873

120583(1205731198731205792119886(1205793minus 1)

120573119873+ 120583

+1205731198761205792119887(1205797minus 1)

120573119876+ 120583

+ 1205792)

gt 0 since 1205792= 1205792119886+ 1205792119887

1198614=1205721198761205796(1205731198761205797+ 120583) + 120583 (120573

119876+ 120583)

120583 (120573119876+ 120583)

gt 0

(59)


4gt 0





1205791(1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119899119901lowast

119899(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(60)

(1 minus 1205791) (1198615119901lowast

119899+ 1198616119904lowast

119902) (1 minus 119861

3119901lowast

119899minus 1198614119904lowast

119902)

= 120572119902119904lowast

119902(1 minus 119861

1119901lowast

119899minus 1198612119904lowast

119902)

(61)

where

1198615= 120588119899+ 120588119902+ 120591119899+ 1205911199021198601+ 120585119899+ 1205851199021198603+ 1198861198891198607

1198616= 1205881199021198861+ 1205911199021198602+ 1205851199021198604+ 1198861198891198608

(62)



(1 minus 1198611119901lowast

119899minus 1198612119904lowast

119902) (120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast

1199021205791) = 0 (63)


119899minus 1198612119904lowast

119902= 0 or (ii)

120572119899119901lowast

119899(1 minus 120579

1) minus 120572119902119904lowast


system


119899minus 1198614119904lowast

119902= 0


119899minus 1198612119904lowast

119902= 0

(64)



119901lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902 (65)



119904lowast

119902= 0

or 119904lowast119902=1198617

1198618

=(1 minus 120579

1) 120572119899(1198770minus 1)

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612) (R minus 1)

= 119909 (1198770minus 1

R minus 1)

(66)

where

1198617= 120572119902120572119899(1 minus 120579

1) ((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

) minus 1)

= 120572119902120572119899(1 minus 120579

1) (1198770minus 1)

1198618= 120572119902[(12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614)

minus (12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612)] = 120572

119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612)((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (

12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614

12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612

) minus 1) = 120572119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612) (1198770119911 minus 1)

(67)

with

119909 =(1 minus 120579

1) 120572119899

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612)

119910 =

1205791120572119902119909

(1 minus 1205791) 120572119899

119911 =

(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

(68)

1198770=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

R =1198770(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

= 1199111198770

(69)


1198612= 11988781198868119886411988611198862=1205796120572119876

120583

1205797120573119876

(120573119876+ 120583)

((1 minus120583

119887)

sdot120575119876

(119903119871119876+ 120575119876+ 120583)

120574119876

(119903119864119876+ 120574119876+ 120583)) lt

1205796120572119876

120583

sdot1205797120573119876

(120573119876+ 120583)

lt1205796120572119876

120583

(1205797120573119876+ 120583)

(120573119876+ 120583)

+ 1 = 1198614

(70)



3lt 1198611 we will


1120572119902(1minus120579

1)120572119899)(1198611minus1198613)


system as follows




0gt 1 and the


0




119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119894lowast

119899= 119888lowast

119899= 119904lowast

119899= 119901lowast

119899= 119910(

1198770minus 1

R minus 1)

119901lowast

119902= (119910 + 119886

1119909) (1198770minus 1

R minus 1)

119888lowast

119902= (1198601119910 + 119860

2119909) (1198770minus 1

R minus 1)

119894lowast

119902= (1198603119910 + 119860

4119909) (1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

119903lowast

119889= (1198607119910 + 119860

8119909) (1198770minus 1

R minus 1)

119904lowast= 1 minus (119861

3119910 + 1198614119909) (1198770minus 1

R minus 1)


1119910 + 1198612119909) (1198770minus 1

R minus 1)

(71)



119889119891119890=


119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast







119869dfe =

((((((((((((((((((((((

(


1198890

0 minus1205721198990 120579

1120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 0 minus120572119902

1205791120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 120573119899

0 minus120573119899

0 0 0 0 0 0 0 0

0 1205731199021198861120573119902

0 minus120573119902

0 0 0 0 0 0 0

0 0 0 120574119899

0 minus1205741198990 0 0 0 0 0

0 0 0 120574119902

1198862120574119902

0 minus1205741199020 0 0 0 0

0 0 0 0 0 120575119899

0 minus1205751198990 0 0 0

0 0 0 0 0 12057511990211988641205751199021198863120575119902minus1205751199020 0 0

0 0 0 0 0 1 11988661198865

1198867minus1 0 0

0 0 0 0 0 0 0 12058311988912058311988911988680 minus120583

1198890

0 0 0 0 0 0 0 0 0 0 minus1198878minus1

))))))))))))))))))))))

)

(72)

where 1205791= 1 minus 120579



119875 (120577) =1003816100381610038161003816119869dfe minus 120577I

1003816100381610038161003816 = (120577 + 1)31198759(120577) = 0 (73)


1198759 (120577) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus120572119899minus 120577 0 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

0 minus120572119902minus 120577 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

120573119899

0 minus120573119899minus 120577 0 0 0 0 0 0

120573119902

1198861120573119902

0 minus120573119902minus 120577 0 0 0 0 0

0 0 120574119899

0 minus120574119899minus 120577 0 0 0 0

0 0 120574119902

1198862120574119902

0 minus120574119902minus 120577 0 0 0

0 0 0 0 120575119899

0 minus120575119899minus 120577 0 0

0 0 0 0 120575119902

1198864120575119902

1198863120575119902minus120575119902minus 120577 0

0 0 0 0 0 0 120583119889

1205831198891198868minus120583119889minus 120577

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(74)


119875 (120577) = (120577 + 1)3

9

sum

119896=0

119888119894120577119894 (75)

where1198889= 1

1198888= 120572119899+ 120572119902+ 120573119899+ 120573119902+ 120574119899+ 120574119902+ 120575119899+ 120575119902+ 120583119889

1198880= 1205731198991205731199021205741198991205741199021205751198991205751199021205831198891205721198991205721199021 minus

12057911198615

120572119899


1) 1198616

120572119902

= 120573119899120573119902120574119899120574119902120575119899120575119902120583119889120572119899120572119902(1 minus 119877

0)

(76)




0= 1 to





0 1198770




0= 1


0specifically


0which is fixedRmay






0is the


119899 119901119902 119888119899 119888119902 119894119899 119894119902 119903119903 119903119889 and ℎ The




F =

(((((((((((((((((

(

0

1205791120582119904

(1 minus 1205791) 120582119904

0

0

0

0

0

0

0

0

0

)))))))))))))))))

)

V =

((((((((((((((((((((((((((((((((

(


120572119899119904119899

120572119902119904119902

120573119899(minus119904119899+ 119901119899)

120573119902(minus119904119899minus 1198861119904119902+ 119901119902)

120574119899(minus119901119899+ 119888119899)

120574119902(minus119901119899minus 1198862119901119902+ 119888119902)

120575119899(minus119888119899+ 119894119899)



minus120583119889119894119899minus 1205831198891198868119894119902+ 120583119889119903119889

minus1 + ℎ + 1198878119903119889

))))))))))))))))))))))))))))))))

)

(77)


119894119895=

120597F119894120597119909119895and (119881)

119894119895= 120597V



119901119902= 119888119899= 119888119902= 119894119899= 119894119902= 119903119903= 119903119889= 0 The basic


1198770= 984858 (119865119881

minus1) =

1

120572119899120572119902

[1205721199021205791(1

+ (1 + 1198863+ (1 + 119886

2) 1198864) 119886119889

+ 120585119902(11988621198864+ 1198863+ 1198864+ 1) + 119886

2120591119902+ 120585119899+ 120588119899+ 120588119902

+ 120591119899+ 120591119902) minus 120572

119899(1205791minus 1) (119886

1120588119902

+ 1198861[11988621198864(1198868119886119889+ 120585119902) + 1198862120591119902])]

=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

(78)






119899(which








120583


120583 + rate120583

asymp large (79)


119899 1120573119902



119901119899= 119888119899= 119894119899= 119904119899

119901119902= 119904119899+ 1198861119904119902

119888119902= 1198601119904119899+ 1198602119904119902

119894119902= 1198603119904119899+ 1198604119904119902

119903119889= 1198607119904119899+ 1198608119904119902

(80)


119889119905= 1 minus 120582119904 + (120572


119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (82)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (83)

119889119903119903

119889119905= 1198605119904119899+ 1198606119904119902minus 119903119903 (84)

119889ℎ

119889119905= 1 minus ℎ minus 119861

1119904119899minus 1198612119904119902 (85)


120582119904 = (120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)) 119904 = (119861

5(119904119899

ℎ)

+ 1198616(

119904119902

ℎ)) 119904

(86)





119899 119904lowast

119902 119903lowast


119864dfe = (1 0 0 0 1)

119864119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast)

(87)


119904lowast

119902=1198617

1198618

119904lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902

119903lowast

119903= 1198605119904lowast

119899+ 1198606119904lowast

119902

119904lowast= 1 minus 119861

3119904lowast

119899minus 1198614119904lowast

119902


1119904lowast

119899minus 1198612119904lowast

119902

(88)


3119910 + 119861

4119909 and

1198611119910 + 1198612119909 = 1 we get

119904lowast= (119911 minus 1

R minus 1)

119904lowast

119899= 119910(

1198770minus 1

R minus 1)

119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

ℎlowast= ((119911 minus 1) 119877

0

R minus 1)

(89)


119899 119904lowast

119902 119903lowast




119869 (xlowast) =((

(


4(xlowast) + 120572


119902minus 11986140 minus119862

6(xlowast)

12057911198623(xlowast) 120579


11989912057911198625(xlowast) 0 120579

11198626(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) 120579


1199020 12057911198626(xlowast)

0 1198605

1198606

minus1 0

0 minus1198611

minus1198612

0 minus1

))

)

(90)

where 1205791= 1 minus 120579

1and

1198623(xlowast) = 119861

5

119904lowast

119899

ℎlowast+ 1198616

119904lowast

119902

ℎlowast=120572119899119910 (1198770minus 1)

1205791(119911 minus 1)

1198624(xlowast) = 119861

5

119904lowast

ℎlowast=1198615

1198770

1198625(xlowast) = 119861

6

119904lowast

ℎlowast=1198616

1198770

1198626(xlowast) = minus(119861

5


119899

ℎlowast2+ 1198616


119902

ℎlowast2) = minus

120572119899119910 (1198770minus 1)

1205791 (119911 minus 1) 1198770

(91)


1198755(120577 xlowast)

= (120577 + 1)2(1205773+ 1198762(xlowast) 1205772 + 119876

1(xlowast) 120577 + 119876

0(xlowast))

= 0

(92)

where

1198762(xlowast) = 120572

119899+ 120572119902+ 1198623(xlowast) minus 119862

4(xlowast) 120579


1

+ 1

1198761(xlowast) = 120572

119902


1199021198624(xlowast) + 119862

4(xlowast))

+ 11986121198626(xlowast) 120579

1+ 1198623(xlowast) (120572

1199021205791+ 11986131205791+ 11986141205791)

+ 120572119899(120572119902+ 12057911198623(xlowast) minus 119862

5(xlowast) 120579

1+ 1) minus 119862

5(xlowast)

sdot 1205791

1198760(xlowast) = 120572

1199021205791(11986111198626(xlowast) + 119861


4(xlowast))

+ 120572119899(1205791(11986121198626(xlowast) + 119861


5(xlowast)) + 120572

119902)

(93)




119902=

119903lowast




1198616 1198626(xdfe) = 0 and (92) becomes

1198755(120577 xdfe) = (120577 + 1)

3(1205772+ 119876dfe120577 + 119877dfe) = 0 (94)

where


= 120572119899(1 minus 119877

0) + 120572119902+ (1 minus 120579

1) 1198616(

120572119899minus 120572119902

120572119902

)

119877dfe = 120572119902 (120572119899 minus 11986151205791) + 1205721198991198616 (1205791 minus 1)

= 120572119902120572119899(1 minus 119877

0)

(95)


radic1198762







=


119899 119904lowast

119902 119903lowast


and we have

1198755(120577 xlowast) = (120577 + 1)2 (1205773 + 119875x

119890119890

1205772+ 119876x

119890119890

120577 + 119877x119890119890

) = 0 (96)

where


119875x119890119890

=


12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)


119890119890will be the


119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)




119899= 0 and we


1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)





grouping 1198616




1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)





CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References





































119904lowast

119902= 0

or 119904lowast119902=1198617

1198618

=(1 minus 120579

1) 120572119899(1198770minus 1)

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612) (R minus 1)

= 119909 (1198770minus 1

R minus 1)

(66)

where

1198617= 120572119902120572119899(1 minus 120579

1) ((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

) minus 1)

= 120572119902120572119899(1 minus 120579

1) (1198770minus 1)

1198618= 120572119902[(12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614)

minus (12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612)] = 120572

119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612)((

12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

)

sdot (

12057211990212057911198613+ 120572119899(1 minus 120579

1) 1198614

12057911198611120572119902+ 120572119899(1 minus 120579

1) 1198612

) minus 1) = 120572119902(12057911198611120572119902

+ 120572119899(1 minus 120579

1) 1198612) (1198770119911 minus 1)

(67)

with

119909 =(1 minus 120579

1) 120572119899

(12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612)

119910 =

1205791120572119902119909

(1 minus 1205791) 120572119899

119911 =

(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

(68)

1198770=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

R =1198770(12057911205721199021198613+ (1 minus 120579

1) 1205721198991198614)

12057911205721199021198611+ (1 minus 120579

1) 1205721198991198612

= 1199111198770

(69)


1198612= 11988781198868119886411988611198862=1205796120572119876

120583

1205797120573119876

(120573119876+ 120583)

((1 minus120583

119887)

sdot120575119876

(119903119871119876+ 120575119876+ 120583)

120574119876

(119903119864119876+ 120574119876+ 120583)) lt

1205796120572119876

120583

sdot1205797120573119876

(120573119876+ 120583)

lt1205796120572119876

120583

(1205797120573119876+ 120583)

(120573119876+ 120583)

+ 1 = 1198614

(70)



3lt 1198611 we will


1120572119902(1minus120579

1)120572119899)(1198611minus1198613)


system as follows




0gt 1 and the


0




119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119894lowast

119899= 119888lowast

119899= 119904lowast

119899= 119901lowast

119899= 119910(

1198770minus 1

R minus 1)

119901lowast

119902= (119910 + 119886

1119909) (1198770minus 1

R minus 1)

119888lowast

119902= (1198601119910 + 119860

2119909) (1198770minus 1

R minus 1)

119894lowast

119902= (1198603119910 + 119860

4119909) (1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

119903lowast

119889= (1198607119910 + 119860

8119909) (1198770minus 1

R minus 1)

119904lowast= 1 minus (119861

3119910 + 1198614119909) (1198770minus 1

R minus 1)


1119910 + 1198612119909) (1198770minus 1

R minus 1)

(71)



119889119891119890=


119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119901lowast

119899 119901lowast

119902 119888lowast

119899 119888lowast

119902 119894lowast

119899 119894lowast

119902 119903lowast

119903 119903lowast







119869dfe =

((((((((((((((((((((((

(


1198890

0 minus1205721198990 120579

1120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 0 minus120572119902

1205791120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 120573119899

0 minus120573119899

0 0 0 0 0 0 0 0

0 1205731199021198861120573119902

0 minus120573119902

0 0 0 0 0 0 0

0 0 0 120574119899

0 minus1205741198990 0 0 0 0 0

0 0 0 120574119902

1198862120574119902

0 minus1205741199020 0 0 0 0

0 0 0 0 0 120575119899

0 minus1205751198990 0 0 0

0 0 0 0 0 12057511990211988641205751199021198863120575119902minus1205751199020 0 0

0 0 0 0 0 1 11988661198865

1198867minus1 0 0

0 0 0 0 0 0 0 12058311988912058311988911988680 minus120583

1198890

0 0 0 0 0 0 0 0 0 0 minus1198878minus1

))))))))))))))))))))))

)

(72)

where 1205791= 1 minus 120579



119875 (120577) =1003816100381610038161003816119869dfe minus 120577I

1003816100381610038161003816 = (120577 + 1)31198759(120577) = 0 (73)


1198759 (120577) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus120572119899minus 120577 0 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

0 minus120572119902minus 120577 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

120573119899

0 minus120573119899minus 120577 0 0 0 0 0 0

120573119902

1198861120573119902

0 minus120573119902minus 120577 0 0 0 0 0

0 0 120574119899

0 minus120574119899minus 120577 0 0 0 0

0 0 120574119902

1198862120574119902

0 minus120574119902minus 120577 0 0 0

0 0 0 0 120575119899

0 minus120575119899minus 120577 0 0

0 0 0 0 120575119902

1198864120575119902

1198863120575119902minus120575119902minus 120577 0

0 0 0 0 0 0 120583119889

1205831198891198868minus120583119889minus 120577

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(74)


119875 (120577) = (120577 + 1)3

9

sum

119896=0

119888119894120577119894 (75)

where1198889= 1

1198888= 120572119899+ 120572119902+ 120573119899+ 120573119902+ 120574119899+ 120574119902+ 120575119899+ 120575119902+ 120583119889

1198880= 1205731198991205731199021205741198991205741199021205751198991205751199021205831198891205721198991205721199021 minus

12057911198615

120572119899


1) 1198616

120572119902

= 120573119899120573119902120574119899120574119902120575119899120575119902120583119889120572119899120572119902(1 minus 119877

0)

(76)




0= 1 to





0 1198770




0= 1


0specifically


0which is fixedRmay






0is the


119899 119901119902 119888119899 119888119902 119894119899 119894119902 119903119903 119903119889 and ℎ The




F =

(((((((((((((((((

(

0

1205791120582119904

(1 minus 1205791) 120582119904

0

0

0

0

0

0

0

0

0

)))))))))))))))))

)

V =

((((((((((((((((((((((((((((((((

(


120572119899119904119899

120572119902119904119902

120573119899(minus119904119899+ 119901119899)

120573119902(minus119904119899minus 1198861119904119902+ 119901119902)

120574119899(minus119901119899+ 119888119899)

120574119902(minus119901119899minus 1198862119901119902+ 119888119902)

120575119899(minus119888119899+ 119894119899)



minus120583119889119894119899minus 1205831198891198868119894119902+ 120583119889119903119889

minus1 + ℎ + 1198878119903119889

))))))))))))))))))))))))))))))))

)

(77)


119894119895=

120597F119894120597119909119895and (119881)

119894119895= 120597V



119901119902= 119888119899= 119888119902= 119894119899= 119894119902= 119903119903= 119903119889= 0 The basic


1198770= 984858 (119865119881

minus1) =

1

120572119899120572119902

[1205721199021205791(1

+ (1 + 1198863+ (1 + 119886

2) 1198864) 119886119889

+ 120585119902(11988621198864+ 1198863+ 1198864+ 1) + 119886

2120591119902+ 120585119899+ 120588119899+ 120588119902

+ 120591119899+ 120591119902) minus 120572

119899(1205791minus 1) (119886

1120588119902

+ 1198861[11988621198864(1198868119886119889+ 120585119902) + 1198862120591119902])]

=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

(78)






119899(which








120583


120583 + rate120583

asymp large (79)


119899 1120573119902



119901119899= 119888119899= 119894119899= 119904119899

119901119902= 119904119899+ 1198861119904119902

119888119902= 1198601119904119899+ 1198602119904119902

119894119902= 1198603119904119899+ 1198604119904119902

119903119889= 1198607119904119899+ 1198608119904119902

(80)


119889119905= 1 minus 120582119904 + (120572


119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (82)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (83)

119889119903119903

119889119905= 1198605119904119899+ 1198606119904119902minus 119903119903 (84)

119889ℎ

119889119905= 1 minus ℎ minus 119861

1119904119899minus 1198612119904119902 (85)


120582119904 = (120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)) 119904 = (119861

5(119904119899

ℎ)

+ 1198616(

119904119902

ℎ)) 119904

(86)





119899 119904lowast

119902 119903lowast


119864dfe = (1 0 0 0 1)

119864119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast)

(87)


119904lowast

119902=1198617

1198618

119904lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902

119903lowast

119903= 1198605119904lowast

119899+ 1198606119904lowast

119902

119904lowast= 1 minus 119861

3119904lowast

119899minus 1198614119904lowast

119902


1119904lowast

119899minus 1198612119904lowast

119902

(88)


3119910 + 119861

4119909 and

1198611119910 + 1198612119909 = 1 we get

119904lowast= (119911 minus 1

R minus 1)

119904lowast

119899= 119910(

1198770minus 1

R minus 1)

119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

ℎlowast= ((119911 minus 1) 119877

0

R minus 1)

(89)


119899 119904lowast

119902 119903lowast




119869 (xlowast) =((

(


4(xlowast) + 120572


119902minus 11986140 minus119862

6(xlowast)

12057911198623(xlowast) 120579


11989912057911198625(xlowast) 0 120579

11198626(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) 120579


1199020 12057911198626(xlowast)

0 1198605

1198606

minus1 0

0 minus1198611

minus1198612

0 minus1

))

)

(90)

where 1205791= 1 minus 120579

1and

1198623(xlowast) = 119861

5

119904lowast

119899

ℎlowast+ 1198616

119904lowast

119902

ℎlowast=120572119899119910 (1198770minus 1)

1205791(119911 minus 1)

1198624(xlowast) = 119861

5

119904lowast

ℎlowast=1198615

1198770

1198625(xlowast) = 119861

6

119904lowast

ℎlowast=1198616

1198770

1198626(xlowast) = minus(119861

5


119899

ℎlowast2+ 1198616


119902

ℎlowast2) = minus

120572119899119910 (1198770minus 1)

1205791 (119911 minus 1) 1198770

(91)


1198755(120577 xlowast)

= (120577 + 1)2(1205773+ 1198762(xlowast) 1205772 + 119876

1(xlowast) 120577 + 119876

0(xlowast))

= 0

(92)

where

1198762(xlowast) = 120572

119899+ 120572119902+ 1198623(xlowast) minus 119862

4(xlowast) 120579


1

+ 1

1198761(xlowast) = 120572

119902


1199021198624(xlowast) + 119862

4(xlowast))

+ 11986121198626(xlowast) 120579

1+ 1198623(xlowast) (120572

1199021205791+ 11986131205791+ 11986141205791)

+ 120572119899(120572119902+ 12057911198623(xlowast) minus 119862

5(xlowast) 120579

1+ 1) minus 119862

5(xlowast)

sdot 1205791

1198760(xlowast) = 120572

1199021205791(11986111198626(xlowast) + 119861


4(xlowast))

+ 120572119899(1205791(11986121198626(xlowast) + 119861


5(xlowast)) + 120572

119902)

(93)




119902=

119903lowast




1198616 1198626(xdfe) = 0 and (92) becomes

1198755(120577 xdfe) = (120577 + 1)

3(1205772+ 119876dfe120577 + 119877dfe) = 0 (94)

where


= 120572119899(1 minus 119877

0) + 120572119902+ (1 minus 120579

1) 1198616(

120572119899minus 120572119902

120572119902

)

119877dfe = 120572119902 (120572119899 minus 11986151205791) + 1205721198991198616 (1205791 minus 1)

= 120572119902120572119899(1 minus 119877

0)

(95)


radic1198762







=


119899 119904lowast

119902 119903lowast


and we have

1198755(120577 xlowast) = (120577 + 1)2 (1205773 + 119875x

119890119890

1205772+ 119876x

119890119890

120577 + 119877x119890119890

) = 0 (96)

where


119875x119890119890

=


12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)


119890119890will be the


119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)




119899= 0 and we


1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)





grouping 1198616




1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)





CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References




































119869dfe =

((((((((((((((((((((((

(


1198890

0 minus1205721198990 120579

1120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 0 minus120572119902

1205791120588119899

1205791120588119902

12057911205911198991205791120591119902120579112058511989912057911205851199020 11988611988912057910

0 120573119899

0 minus120573119899

0 0 0 0 0 0 0 0

0 1205731199021198861120573119902

0 minus120573119902

0 0 0 0 0 0 0

0 0 0 120574119899

0 minus1205741198990 0 0 0 0 0

0 0 0 120574119902

1198862120574119902

0 minus1205741199020 0 0 0 0

0 0 0 0 0 120575119899

0 minus1205751198990 0 0 0

0 0 0 0 0 12057511990211988641205751199021198863120575119902minus1205751199020 0 0

0 0 0 0 0 1 11988661198865

1198867minus1 0 0

0 0 0 0 0 0 0 12058311988912058311988911988680 minus120583

1198890

0 0 0 0 0 0 0 0 0 0 minus1198878minus1

))))))))))))))))))))))

)

(72)

where 1205791= 1 minus 120579



119875 (120577) =1003816100381610038161003816119869dfe minus 120577I

1003816100381610038161003816 = (120577 + 1)31198759(120577) = 0 (73)


1198759 (120577) =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus120572119899minus 120577 0 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

0 minus120572119902minus 120577 120579

1120588119899

1205791120588119902

1205791120591119899

1205791120591119902

1205791120585119899

1205791120585119902

1198861198891205791

120573119899

0 minus120573119899minus 120577 0 0 0 0 0 0

120573119902

1198861120573119902

0 minus120573119902minus 120577 0 0 0 0 0

0 0 120574119899

0 minus120574119899minus 120577 0 0 0 0

0 0 120574119902

1198862120574119902

0 minus120574119902minus 120577 0 0 0

0 0 0 0 120575119899

0 minus120575119899minus 120577 0 0

0 0 0 0 120575119902

1198864120575119902

1198863120575119902minus120575119902minus 120577 0

0 0 0 0 0 0 120583119889

1205831198891198868minus120583119889minus 120577

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(74)


119875 (120577) = (120577 + 1)3

9

sum

119896=0

119888119894120577119894 (75)

where1198889= 1

1198888= 120572119899+ 120572119902+ 120573119899+ 120573119902+ 120574119899+ 120574119902+ 120575119899+ 120575119902+ 120583119889

1198880= 1205731198991205731199021205741198991205741199021205751198991205751199021205831198891205721198991205721199021 minus

12057911198615

120572119899


1) 1198616

120572119902

= 120573119899120573119902120574119899120574119902120575119899120575119902120583119889120572119899120572119902(1 minus 119877

0)

(76)




0= 1 to





0 1198770




0= 1


0specifically


0which is fixedRmay






0is the


119899 119901119902 119888119899 119888119902 119894119899 119894119902 119903119903 119903119889 and ℎ The




F =

(((((((((((((((((

(

0

1205791120582119904

(1 minus 1205791) 120582119904

0

0

0

0

0

0

0

0

0

)))))))))))))))))

)

V =

((((((((((((((((((((((((((((((((

(


120572119899119904119899

120572119902119904119902

120573119899(minus119904119899+ 119901119899)

120573119902(minus119904119899minus 1198861119904119902+ 119901119902)

120574119899(minus119901119899+ 119888119899)

120574119902(minus119901119899minus 1198862119901119902+ 119888119902)

120575119899(minus119888119899+ 119894119899)



minus120583119889119894119899minus 1205831198891198868119894119902+ 120583119889119903119889

minus1 + ℎ + 1198878119903119889

))))))))))))))))))))))))))))))))

)

(77)


119894119895=

120597F119894120597119909119895and (119881)

119894119895= 120597V



119901119902= 119888119899= 119888119902= 119894119899= 119894119902= 119903119903= 119903119889= 0 The basic


1198770= 984858 (119865119881

minus1) =

1

120572119899120572119902

[1205721199021205791(1

+ (1 + 1198863+ (1 + 119886

2) 1198864) 119886119889

+ 120585119902(11988621198864+ 1198863+ 1198864+ 1) + 119886

2120591119902+ 120585119899+ 120588119899+ 120588119902

+ 120591119899+ 120591119902) minus 120572

119899(1205791minus 1) (119886

1120588119902

+ 1198861[11988621198864(1198868119886119889+ 120585119902) + 1198862120591119902])]

=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

(78)






119899(which








120583


120583 + rate120583

asymp large (79)


119899 1120573119902



119901119899= 119888119899= 119894119899= 119904119899

119901119902= 119904119899+ 1198861119904119902

119888119902= 1198601119904119899+ 1198602119904119902

119894119902= 1198603119904119899+ 1198604119904119902

119903119889= 1198607119904119899+ 1198608119904119902

(80)


119889119905= 1 minus 120582119904 + (120572


119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (82)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (83)

119889119903119903

119889119905= 1198605119904119899+ 1198606119904119902minus 119903119903 (84)

119889ℎ

119889119905= 1 minus ℎ minus 119861

1119904119899minus 1198612119904119902 (85)


120582119904 = (120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)) 119904 = (119861

5(119904119899

ℎ)

+ 1198616(

119904119902

ℎ)) 119904

(86)





119899 119904lowast

119902 119903lowast


119864dfe = (1 0 0 0 1)

119864119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast)

(87)


119904lowast

119902=1198617

1198618

119904lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902

119903lowast

119903= 1198605119904lowast

119899+ 1198606119904lowast

119902

119904lowast= 1 minus 119861

3119904lowast

119899minus 1198614119904lowast

119902


1119904lowast

119899minus 1198612119904lowast

119902

(88)


3119910 + 119861

4119909 and

1198611119910 + 1198612119909 = 1 we get

119904lowast= (119911 minus 1

R minus 1)

119904lowast

119899= 119910(

1198770minus 1

R minus 1)

119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

ℎlowast= ((119911 minus 1) 119877

0

R minus 1)

(89)


119899 119904lowast

119902 119903lowast




119869 (xlowast) =((

(


4(xlowast) + 120572


119902minus 11986140 minus119862

6(xlowast)

12057911198623(xlowast) 120579


11989912057911198625(xlowast) 0 120579

11198626(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) 120579


1199020 12057911198626(xlowast)

0 1198605

1198606

minus1 0

0 minus1198611

minus1198612

0 minus1

))

)

(90)

where 1205791= 1 minus 120579

1and

1198623(xlowast) = 119861

5

119904lowast

119899

ℎlowast+ 1198616

119904lowast

119902

ℎlowast=120572119899119910 (1198770minus 1)

1205791(119911 minus 1)

1198624(xlowast) = 119861

5

119904lowast

ℎlowast=1198615

1198770

1198625(xlowast) = 119861

6

119904lowast

ℎlowast=1198616

1198770

1198626(xlowast) = minus(119861

5


119899

ℎlowast2+ 1198616


119902

ℎlowast2) = minus

120572119899119910 (1198770minus 1)

1205791 (119911 minus 1) 1198770

(91)


1198755(120577 xlowast)

= (120577 + 1)2(1205773+ 1198762(xlowast) 1205772 + 119876

1(xlowast) 120577 + 119876

0(xlowast))

= 0

(92)

where

1198762(xlowast) = 120572

119899+ 120572119902+ 1198623(xlowast) minus 119862

4(xlowast) 120579


1

+ 1

1198761(xlowast) = 120572

119902


1199021198624(xlowast) + 119862

4(xlowast))

+ 11986121198626(xlowast) 120579

1+ 1198623(xlowast) (120572

1199021205791+ 11986131205791+ 11986141205791)

+ 120572119899(120572119902+ 12057911198623(xlowast) minus 119862

5(xlowast) 120579

1+ 1) minus 119862

5(xlowast)

sdot 1205791

1198760(xlowast) = 120572

1199021205791(11986111198626(xlowast) + 119861


4(xlowast))

+ 120572119899(1205791(11986121198626(xlowast) + 119861


5(xlowast)) + 120572

119902)

(93)




119902=

119903lowast




1198616 1198626(xdfe) = 0 and (92) becomes

1198755(120577 xdfe) = (120577 + 1)

3(1205772+ 119876dfe120577 + 119877dfe) = 0 (94)

where


= 120572119899(1 minus 119877

0) + 120572119902+ (1 minus 120579

1) 1198616(

120572119899minus 120572119902

120572119902

)

119877dfe = 120572119902 (120572119899 minus 11986151205791) + 1205721198991198616 (1205791 minus 1)

= 120572119902120572119899(1 minus 119877

0)

(95)


radic1198762







=


119899 119904lowast

119902 119903lowast


and we have

1198755(120577 xlowast) = (120577 + 1)2 (1205773 + 119875x

119890119890

1205772+ 119876x

119890119890

120577 + 119877x119890119890

) = 0 (96)

where


119875x119890119890

=


12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)


119890119890will be the


119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)




119899= 0 and we


1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)





grouping 1198616




1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)





CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References





































0 1198770




0= 1


0specifically


0which is fixedRmay






0is the


119899 119901119902 119888119899 119888119902 119894119899 119894119902 119903119903 119903119889 and ℎ The




F =

(((((((((((((((((

(

0

1205791120582119904

(1 minus 1205791) 120582119904

0

0

0

0

0

0

0

0

0

)))))))))))))))))

)

V =

((((((((((((((((((((((((((((((((

(


120572119899119904119899

120572119902119904119902

120573119899(minus119904119899+ 119901119899)

120573119902(minus119904119899minus 1198861119904119902+ 119901119902)

120574119899(minus119901119899+ 119888119899)

120574119902(minus119901119899minus 1198862119901119902+ 119888119902)

120575119899(minus119888119899+ 119894119899)



minus120583119889119894119899minus 1205831198891198868119894119902+ 120583119889119903119889

minus1 + ℎ + 1198878119903119889

))))))))))))))))))))))))))))))))

)

(77)


119894119895=

120597F119894120597119909119895and (119881)

119894119895= 120597V



119901119902= 119888119899= 119888119902= 119894119899= 119894119902= 119903119903= 119903119889= 0 The basic


1198770= 984858 (119865119881

minus1) =

1

120572119899120572119902

[1205721199021205791(1

+ (1 + 1198863+ (1 + 119886

2) 1198864) 119886119889

+ 120585119902(11988621198864+ 1198863+ 1198864+ 1) + 119886

2120591119902+ 120585119899+ 120588119899+ 120588119902

+ 120591119899+ 120591119902) minus 120572

119899(1205791minus 1) (119886

1120588119902

+ 1198861[11988621198864(1198868119886119889+ 120585119902) + 1198862120591119902])]

=12057911198615

120572119899

+(1 minus 120579

1) 1198616

120572119902

(78)






119899(which








120583


120583 + rate120583

asymp large (79)


119899 1120573119902



119901119899= 119888119899= 119894119899= 119904119899

119901119902= 119904119899+ 1198861119904119902

119888119902= 1198601119904119899+ 1198602119904119902

119894119902= 1198603119904119899+ 1198604119904119902

119903119889= 1198607119904119899+ 1198608119904119902

(80)


119889119905= 1 minus 120582119904 + (120572


119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (82)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (83)

119889119903119903

119889119905= 1198605119904119899+ 1198606119904119902minus 119903119903 (84)

119889ℎ

119889119905= 1 minus ℎ minus 119861

1119904119899minus 1198612119904119902 (85)


120582119904 = (120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)) 119904 = (119861

5(119904119899

ℎ)

+ 1198616(

119904119902

ℎ)) 119904

(86)





119899 119904lowast

119902 119903lowast


119864dfe = (1 0 0 0 1)

119864119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast)

(87)


119904lowast

119902=1198617

1198618

119904lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902

119903lowast

119903= 1198605119904lowast

119899+ 1198606119904lowast

119902

119904lowast= 1 minus 119861

3119904lowast

119899minus 1198614119904lowast

119902


1119904lowast

119899minus 1198612119904lowast

119902

(88)


3119910 + 119861

4119909 and

1198611119910 + 1198612119909 = 1 we get

119904lowast= (119911 minus 1

R minus 1)

119904lowast

119899= 119910(

1198770minus 1

R minus 1)

119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

ℎlowast= ((119911 minus 1) 119877

0

R minus 1)

(89)


119899 119904lowast

119902 119903lowast




119869 (xlowast) =((

(


4(xlowast) + 120572


119902minus 11986140 minus119862

6(xlowast)

12057911198623(xlowast) 120579


11989912057911198625(xlowast) 0 120579

11198626(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) 120579


1199020 12057911198626(xlowast)

0 1198605

1198606

minus1 0

0 minus1198611

minus1198612

0 minus1

))

)

(90)

where 1205791= 1 minus 120579

1and

1198623(xlowast) = 119861

5

119904lowast

119899

ℎlowast+ 1198616

119904lowast

119902

ℎlowast=120572119899119910 (1198770minus 1)

1205791(119911 minus 1)

1198624(xlowast) = 119861

5

119904lowast

ℎlowast=1198615

1198770

1198625(xlowast) = 119861

6

119904lowast

ℎlowast=1198616

1198770

1198626(xlowast) = minus(119861

5


119899

ℎlowast2+ 1198616


119902

ℎlowast2) = minus

120572119899119910 (1198770minus 1)

1205791 (119911 minus 1) 1198770

(91)


1198755(120577 xlowast)

= (120577 + 1)2(1205773+ 1198762(xlowast) 1205772 + 119876

1(xlowast) 120577 + 119876

0(xlowast))

= 0

(92)

where

1198762(xlowast) = 120572

119899+ 120572119902+ 1198623(xlowast) minus 119862

4(xlowast) 120579


1

+ 1

1198761(xlowast) = 120572

119902


1199021198624(xlowast) + 119862

4(xlowast))

+ 11986121198626(xlowast) 120579

1+ 1198623(xlowast) (120572

1199021205791+ 11986131205791+ 11986141205791)

+ 120572119899(120572119902+ 12057911198623(xlowast) minus 119862

5(xlowast) 120579

1+ 1) minus 119862

5(xlowast)

sdot 1205791

1198760(xlowast) = 120572

1199021205791(11986111198626(xlowast) + 119861


4(xlowast))

+ 120572119899(1205791(11986121198626(xlowast) + 119861


5(xlowast)) + 120572

119902)

(93)




119902=

119903lowast




1198616 1198626(xdfe) = 0 and (92) becomes

1198755(120577 xdfe) = (120577 + 1)

3(1205772+ 119876dfe120577 + 119877dfe) = 0 (94)

where


= 120572119899(1 minus 119877

0) + 120572119902+ (1 minus 120579

1) 1198616(

120572119899minus 120572119902

120572119902

)

119877dfe = 120572119902 (120572119899 minus 11986151205791) + 1205721198991198616 (1205791 minus 1)

= 120572119902120572119899(1 minus 119877

0)

(95)


radic1198762







=


119899 119904lowast

119902 119903lowast


and we have

1198755(120577 xlowast) = (120577 + 1)2 (1205773 + 119875x

119890119890

1205772+ 119876x

119890119890

120577 + 119877x119890119890

) = 0 (96)

where


119875x119890119890

=


12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)


119890119890will be the


119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)




119899= 0 and we


1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)





grouping 1198616




1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)





CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References








































120583


120583 + rate120583

asymp large (79)


119899 1120573119902



119901119899= 119888119899= 119894119899= 119904119899

119901119902= 119904119899+ 1198861119904119902

119888119902= 1198601119904119899+ 1198602119904119902

119894119902= 1198603119904119899+ 1198604119904119902

119903119889= 1198607119904119899+ 1198608119904119902

(80)


119889119905= 1 minus 120582119904 + (120572


119889119904119899

119889119905= 1205791120582119904 minus 120572

119899119904119899 (82)

119889119904119902

119889119905= (1 minus 120579

1) 120582119904 minus 120572

119902119904119902 (83)

119889119903119903

119889119905= 1198605119904119899+ 1198606119904119902minus 119903119903 (84)

119889ℎ

119889119905= 1 minus ℎ minus 119861

1119904119899minus 1198612119904119902 (85)


120582119904 = (120588119899(119901119899

ℎ) + 120588119902(

119901119902

ℎ) + 120591119899(119888119899

ℎ) + 120591119902(

119888119902

ℎ)

+ 120585119899(119894119899

ℎ) + 120585119902(

119894119902

ℎ) + 119886119889(119903119889

ℎ)) 119904 = (119861

5(119904119899

ℎ)

+ 1198616(

119904119902

ℎ)) 119904

(86)





119899 119904lowast

119902 119903lowast


119864dfe = (1 0 0 0 1)

119864119890119890= (119904lowast 119904lowast

119899 119904lowast

119902 119903lowast

119903 ℎlowast)

(87)


119904lowast

119902=1198617

1198618

119904lowast

119899= (

1205721199021205791

120572119899(1 minus 120579

1)) 119904lowast

119902

119903lowast

119903= 1198605119904lowast

119899+ 1198606119904lowast

119902

119904lowast= 1 minus 119861

3119904lowast

119899minus 1198614119904lowast

119902


1119904lowast

119899minus 1198612119904lowast

119902

(88)


3119910 + 119861

4119909 and

1198611119910 + 1198612119909 = 1 we get

119904lowast= (119911 minus 1

R minus 1)

119904lowast

119899= 119910(

1198770minus 1

R minus 1)

119904lowast

119902= 119909(

1198770minus 1

R minus 1)

119903lowast

119903= (1198605119910 + 119860

6119909) (1198770minus 1

R minus 1)

ℎlowast= ((119911 minus 1) 119877

0

R minus 1)

(89)


119899 119904lowast

119902 119903lowast




119869 (xlowast) =((

(


4(xlowast) + 120572


119902minus 11986140 minus119862

6(xlowast)

12057911198623(xlowast) 120579


11989912057911198625(xlowast) 0 120579

11198626(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) 120579


1199020 12057911198626(xlowast)

0 1198605

1198606

minus1 0

0 minus1198611

minus1198612

0 minus1

))

)

(90)

where 1205791= 1 minus 120579

1and

1198623(xlowast) = 119861

5

119904lowast

119899

ℎlowast+ 1198616

119904lowast

119902

ℎlowast=120572119899119910 (1198770minus 1)

1205791(119911 minus 1)

1198624(xlowast) = 119861

5

119904lowast

ℎlowast=1198615

1198770

1198625(xlowast) = 119861

6

119904lowast

ℎlowast=1198616

1198770

1198626(xlowast) = minus(119861

5


119899

ℎlowast2+ 1198616


119902

ℎlowast2) = minus

120572119899119910 (1198770minus 1)

1205791 (119911 minus 1) 1198770

(91)


1198755(120577 xlowast)

= (120577 + 1)2(1205773+ 1198762(xlowast) 1205772 + 119876

1(xlowast) 120577 + 119876

0(xlowast))

= 0

(92)

where

1198762(xlowast) = 120572

119899+ 120572119902+ 1198623(xlowast) minus 119862

4(xlowast) 120579


1

+ 1

1198761(xlowast) = 120572

119902


1199021198624(xlowast) + 119862

4(xlowast))

+ 11986121198626(xlowast) 120579

1+ 1198623(xlowast) (120572

1199021205791+ 11986131205791+ 11986141205791)

+ 120572119899(120572119902+ 12057911198623(xlowast) minus 119862

5(xlowast) 120579

1+ 1) minus 119862

5(xlowast)

sdot 1205791

1198760(xlowast) = 120572

1199021205791(11986111198626(xlowast) + 119861


4(xlowast))

+ 120572119899(1205791(11986121198626(xlowast) + 119861


5(xlowast)) + 120572

119902)

(93)




119902=

119903lowast




1198616 1198626(xdfe) = 0 and (92) becomes

1198755(120577 xdfe) = (120577 + 1)

3(1205772+ 119876dfe120577 + 119877dfe) = 0 (94)

where


= 120572119899(1 minus 119877

0) + 120572119902+ (1 minus 120579

1) 1198616(

120572119899minus 120572119902

120572119902

)

119877dfe = 120572119902 (120572119899 minus 11986151205791) + 1205721198991198616 (1205791 minus 1)

= 120572119902120572119899(1 minus 119877

0)

(95)


radic1198762







=


119899 119904lowast

119902 119903lowast


and we have

1198755(120577 xlowast) = (120577 + 1)2 (1205773 + 119875x

119890119890

1205772+ 119876x

119890119890

120577 + 119877x119890119890

) = 0 (96)

where


119875x119890119890

=


12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)


119890119890will be the


119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)




119899= 0 and we


1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)





grouping 1198616




1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)





CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References




































119869 (xlowast) =((

(


4(xlowast) + 120572


119902minus 11986140 minus119862

6(xlowast)

12057911198623(xlowast) 120579


11989912057911198625(xlowast) 0 120579

11198626(xlowast)

12057911198623(xlowast) 120579

11198624(xlowast) 120579


1199020 12057911198626(xlowast)

0 1198605

1198606

minus1 0

0 minus1198611

minus1198612

0 minus1

))

)

(90)

where 1205791= 1 minus 120579

1and

1198623(xlowast) = 119861

5

119904lowast

119899

ℎlowast+ 1198616

119904lowast

119902

ℎlowast=120572119899119910 (1198770minus 1)

1205791(119911 minus 1)

1198624(xlowast) = 119861

5

119904lowast

ℎlowast=1198615

1198770

1198625(xlowast) = 119861

6

119904lowast

ℎlowast=1198616

1198770

1198626(xlowast) = minus(119861

5


119899

ℎlowast2+ 1198616


119902

ℎlowast2) = minus

120572119899119910 (1198770minus 1)

1205791 (119911 minus 1) 1198770

(91)


1198755(120577 xlowast)

= (120577 + 1)2(1205773+ 1198762(xlowast) 1205772 + 119876

1(xlowast) 120577 + 119876

0(xlowast))

= 0

(92)

where

1198762(xlowast) = 120572

119899+ 120572119902+ 1198623(xlowast) minus 119862

4(xlowast) 120579


1

+ 1

1198761(xlowast) = 120572

119902


1199021198624(xlowast) + 119862

4(xlowast))

+ 11986121198626(xlowast) 120579

1+ 1198623(xlowast) (120572

1199021205791+ 11986131205791+ 11986141205791)

+ 120572119899(120572119902+ 12057911198623(xlowast) minus 119862

5(xlowast) 120579

1+ 1) minus 119862

5(xlowast)

sdot 1205791

1198760(xlowast) = 120572

1199021205791(11986111198626(xlowast) + 119861


4(xlowast))

+ 120572119899(1205791(11986121198626(xlowast) + 119861


5(xlowast)) + 120572

119902)

(93)




119902=

119903lowast




1198616 1198626(xdfe) = 0 and (92) becomes

1198755(120577 xdfe) = (120577 + 1)

3(1205772+ 119876dfe120577 + 119877dfe) = 0 (94)

where


= 120572119899(1 minus 119877

0) + 120572119902+ (1 minus 120579

1) 1198616(

120572119899minus 120572119902

120572119902

)

119877dfe = 120572119902 (120572119899 minus 11986151205791) + 1205721198991198616 (1205791 minus 1)

= 120572119902120572119899(1 minus 119877

0)

(95)


radic1198762







=


119899 119904lowast

119902 119903lowast


and we have

1198755(120577 xlowast) = (120577 + 1)2 (1205773 + 119875x

119890119890

1205772+ 119876x

119890119890

120577 + 119877x119890119890

) = 0 (96)

where


119875x119890119890

=


12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)


119890119890will be the


119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)




119899= 0 and we


1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)





grouping 1198616




1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)





CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References




































119875x119890119890

=


12057211990212057911198770 (119911 minus 1)

119876x119890119890

=

120572119899120579111987611+ 1205722

119902119877011987612

1205721198991205721199021198770(119911 minus 1) 120579

1

119877x119890119890

=

(1198770minus 1) (R minus 1) 120572

119899120572119902

(119911 minus 1) 1198770

(97)

where

11987611= (1198616(119911 minus 1) 120579

1(120572119902minus 120572119899) + 120572119902(120572119902(1198612119909 + 1198772

0119911)

+ 120572119899(1198770minus 1) (119861

2119909 + 1205721199021198770119909 minus 1)))

11987612= (120572119899(minus1205791(1198612119909 + (119877

0minus 1) 119909 (120572

119902+ 1198613) + 1)

+ 1198612119909 + 1) + 120572

11990211986131205791(1198770minus 1) 119909)

(98)


119890119890will be the


119875x119890119890

gt 0

119876x119890119890

gt 0

119877x119890119890

gt 0

119875x119890119890

119876x119890119890

minus 119877x119890119890

gt 0

(99)




119899= 0 and we


1198770=1198616

120572119902

119910 = 0

119909 =1

1198612

119911 =1198614

1198612

1198623=

120572119902119909 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(100)

and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

119911 minus 1) 120577

+ (

(R minus 1) (1198770minus 1) 120572

119902

1198770(119911 minus 1)

))

(101)





grouping 1198616




1198770=1198615

120572119899

119909 = 0

119910 =1

1198611

119911 =1198613

1198611

1198623=120572119899119910 (1198770minus 1)

119911 minus 1

1198626= minus1198623

1198770

(102)





CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References







































CN

(1 minus 1205792)120572NSN

1205792a120572NSN

1205793a120573NPN

1205794120574NCN


IN


120579 1120582S

(1 minus 1205791 )120582S

1205792b 120572NSN

1205793b120573NPN


rEQCQ

rLNIN

120590NCN

(1 minus 1205794)120574NCN

rLQ I

Q

120575NIN 120575QIQ


SQ

(1 minus 1205797)120573QPQ

(1 minus 1205796)120572QSQ


PQ

1205797120573QPQ


CQ

120574QCQ


(IQ)

(1 minus 1205793)120573NPN


bRD

1205796120572QSQ

Non

quar

antin

ed

Qua

rant

ine

Π


and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119899) (120577 + 1)

2(1205772

+ (1 +(1198770minus 1) 119910120572

119899

119911 minus 1) 120577

+ ((R minus 1) (119877

0minus 1) 120572

119899

1198770(119911 minus 1)

))

(103)



0gt 1 In this







119899= 120572119902and (96) becomes

1198755(120577 xlowast) = (120577 + 120572

119902) (120577 + 1)

2(1205772

+ (1 +

(1198770minus 1) 119909120572

119902

(119911 minus 1) (1 minus 1205791)) 120577

+ (

(R minus 1) (1198770 minus 1) 120572119902

1198770(119911 minus 1)

))

(104)






6





119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References





































119873isin


119876= 120601119876120585119873 where












stant 119902119902isin [0 1] Thus 120591

119873= 119902119873120585119873and 120591

119876= 119902119876120585119876 with

0 lt 119902119873 119902119876lt 1




119873and120572119876are in the


119873) and




119876)






119873and 120575

119876are



119864119876isin [04829 05903] If we assume





and 119903119871119876



consider that 119903119871119873= 120581119903119864119873

and 119903119871119876= 120581119903119864119876 where 0 lt 120581 ≪ 1


119873= 05 per day



6 1205797 1205794 and 120579




119873and 120588



0









120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References









































120591119873

119902119873120585119873 119902119873= 075 119902


120585119876

120601119876120585119873 120601119876= 05 120601


120591119876

119902119876120585119876 119902119876= 075 119902


120579119894

120579119894isin [0 1] 119894 = 1 2 7 120579

119894isin [0 1] 119894 = 1 2 7 Variable

120572119873

110 [111 14] [8ndash10]120572119876

110 [111 14] [8ndash10]120573119873

05 [13 1] [10]120573119876

12 [13 1] [10]120574119873

13 [14 12] [10 12]120574119876

13 [14 12] [10 12]120575119873

13 [14 12] [10 12]120575119876


03000 [0111 0489] dayminus1 [15]119903119864119876

05366 [04829 05903] [16]119903119864119873

120596119903119864119876 120596 = 088 120596 isin [0 1] Estimate

119903119871119873

120581119903119864119873

120581 = 002 120581 ≪ 1 Estimate119903119871119876

120581119903119864119876 120581 = 002 120581 ≪ 1 Estimate

120590119873





0would




0gt 1 the







0can be shown




1199030+1199031120588119876+1199032120588119873 where 119903



120588119876and 120588



0being














119873on 1198770and the



00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References




































00020

00015

00010

00005

000005 10 15 20

t

s n

(a)

00006

00005

00004

00003

00002

00001

00000

s q

5 10 15 20

t

(b)

00010

00008

00006

00004

00002

00000

r r

5 10 15 20

t

(c)

100

095

090

085

080

075

s

5 10 15 20

t

(d)100

095

090

085

080

075

h

5 10 15 20

t

(e)

00010

00008

00006

00004

00002

00000

t

s n

001 002 003 004

(f)


1= 085 and 120588

119873= 115 and 120588




119873= 120588119876= 08 and


stable


119876









10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References




































10

08

06

04

02

002 4 6 8 10

s infin

R0

(a)

2 4 6 8 10

R0

06

05

04

02

03

01

00

s ninfin

(b)








0increases

10000

09995

09990

09985

09980

san

dh

10 20 30 40

t

h

s

(a)

10 20 30 40

t

25

20

15

10

05

times10minus6s n

ands q

sn

sq

(b)

san

dh

10 20 30 40

t

098

097

096

095

094

h

s

(c)

10 20 30 40

t

sn

sq

s nan

ds q

00001

000008

000006

000004

000002

(d)


119873= 120588119876= 08 and 120579



119873is increased







09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References




































09

08

07

06

10 20 30 40

h

s

san

dh

t

(a)

s nan

ds q

00007

00006

00005

00004

00003

00001

00002

sn

sq

10 20 30 40

t

(b)


119873= 115 120588

119876= 085 and 120579



0value is



















1= 085 and

120588119873= 115 and 120588





1= 085 and 120588

119873= 2 and 120588





S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References




































S Nan

dS Q

SN

SQ

7000

6000

5000

4000

3000

2000

1000

00

t

10000 20000 30000 40000 50000

(a)

CN

andC

Q

CN

CQ

t

10000 20000 30000 40000 50000

250

200

150

100

50

(b)

IN

IQ

t

10000 20000 30000 40000 50000

I Nan

dI Q

400

300

200

100

0

(c)

RD

t

10000 20000 30000 40000 50000

400

300

200

100

0

500

(d)


1= 085 and 120588

119873= 115 and 120588










S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References




































S Nan

dS Q

2000

1500

1000

500

0

t

100 200 300 400 500

SN

SQ

(a)

CN

andC

Q CN

CQ

100

80

60

40

20

t

100 200 300 400 500

(b)

I Nan

dI Q

IN

IQ

t

0100 200 300 400 500

120

100

80

60

40

20

(c)

Cum

ulat

iveI

Nan

dI Q

IN

IQ

t

35000

30000

25000

20000

15000

10000

5000

02000 4000 6000 8000 10000 12000 14000

(d)


1= 085 and 120588

119873= 2 and 120588



that fades out





3119886and 1205792119886



0gt 1



0





Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References





































Notations








119873or





119875119876



119873




119873or













3of these contacts



7of these contacts








120579119894 Proportions 0 le 120579

119894le 1 119894 = 1 2 7 1












buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References












































buried 119879minus1119903119864119873


119903119871119873





Competing Interests


Acknowledgments


References



































A Mathematical Model with Quarantine States for the ......2...

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