Research Article Dynamical Analysis of an SEIT Epidemic...

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Research Article Dynamical Analysis of an SEIT Epidemic Model with Application to Ebola Virus Transmission in Guinea Zhiming Li, 1 Zhidong Teng, 1 Xiaomei Feng, 2 Yingke Li, 3 and Huiguo Zhang 1 1 College of Mathematics and System Sciences, Xinjiang University, Urumqi 834800, China 2 Department of Mathematics, Yuncheng University, Yuncheng 044000, China 3 College of Mathematics and Physics, Xinjiang Agricultural University, Urumqi 830052, China Correspondence should be addressed to Zhidong Teng; zhidong [email protected] Received 4 April 2015; Revised 27 May 2015; Accepted 10 June 2015 Academic Editor: Konstantin Blyuss Copyright © 2015 Zhiming Li et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In order to investigate the transmission mechanism of the infectious individual with Ebola virus, we establish an SEIT (susceptible, exposed in the latent period, infectious, and treated/recovery) epidemic model. e basic reproduction number is defined. e mathematical analysis on the existence and stability of the disease-free equilibrium and endemic equilibrium is given. As the applications of the model, we use the recognized infectious and death cases in Guinea to estimate parameters of the model by the least square method. With suitable parameter values, we obtain the estimated value of the basic reproduction number and analyze the sensitivity and uncertainty property by partial rank correlation coefficients. 1. Introduction Ebola virus disease (EVD) is first identified in the Democratic Republic of Congo (formerly Z¨ aire) in 1976. It is a lethal viral hemorrhagic fever and can cause a high case fatality rate lying between 50 and 90% [1, 2]. Since 1976, the outbreak of Ebola epidemic has happened more than 20 times, most of which appeared in Africa. Sudan (1976, 1979, and 2004), Democratic Republic of Congo (1976, 1977, 1995, 2007, 2008, and 2012), Gabon (1994, 1996, 2001, and 2002), Republic of Cˆ ote d’Ivoire (1994), Uganda (2000, 2007, 2011, and 2012), and Republic of the Congo (2001, 2002, 2003, and 2005) have reported EVD epidemics [3, 4]. Mathematical modeling has emerged as an important tool for gaining understanding of the dynamics of the spread of EVD. Reference [5] established a stochastic discrete-time susceptible-exposed-infectious-recovered (SEIR) model to estimate parameters from daily incidence and mortality time series for an outbreak of Ebola in the Democratic Republic of Congo in 1995. Different from [5], reference [1] used both onset and death date to constrain the optimization of SEIR model parameters by Bayesian inference. To develop a better understanding of Ebola transmission dynamics, [6] intro- duced a compartmental model to quantify transmission in different settings (illness in the community, hospitalization, and traditional burial). Further, [7] used the above model to analyze the temporal dynamics of Ebola. e current Ebola outbreak began in December 2013 in Guinea [8], initially in the Prefecture of Gueckedou, and shortly spread to other West African countries such as Liberia, Sierra Leone, Nigeria, and Senegal [6, 9–11]. e outbreak is the largest to date: as of 25th January 2015, 22,092 cases have been reported by World Health Organisation (WHO), as well as 8,810 deaths [12], which contain 2,917 infected cases and 1,910 deaths of Guinea. Reference [13] analyzed transmission dynamics of EVD in Nigeria and showed the time window for successful containment of EVD outbreaks caused by infected air travelers. Reference [14] designed a model that is formulated by splitting the total population into two main subgroups, namely a subgroup of individuals in the community and another for those in health-care settings. To assess the effect these various intervention strate- gies could have on controlling the spread of Ebola virus, we develop a mathematical model for transmission, fitted probabilistically to epidemiological data of reported cases in Guinea. Experimental vaccines and treatments for Ebola are under development, but they have not yet been fully tested Hindawi Publishing Corporation Computational and Mathematical Methods in Medicine Volume 2015, Article ID 582625, 11 pages http://dx.doi.org/10.1155/2015/582625

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Research ArticleDynamical Analysis of an SEIT Epidemic Model with Applicationto Ebola Virus Transmission in Guinea

Zhiming Li1 Zhidong Teng1 Xiaomei Feng2 Yingke Li3 and Huiguo Zhang1

1College of Mathematics and System Sciences Xinjiang University Urumqi 834800 China2Department of Mathematics Yuncheng University Yuncheng 044000 China3College of Mathematics and Physics Xinjiang Agricultural University Urumqi 830052 China

Correspondence should be addressed to Zhidong Teng zhidong tengsinacom

Received 4 April 2015 Revised 27 May 2015 Accepted 10 June 2015

Academic Editor Konstantin Blyuss

Copyright copy 2015 Zhiming Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to investigate the transmission mechanism of the infectious individual with Ebola virus we establish an SEIT (susceptibleexposed in the latent period infectious and treatedrecovery) epidemic model The basic reproduction number is defined Themathematical analysis on the existence and stability of the disease-free equilibrium and endemic equilibrium is given As theapplications of the model we use the recognized infectious and death cases in Guinea to estimate parameters of the model bythe least square method With suitable parameter values we obtain the estimated value of the basic reproduction number andanalyze the sensitivity and uncertainty property by partial rank correlation coefficients

1 Introduction

Ebola virus disease (EVD) is first identified in theDemocraticRepublic of Congo (formerly Zaire) in 1976 It is a lethal viralhemorrhagic fever and can cause a high case fatality rate lyingbetween 50 and 90 [1 2] Since 1976 the outbreak of Ebolaepidemic has happened more than 20 times most of whichappeared inAfrica Sudan (1976 1979 and 2004) DemocraticRepublic of Congo (1976 1977 1995 2007 2008 and 2012)Gabon (1994 1996 2001 and 2002) Republic of Cote drsquoIvoire(1994) Uganda (2000 2007 2011 and 2012) and Republic ofthe Congo (2001 2002 2003 and 2005) have reported EVDepidemics [3 4]

Mathematicalmodeling has emerged as an important toolfor gaining understanding of the dynamics of the spreadof EVD Reference [5] established a stochastic discrete-timesusceptible-exposed-infectious-recovered (SEIR) model toestimate parameters from daily incidence and mortality timeseries for an outbreak of Ebola in the Democratic Republicof Congo in 1995 Different from [5] reference [1] used bothonset and death date to constrain the optimization of SEIRmodel parameters by Bayesian inference To develop a betterunderstanding of Ebola transmission dynamics [6] intro-duced a compartmental model to quantify transmission in

different settings (illness in the community hospitalizationand traditional burial) Further [7] used the above model toanalyze the temporal dynamics of Ebola

The current Ebola outbreak began in December 2013in Guinea [8] initially in the Prefecture of Gueckedouand shortly spread to other West African countries such asLiberia Sierra Leone Nigeria and Senegal [6 9ndash11] Theoutbreak is the largest to date as of 25th January 2015 22092cases have been reported by World Health Organisation(WHO) as well as 8810 deaths [12] which contain 2917infected cases and 1910 deaths of Guinea Reference [13]analyzed transmission dynamics of EVD in Nigeria andshowed the time window for successful containment of EVDoutbreaks caused by infected air travelers Reference [14]designed a model that is formulated by splitting the totalpopulation into two main subgroups namely a subgroupof individuals in the community and another for those inhealth-care settings

To assess the effect these various intervention strate-gies could have on controlling the spread of Ebola viruswe develop a mathematical model for transmission fittedprobabilistically to epidemiological data of reported cases inGuinea Experimental vaccines and treatments for Ebola areunder development but they have not yet been fully tested

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2015 Article ID 582625 11 pageshttpdxdoiorg1011552015582625

2 Computational and Mathematical Methods in Medicine

Susceptible Exposed Infectious TreatedrecoveryS E I TΛ

120583S

12057310IS + 12057320ES

(120583 + 12058310)E

1205760E

(120583 + 12058320)I

0I

(120583 + 12058330)T

1205740T

Figure 1 Transfer diagram of SEIT epidemic model

for safety or effectiveness Thus we establish an SEIT modeland apply it to analyze the spread of EVD in Guinea where119878 119864 119868 and 119879 denote the number of the susceptible exposedin the latent period infectious and treatedrecovered popu-lation respectively For the SEIT model [15ndash17] constructeda discrete version with latent age structure to investigatetuberculosis transmission Along the paper we assume thatinfected individuals in SEIT model can develop EVD bycontacting the exposed in the latent period or infectiousindividuals

This paper is organized as follows Section 2 introducesan SEIT model by a four-dimensional differential equationssystem In Section 3 the basic reproduction number 1198770 isdefined Based on 1198770 lt 1 and 1198770 gt 1 the local stabilityof the disease-free equilibrium and endemic equilibrium isobtained according to Hurwitz criterion In Section 4 by theleast square method we propose the estimation of parametersby onset and death data in Guinea which can be usedto calculate 1198770 Moreover we provide the sensitivity anduncertainty analysis of 1198770 Section 5 presents our concludingremark

2 Model Formulation

The transmission of EVD is by direct contact with body fluidssecretions tissues or semen from the infectious individuals[1] which starts with acute fever diarrhea that can be bloodyand vomiting followed by headache nausea and abdominalpain [2] Its incubation period ranges from 2 to 21 days (5ndash12 days in most cases) [1] Since the diseases had causedthe loss of thousands of lives and brought great pain tofamilies there have been lots of mathematical models forgaining understanding of the dynamics of the spread of EVD[1 2 6 7 18 19] In this work we establish an SEIT modelto extend the SEIR and SEIRS types and apply the modelto describe the dynamics of EVD during 2014 outbreak inGuinea

The total host population is partitioned into susceptiblesexposed (in the latent period) infectious and treatedrecov-ered individuals respectively denoted by 119878(119905) 119864(119905) 119868(119905) and119879(119905) at time 119905 After one unit time a susceptible individual canbe infected through contactingwith the exposed or infectiousindividuals and enter the latent class or is still in the suscep-tible class or dies A latent individual may become infectiousand enter the infectious class or still stay in the latent classor die An infectious individual may be treated and enter thetreatedrecovery class or stay in the infectious class or dieFor a treated individual he (or she) may recover by effectivetreatment The recovered individual from Ebola depends ongood supportive care and the patients immune response

People who recover from Ebola infection develop antibodiesthat last for at least 10 years possibly longer However itis not known whether people who recover are immune forlife or whether they can become infected with a differentspecies of Ebola [20] Thus we consider that the recoveredindividual may enter the susceptible class Otherwise thetreated individual may stay in the treatedrecovered class ordie Figure 1 shows the relationship between the four variablesof our SEIT model

Using Figure 1 we formulate the following SEIT modelwhich is a four-dimensional differential equations model

119878 = Λ minus 12057310119868119878 minus 12057320119864119878 minus 120583119878 + 1205740119879

= 12057310119868119878 + 12057320119864119878 minus 1205760119864 minus (120583 + 12058310) 119864

119868 = 1205760119864 minus V0119868 minus (120583 + 12058320) 119868

= V0119868 minus 1205740119879 minus (120583 + 12058330) 119879

(1)

where Λ is the recruitment rate the positive parameter 120583

is the rate of natural death 12058310 12058320 and 12058330 are nonneg-ative constants and denote rates of disease-caused deathParameters 12057310 and 12057320 are the rate of the efficient contactin the infected latent and exposed period 1205760 and V0respectively denote the transfer rates between the exposedand the infectious the infectious and treated infectious and1205740 denotes the rate of the effectively treated individuals Inmodel (1) when 1205740 = 0 the SEIT model is an SEIR typeIf 1205740 = 0 the SEIT model is an SEIRS type Thus the SEITmodel is a general version of SEIR or SEIRS type

For convenience denote 1205731 = 12057310120583 1205732 = 12057320120583 1205831 =

12058310120583 1205832 = 12058320120583 1205833 = 12058330120583 120576 = 1205760120583 V = V0120583 and 120574 =

1205740120583 Further define

1205961 = 1+ 120576 + 1205831

1205962 = 1+ V+ 1205832

1205963 = 1+ 120574 + 1205833

(2)

Let 120591 = 120583119905 thenmodel (1) is equivalent to the following form

119889119878

119889120591=

Λ

120583minus 1205731119868119878 minus 1205732119864119878 minus 119878 + 120574119879

119889119864

119889120591= 1205731119868119878 + 1205732119864119878 minus 1205961119864

119889119868

119889120591= 120576119864 minus 1205962119868

119889119879

119889120591= V119868 minus 1205963119879

(3)

Computational and Mathematical Methods in Medicine 3

3 Equilibria and Local Stability

It is clear that model (3) always has a disease-free equilibrium1198750(1198780 0 0 0) with 1198780 = Λ120583 Let 119875

lowast

= (119878lowast

119864lowast

119868lowast

119879lowast

) be theendemic equilibrium then we have

Λ

120583minus 1205731119868lowast

119878lowast

minus 1205732119864lowast

119878lowast

minus 119878lowast

+ 120574119879lowast

= 0

1205731119868lowast

119878lowast

+ 1205732119864lowast

119878lowast

minus 1205961119864lowast

= 0

120576119864lowast

minus 1205962119868lowast

= 0

V119868lowast

minus 1205963119879lowast

= 0

(4)

Define the basic reproduction number as

1198770 =Λ (1205761205731 + 12059621205732)

12058312059611205962 (5)

Based on (2) we know that 1205961 1205962 1205963 gt 0 and 120596112059621205963 minus

120576120574V gt 0 When 1198770 gt 1 solving (4) we obtain the endemicequilibrium 119875

lowast

119878lowast

=12059611205962

1205761205731 + 12059621205732

119864lowast

=12059621205963 (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)

120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)

119868lowast

=1205761205963 (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)

120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)

119879lowast

=120576V (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)

120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)

(6)

Now we discuss the local stability of equilibria Firstlyon the stability of disease-free equilibrium 1198750 we have thefollowing result

Theorem 1 If 1198770 lt 1 then disease-free equilibrium 1198750 =

(Λ120583 0 0 0) is locally asymptotically stable

Proof The Jacobian of model (3) is

119869 = (

minus1205731119868 minus 1205732119864 minus 1 minus1205732119878 minus1205731119878 120574

1205731119868 + 1205732119864 1205732119878 minus 1205961 1205731119878 00 120576 minus1205962 00 0 V minus1205963

) (7)

Thus the Jacobian at point 1198750 is

119869 (1198750) =(((

(

minus1 minus1205732Λ

120583minus

1205731Λ

120583120574

01205732Λ

120583minus 1205961

1205731Λ

1205830

0 120576 minus1205962 00 0 V minus1205963

)))

)

(8)

and its characteristic equation is given by det(120582119868 minus 119869(1198750)) = 0where 119868 is the unit matrix since

det (120582119868 minus 119869 (1198750))

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120582 + 11205732Λ

120583

1205731Λ

120583minus120574

0 120582 minus1205732Λ

120583+ 1205961 minus

1205731Λ

1205830

0 minus120576 120582 + 1205962 00 0 minusV 120582 + 1205963

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= (120582 + 1)

sdot (120582 + 1205963) [(120582 minus1205732Λ

120583+ 1205961) (120582 + 1205962) minus

1205761205731Λ

120583]

(9)

Clearly there exist two roots 1205821 = minus1 and 1205822 = minus1205963 andother roots satisfy

1198911 (120582) ≜ (120582 minus1205732Λ

120583+ 1205961) (120582 + 1205962) minus

1205761205731Λ

120583

= 1205822

+ 1198861120582 + 1198862

(10)

where

1198861 = 1205961 + 1205962 minus1205732Λ

120583

1198862 = 12059611205962 minusΛ

120583(12057321205962 + 1205761205731)

(11)

If 1198770 = Λ(1205761205731 + 12059621205732)12058312059611205962 lt 1 then 1205961 gt

Λ12057612057311205831205962 + Λ1205732120583 and 12059611205962 gt Λ1205761205731120583 + (Λ1205732120583)1205962 Clearly1198861 1198862 gt 0 and 11988611198862 gt 0 According to Hurwitz criterion allroots of 1198911(120582) have negative real parts Hence disease-freeequilibrium 1198750 is local asymptotical stability

Next on the stability of endemic equilibrium 119875lowast we have

the result as follows

Theorem 2 If 1198770 gt 1 then endemic equilibrium 119875lowast is locally

asymptotically stable

Proof By (7) the matrix of the linearization of model (3) atequilibrium 119875

lowast is

119869 (119875lowast

)

= (

minus1205731119868lowast

minus 1205732119864lowast

minus 1 minus1205732119878lowast

minus1205731119878lowast

120574

1205731119868lowast

+ 1205732119864lowast

1205732119878lowast

minus 1205961 1205731119878lowast 0

0 120576 minus1205962 00 0 V minus1205963

)

(12)

The corresponding characteristic equation is

1198912 (120582) ≜ det (120582119868 minus 119869 (119875lowast

))

= 1205824

+ 11988711205823

+ 11988721205822

+ 1198873120582 + 1198874

(13)

4 Computational and Mathematical Methods in Medicine

where

1198871 = 1205731119868lowast

+ 1205732119864lowast

minus 1205732119878lowast

+ 1205961 + 1205962 + 1205963 + 1

1198872 = (1205961 + 1205962 + 1205963) (1205731119868lowast

+ 1205732119864lowast

+ 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 1205961 (1205962 + 1205963)

+ 12059621205963

1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast

+ 1205732119864lowast

)

minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast

+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963

1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast

+ 1205732119864lowast

)

(14)

Denote119860 = (Λ(12057612059631205731+120596212059631205732)minus120583120596112059621205963)120583(120596112059621205963minus120576120574V)Based on (6) we have

1205731119868lowast

+ 1205732119864lowast

=1205761205731 + 12059621205732

120576119868lowast

= 119860 (15)

Thus

1198871 = 119860 +12057612059611205731

1205761205731 + 12059621205732+ 1205962 + 1205963 + 1

1198872 = (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732+ 1205962 + 1205963

+ 12059621205963

1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731

1205761205731 + 12059621205732

1198874 = (120596112059621205963 minus 120576120574V) 119860

(16)

The calculation of 119887119894 119894 = 1 2 3 4 is placed in Appendix A

In order to obtain the local asymptotical stability of119875lowast we

need to verify the following conditions according to Hurwitzcriterion (i) 119887

119894gt 0 for 119894 = 1 2 3 4 (ii) 11988711198872 minus 1198873 gt 0 and (iii)

1198873(11988711198872 minus 1198873) minus 119887211198874 gt 0

(i) Since 1205731 1205732 1205961 1205962 1205963 gt 0 and 120596112059621205963 minus 120576120574V gt 0 it iseasy to get 1198871 1198872 1198873 1198874 gt 0

(ii) For 11988711198872 minus 1198873 we have

11988711198872 minus 1198873 = (1205961 + 1205962 + 1205963) 1198602

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732119860

+ (1205961 + 1205962 + 1205963) 119860

+12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732+

1205761205961120596231205731

1205761205731 + 12059621205732

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732

+ (1205962 + 1205963) (1205962 + 1205963 + 1)

+ 12059621205963 (1205962 + 1205963)

(17)

Clearly 11988711198872 minus 1198873 gt 0(iii) Based on (16) and (ii) we have 1198873(11988711198872 minus 1198873) minus 119887

211198874 gt 0

Because the expression of 1198873(11988711198872 minus 1198873) minus 119887211198874 is too

long we place it inAppendix B According toHurwitzcriterion all roots of 1198912(120582) have negative real partsHence endemic equilibrium 119875

lowast is local asymptoticalstability

4 Application of the Model

In this section model (1) will be applied to analyze thecharacteristics of EVD which is equivalent to system (3)In order to estimate its parameters and calculate the basicreproduced number 1198770 we use the onset and death data ofEVD to fit the observed variables by least square method

41 Data Sets The 2014 Ebola outbreak began in GuineaThepatient zero is a 2-year-old toddler named Emile Ouamounowho lived in the village ofMeliandou sitting close to Guinearsquosborders with Sierra Leone and Liberia The boy was infectedin December 2013 and it is not clear exactly how he gotinfected [8] In December Emile had a fever black stool andstarted vomiting Four days later on December 6 he wasdead Within a month so were his young sister his motherand his grandmother

In March 22 2014 the Ministry of health of Guinea hasreported the acute infectious disease named EVD whichbegan with fever severe diarrhea vomiting and high casefatality rate (59) WHO publicly announced outbreak ofEVD on its web site on March 23 49 cases and 29 deathswere officially reported in March 22 [21] By January 252015 2917 reported cases of Ebola in Guinea were identifiedof which 1910 individuals had died From the first case inDecember 1 2013 to January 18 2015 in Guinea the Ebolaepidemics lasted 400 days but the recorded data starting timeand evolution of the epidemic is March 22 2014 Thereforethe onset and death data were collected fromMarch 22 2014to January 25 2015 [22] see Figures 2 and 3

42 Least Square Method for Parameters Estimation In thissubsection we use the least square method to estimate theparameters of model (1) The spread of EVD started inDecember 2013 inwhich thewhole population ofGuineawas11745000 in this year [23] Based on the data sets (see Figures2 and 3) we know that 119868(0) = 49 119879(0) = 49 minus 29 = 20 and

Computational and Mathematical Methods in Medicine 5

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015N

umbe

r of i

nfec

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case

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Figure 2 Number of recognized infectious cases

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Figure 3 Number of recognized death cases

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Figure 4 Ebola disease-caused death rate of recognized infectious cases

119878(0) = 11 745 000minus49 = 11 744 951Moreover since Ebolaincubation period ranges from 5 to 12 days in most cases[1] then we take 5 days as the incubation period Therefore119864(0) = 37 (that is the reported infected cases at the fifth dayminus the reported infected cases at the first day)

In model (1) there are ten parameters that correspond tothe recruitment rate Λ the natural and disease-caused deathrates 120583 12058310 12058320 and 12058330 the efficient contact rates in theinfected and the latent period 12057310 and 12057320 and the transferrates 1205760 V0 and 1205740 Among these parameters parts of themneed to be fixed and others will be estimated as follows

(P1) The recruitment rate Λ can be given by birth ratewhich is the total number of births per 1000 of apopulation in a year Based on [24ndash26] the 2014 birth

rate of Guinea is 003602 Thus Λ = (003602365) times

11 744 951 = 1 159

(P2) The natural death rate 120583 is typically expressed inunits of deaths per 1000 individuals per year Thuswe use the natural death rate 120583 = 00097365 =

2657 times 10minus5 of Guinea in 2013 [27] For two disease-caused death rates 12058310 in the latent period and 12058330 intreatedrecovery period their values are smaller thanthat of 12058320 in the infectious period Thus we assumethat 12058310 = 12058330 = 0 The rate 12058320 will be estimatedIn order to provide the value range of 12058320 we use themethod in which the accumulative death number isdivided by the accumulative casesThen it is obtainedthat 05 le 12058320 le 08 (see Figure 4)

6 Computational and Mathematical Methods in Medicine

0 50 100 150 200 250 300 3500

500

1000

1500

2000

2500

3000

3500

Day

Accu

mul

ated

infe

ctio

us p

opul

atio

n

Observed infectious valueEstimated value

0 50 100 150 200 250 300 3500

500

1000

1500

2000

2500

Day

Accu

mul

ated

dea

th p

opul

atio

n

Observed death valueEstimated value

Figure 5 Real values and fitting values of the accumulative infectious cases and death cases

(P3) In this paper we consider that Ebola virus during thelatent and exposed period is contagious In generalthe efficient contact rates 12057310 and 12057320 are differentIn order to reduce the risk of human-to-humantransmission WHO raises the awareness of the riskfactors of Ebola infection and the protectivemeasures[28 29]Thus we consider that 0 lt 12057310 le 12057320 lt 1 andall of them will be estimated

(P4) Since Ebola incubation period ranges from 2 to 21days (5ndash12 days in most cases) [1] the transfer rate 1205760is set following the constraint 2 lt 11205760 lt 21 that is004761 lt 1205760 lt 05 For the transfer rates V0 we willestimate it and let it satisfy 0 lt V0 lt 1

(P5) Based on [2 5] we know that the effective treated rate35 lt 11205740 lt 107 that is 0093 lt 1205740 lt 02857

Based on the conditions (P1)ndash(P5) parameters of model(1) to be estimated are 12057310 12057320 12058320 1205760 V0 and 1205740 We list theirestimated values in Table 1

43 Numerical Simulations Using the parameter values inTable 1 numerical simulations of model (1) respectively givethe comparison curves between the real values and fittingvalues of the accumulative infectious cases and accumulativedeath cases fromMarch 22 2014 to January 25 2015 [22] seeFigure 5

Through Figure 5 we observed that the EVD infectionstill does not get the effective control This point can beconfirmed by threshold value 1198770 Based on the estimatedvalues of parameters in Table 1 we have 1198770 = Λ(1205761205731 +

12059621205732)12058312059611205962 = 416 where 120573119894

= 1205731198940120583 119894 = 1 2 Since 1198770 gt 1

byTheorem 2 endemic equilibrium 119875lowast of model (1) is locally

asymptotically stable This indicates that EVD infection stillinfects humans and will be endemic in Guinea without theeffective control measures

Table 1 The estimated values of model parameters by least squaremethod (unit yearminus1)

Parameters Definition Initial values Estimates

12057310

Contact rate ininfectiousperiod

50138 times 10minus11 6474 times 10minus11

12057320

Contact rate inlatent andexposed period

93133 times 10minus8 5685 times 10minus9

12058320

Disease-causeddeath rate ininfectiousperiod

05061 06647

1205760

Transfer ratefrom latent andexposed to theinfectious

04 00596

V0

Transfer rateform theinfectious totreatment state

05140 08613

1205740Effective treatedrate 02 02999

44 Sensitivity and Uncertainty Analysis of 1198770 Because 1198770is an important threshed value for the spread of EVD weperform uncertainty and sensitivity analysis of 1198770 in model(1) using partial rank correlation coefficients (PRCCs) [30]Based on (1) (2) and (5) we get the expression of 1198770 asfollows

1198770 =Λ (120576012057310 + 12057320 (120583 + V0 + 12058320))

120583 (120583 + V0 + 12058320) (120583 + 1205760 + 12058310) (18)

Among all parameters of model (1) we only analyze theinfluence of seven parameters in determining the magnitude

Computational and Mathematical Methods in Medicine 7

0743408429

0090200172

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

minus04145

minus09510

12057310 12057320 12058320 1205760 0 1205740

Figure 6 PRCCs for the effect of six parameters on 1198770

of 1198770 except the fixed numbers Λ 120583 12058310 and 12058330 which havebeen discussed in Section 42 The ordering of these PRCCsdirectly corresponds to the level of statistical influence theimpact that uncertainty in the estimate of a parameter hason the variability of 1198770 A positive PRCC value indicatesthat an increase in that parameter leads to an increase in 1198770while a negative value shows that increasing that parameterdecreases 1198770 For the Ebola SEIT model seven parameterswere significantly different from 0 (119901 value lt 005) Amongthese parameters 12057310 12057320 12058320 and V0 have a positiveinfluence on 1198770 while 1205760 and 1205740 have a negative influence on1198770 see Figure 6

5 Conclusions

In this paper we establish the SEIT model to analyze thedynamical properties of EVD transmission in Guinea Oneof the reasons is that there is still no effective treatments forEVD and the current response is only support treatment [31]In particular if parameters 1205740 = 0 or 1205740 = 0 the SEIT modelwill become the SEIR or SEIRS types

The SEIT model is formed by four-dimensional differ-ential equations In order to understand the transmissionof Ebola virus we discuss the local stability of the disease-free equilibrium and the endemic equilibrium by basicreproduction number1198770 lt 1 and1198770 gt 1 Since there exist tenparameters in the expression of 1198770 it is important to estimatethe value of each parameter By the least square method andthe recorded data in Guinea we obtain the estimation ofseven parameters except for four fixed constants in Table 1In Table 1 we observe that the effective treated rate is only02999 and the disease-caused death rate in infectious periodis 06647Thus it is urgent to provide the effective vaccines tocure the infectious people and protect the susceptible

With suitable parameter values we obtain estimationvalue of 1198770 The result shows that the Ebola virus still infectspeople in Guinea and does not disappear in short timeWithout the effective control measures the EVD may be

endemic in Guinea By the PRCCs method we analyze thesensitivity and uncertainty of 1198770 The result shows that therates of the efficient contact especially in latent and exposedperiod lead to the significant increase in 1198770 (see Figure 6)

On the other hand it would be interesting to study moreproperties of the present model In particular a study involv-ing both stability properties of pulse vaccination strategy andglobal stability is worth pursuing We leave these for futureconsideration

Appendices

A Calculation of 1198871 1198872 1198873 1198874

By (6) (16) and the definition of 119860 we obtain

1198871 = 1205731119868lowast

+ 1205732119864lowast

minus 1205732119878lowast

+ 1205961 + 1205962 + 1205963 + 1

= 119860 minus 120573212059611205962

1205761205731 + 12059621205732+ 1205961 + 1205962 + 1205963 + 1

= 119860 + (1205961 minus120596112059621205732

1205761205731 + 12059621205732) + 1205962 + 1205963 + 1

= 119860 +12057612059611205731

1205761205731 + 12059621205732+ 1205962 + 1205963 + 1

(A1)

Calculating 1198872 we have

1198872 = (1205961 + 1205962 + 1205963) (119860 + 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 1205961 (1205962 + 1205963)

+ 12059621205963

= (1205961 + 1205962 + 1205963) (119860 + 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731)12059611205962

1205761205731 + 12059621205732

+ 1205961 (1205962 + 1205963) + 12059621205963

= (1205961 + 1205962 + 1205963) (119860 + 1) +120576120596112059631205731 minus 120596112059621205731

1205761205731 + 12059621205732

+ 12059621205963

= (1205961 + 1205962 + 1205963) 119860

+1205961 (1205761205731 + 12059621205732) + 120576120596112059631205731 minus 120596112059621205731

1205761205731 + 12059621205732+ 1205962

+ 1205963 + 12059621205963

= (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732+ 1205962

+ 1205963 + 12059621205963

(A2)

8 Computational and Mathematical Methods in Medicine

For 1198873 we have

1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast

+ 1205732119864lowast

)

minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast

+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963

= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)

minus (120596212059631205732 + 12057612059631205731 + (1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 120596112059621205963

= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)

minus ((1205962 + 1205963) 1205732 + 1205761205731)12059611205962

1205761205731 + 12059621205732

= (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731

1205761205731 + 12059621205732

(A3)

Lastly it is easy to get

1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast

+ 1205732119864lowast

)

= (120596112059621205963 minus 120576120574V) 119860

(A4)

B Calculation of 1198873(11988711198872

minus 1198873) minus 1198872

11198874

By calculating we have

1198873 (11988711198872 minus 1198873) = (1205961 + 1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 1198603

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 1198602

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961

+ 1205962 + 1205963) 1198602

+12057612059611205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+ (1205962 + 1205963)

sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963

+ 1) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961

+ 1205962 + 1205963) 120596212059631198602

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963

+ 12059621205963) (12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 11986211198622

(B1)

where

1198621 = 12059621205962 +120576120596112059631205731

1205761205731 + 12059621205732

1198622 =12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732+

1205761205961120596231205731

1205761205731 + 12059621205732

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732

+ (1205962 + 1205963) (1205962 + 1205963 + 1)

+ 12059621205963 (1205962 + 1205963)

(B2)

On the other hand

119887211198874 = 120596112059621205963119860

3+ 2

120576120596112057311205761205731 + 12059621205732

1205961120596212059631198602

+ 2 (1205962 + 1205963 + 1) 1205961120596212059631198602

+ (1205962 + 1205963 + 1)2

120596112059621205963119860

+ (12057612059611205731

1205761205731 + 12059621205732)

2120596112059621205963119860

Computational and Mathematical Methods in Medicine 9

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

minus 120576120574V1198603

minus 212057612059611205731

1205761205731 + 12059621205732120576120574V119860

2

minus 2 (1205962 + 1205963 + 1) 120576120574V1198602

minus (1205962 + 1205963 + 1)2

120576120574V119860

minus (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860

(B3)

Therefore we further have

1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 1198603

minus 1205961120596212059631198603

+12057612059611205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962

+ 1205963) 1198602

minus 212057612059611205731

1205761205731 + 12059621205732120596112059621205963119860

2+ (1205962 + 1205963)

sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963

+ 1) 1198602

minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2

120596112059621205963119860

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731

1205761205731 + 12059621205732)

2120596112059621205963119860

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))

sdot 119860 minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622

gt12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

10 Computational and Mathematical Methods in Medicine

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0

(B4)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)

References

[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013

[2] 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

[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014

[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014

[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006

[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014

[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014

[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1

[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015

[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014

[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014

[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1

[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015

[14] 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

[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012

[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012

[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010

[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014

[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006

[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml

[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen

[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news

Computational and Mathematical Methods in Medicine 11

[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen

[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml

[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates

[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate

[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate

[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen

[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf

[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994

[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377

Submit your manuscripts athttpwwwhindawicom

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Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Page 2: Research Article Dynamical Analysis of an SEIT Epidemic ...downloads.hindawi.com/journals/cmmm/2015/582625.pdf · Research Article Dynamical Analysis of an SEIT Epidemic Model with

2 Computational and Mathematical Methods in Medicine

Susceptible Exposed Infectious TreatedrecoveryS E I TΛ

120583S

12057310IS + 12057320ES

(120583 + 12058310)E

1205760E

(120583 + 12058320)I

0I

(120583 + 12058330)T

1205740T

Figure 1 Transfer diagram of SEIT epidemic model

for safety or effectiveness Thus we establish an SEIT modeland apply it to analyze the spread of EVD in Guinea where119878 119864 119868 and 119879 denote the number of the susceptible exposedin the latent period infectious and treatedrecovered popu-lation respectively For the SEIT model [15ndash17] constructeda discrete version with latent age structure to investigatetuberculosis transmission Along the paper we assume thatinfected individuals in SEIT model can develop EVD bycontacting the exposed in the latent period or infectiousindividuals

This paper is organized as follows Section 2 introducesan SEIT model by a four-dimensional differential equationssystem In Section 3 the basic reproduction number 1198770 isdefined Based on 1198770 lt 1 and 1198770 gt 1 the local stabilityof the disease-free equilibrium and endemic equilibrium isobtained according to Hurwitz criterion In Section 4 by theleast square method we propose the estimation of parametersby onset and death data in Guinea which can be usedto calculate 1198770 Moreover we provide the sensitivity anduncertainty analysis of 1198770 Section 5 presents our concludingremark

2 Model Formulation

The transmission of EVD is by direct contact with body fluidssecretions tissues or semen from the infectious individuals[1] which starts with acute fever diarrhea that can be bloodyand vomiting followed by headache nausea and abdominalpain [2] Its incubation period ranges from 2 to 21 days (5ndash12 days in most cases) [1] Since the diseases had causedthe loss of thousands of lives and brought great pain tofamilies there have been lots of mathematical models forgaining understanding of the dynamics of the spread of EVD[1 2 6 7 18 19] In this work we establish an SEIT modelto extend the SEIR and SEIRS types and apply the modelto describe the dynamics of EVD during 2014 outbreak inGuinea

The total host population is partitioned into susceptiblesexposed (in the latent period) infectious and treatedrecov-ered individuals respectively denoted by 119878(119905) 119864(119905) 119868(119905) and119879(119905) at time 119905 After one unit time a susceptible individual canbe infected through contactingwith the exposed or infectiousindividuals and enter the latent class or is still in the suscep-tible class or dies A latent individual may become infectiousand enter the infectious class or still stay in the latent classor die An infectious individual may be treated and enter thetreatedrecovery class or stay in the infectious class or dieFor a treated individual he (or she) may recover by effectivetreatment The recovered individual from Ebola depends ongood supportive care and the patients immune response

People who recover from Ebola infection develop antibodiesthat last for at least 10 years possibly longer However itis not known whether people who recover are immune forlife or whether they can become infected with a differentspecies of Ebola [20] Thus we consider that the recoveredindividual may enter the susceptible class Otherwise thetreated individual may stay in the treatedrecovered class ordie Figure 1 shows the relationship between the four variablesof our SEIT model

Using Figure 1 we formulate the following SEIT modelwhich is a four-dimensional differential equations model

119878 = Λ minus 12057310119868119878 minus 12057320119864119878 minus 120583119878 + 1205740119879

= 12057310119868119878 + 12057320119864119878 minus 1205760119864 minus (120583 + 12058310) 119864

119868 = 1205760119864 minus V0119868 minus (120583 + 12058320) 119868

= V0119868 minus 1205740119879 minus (120583 + 12058330) 119879

(1)

where Λ is the recruitment rate the positive parameter 120583

is the rate of natural death 12058310 12058320 and 12058330 are nonneg-ative constants and denote rates of disease-caused deathParameters 12057310 and 12057320 are the rate of the efficient contactin the infected latent and exposed period 1205760 and V0respectively denote the transfer rates between the exposedand the infectious the infectious and treated infectious and1205740 denotes the rate of the effectively treated individuals Inmodel (1) when 1205740 = 0 the SEIT model is an SEIR typeIf 1205740 = 0 the SEIT model is an SEIRS type Thus the SEITmodel is a general version of SEIR or SEIRS type

For convenience denote 1205731 = 12057310120583 1205732 = 12057320120583 1205831 =

12058310120583 1205832 = 12058320120583 1205833 = 12058330120583 120576 = 1205760120583 V = V0120583 and 120574 =

1205740120583 Further define

1205961 = 1+ 120576 + 1205831

1205962 = 1+ V+ 1205832

1205963 = 1+ 120574 + 1205833

(2)

Let 120591 = 120583119905 thenmodel (1) is equivalent to the following form

119889119878

119889120591=

Λ

120583minus 1205731119868119878 minus 1205732119864119878 minus 119878 + 120574119879

119889119864

119889120591= 1205731119868119878 + 1205732119864119878 minus 1205961119864

119889119868

119889120591= 120576119864 minus 1205962119868

119889119879

119889120591= V119868 minus 1205963119879

(3)

Computational and Mathematical Methods in Medicine 3

3 Equilibria and Local Stability

It is clear that model (3) always has a disease-free equilibrium1198750(1198780 0 0 0) with 1198780 = Λ120583 Let 119875

lowast

= (119878lowast

119864lowast

119868lowast

119879lowast

) be theendemic equilibrium then we have

Λ

120583minus 1205731119868lowast

119878lowast

minus 1205732119864lowast

119878lowast

minus 119878lowast

+ 120574119879lowast

= 0

1205731119868lowast

119878lowast

+ 1205732119864lowast

119878lowast

minus 1205961119864lowast

= 0

120576119864lowast

minus 1205962119868lowast

= 0

V119868lowast

minus 1205963119879lowast

= 0

(4)

Define the basic reproduction number as

1198770 =Λ (1205761205731 + 12059621205732)

12058312059611205962 (5)

Based on (2) we know that 1205961 1205962 1205963 gt 0 and 120596112059621205963 minus

120576120574V gt 0 When 1198770 gt 1 solving (4) we obtain the endemicequilibrium 119875

lowast

119878lowast

=12059611205962

1205761205731 + 12059621205732

119864lowast

=12059621205963 (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)

120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)

119868lowast

=1205761205963 (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)

120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)

119879lowast

=120576V (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)

120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)

(6)

Now we discuss the local stability of equilibria Firstlyon the stability of disease-free equilibrium 1198750 we have thefollowing result

Theorem 1 If 1198770 lt 1 then disease-free equilibrium 1198750 =

(Λ120583 0 0 0) is locally asymptotically stable

Proof The Jacobian of model (3) is

119869 = (

minus1205731119868 minus 1205732119864 minus 1 minus1205732119878 minus1205731119878 120574

1205731119868 + 1205732119864 1205732119878 minus 1205961 1205731119878 00 120576 minus1205962 00 0 V minus1205963

) (7)

Thus the Jacobian at point 1198750 is

119869 (1198750) =(((

(

minus1 minus1205732Λ

120583minus

1205731Λ

120583120574

01205732Λ

120583minus 1205961

1205731Λ

1205830

0 120576 minus1205962 00 0 V minus1205963

)))

)

(8)

and its characteristic equation is given by det(120582119868 minus 119869(1198750)) = 0where 119868 is the unit matrix since

det (120582119868 minus 119869 (1198750))

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120582 + 11205732Λ

120583

1205731Λ

120583minus120574

0 120582 minus1205732Λ

120583+ 1205961 minus

1205731Λ

1205830

0 minus120576 120582 + 1205962 00 0 minusV 120582 + 1205963

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= (120582 + 1)

sdot (120582 + 1205963) [(120582 minus1205732Λ

120583+ 1205961) (120582 + 1205962) minus

1205761205731Λ

120583]

(9)

Clearly there exist two roots 1205821 = minus1 and 1205822 = minus1205963 andother roots satisfy

1198911 (120582) ≜ (120582 minus1205732Λ

120583+ 1205961) (120582 + 1205962) minus

1205761205731Λ

120583

= 1205822

+ 1198861120582 + 1198862

(10)

where

1198861 = 1205961 + 1205962 minus1205732Λ

120583

1198862 = 12059611205962 minusΛ

120583(12057321205962 + 1205761205731)

(11)

If 1198770 = Λ(1205761205731 + 12059621205732)12058312059611205962 lt 1 then 1205961 gt

Λ12057612057311205831205962 + Λ1205732120583 and 12059611205962 gt Λ1205761205731120583 + (Λ1205732120583)1205962 Clearly1198861 1198862 gt 0 and 11988611198862 gt 0 According to Hurwitz criterion allroots of 1198911(120582) have negative real parts Hence disease-freeequilibrium 1198750 is local asymptotical stability

Next on the stability of endemic equilibrium 119875lowast we have

the result as follows

Theorem 2 If 1198770 gt 1 then endemic equilibrium 119875lowast is locally

asymptotically stable

Proof By (7) the matrix of the linearization of model (3) atequilibrium 119875

lowast is

119869 (119875lowast

)

= (

minus1205731119868lowast

minus 1205732119864lowast

minus 1 minus1205732119878lowast

minus1205731119878lowast

120574

1205731119868lowast

+ 1205732119864lowast

1205732119878lowast

minus 1205961 1205731119878lowast 0

0 120576 minus1205962 00 0 V minus1205963

)

(12)

The corresponding characteristic equation is

1198912 (120582) ≜ det (120582119868 minus 119869 (119875lowast

))

= 1205824

+ 11988711205823

+ 11988721205822

+ 1198873120582 + 1198874

(13)

4 Computational and Mathematical Methods in Medicine

where

1198871 = 1205731119868lowast

+ 1205732119864lowast

minus 1205732119878lowast

+ 1205961 + 1205962 + 1205963 + 1

1198872 = (1205961 + 1205962 + 1205963) (1205731119868lowast

+ 1205732119864lowast

+ 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 1205961 (1205962 + 1205963)

+ 12059621205963

1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast

+ 1205732119864lowast

)

minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast

+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963

1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast

+ 1205732119864lowast

)

(14)

Denote119860 = (Λ(12057612059631205731+120596212059631205732)minus120583120596112059621205963)120583(120596112059621205963minus120576120574V)Based on (6) we have

1205731119868lowast

+ 1205732119864lowast

=1205761205731 + 12059621205732

120576119868lowast

= 119860 (15)

Thus

1198871 = 119860 +12057612059611205731

1205761205731 + 12059621205732+ 1205962 + 1205963 + 1

1198872 = (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732+ 1205962 + 1205963

+ 12059621205963

1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731

1205761205731 + 12059621205732

1198874 = (120596112059621205963 minus 120576120574V) 119860

(16)

The calculation of 119887119894 119894 = 1 2 3 4 is placed in Appendix A

In order to obtain the local asymptotical stability of119875lowast we

need to verify the following conditions according to Hurwitzcriterion (i) 119887

119894gt 0 for 119894 = 1 2 3 4 (ii) 11988711198872 minus 1198873 gt 0 and (iii)

1198873(11988711198872 minus 1198873) minus 119887211198874 gt 0

(i) Since 1205731 1205732 1205961 1205962 1205963 gt 0 and 120596112059621205963 minus 120576120574V gt 0 it iseasy to get 1198871 1198872 1198873 1198874 gt 0

(ii) For 11988711198872 minus 1198873 we have

11988711198872 minus 1198873 = (1205961 + 1205962 + 1205963) 1198602

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732119860

+ (1205961 + 1205962 + 1205963) 119860

+12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732+

1205761205961120596231205731

1205761205731 + 12059621205732

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732

+ (1205962 + 1205963) (1205962 + 1205963 + 1)

+ 12059621205963 (1205962 + 1205963)

(17)

Clearly 11988711198872 minus 1198873 gt 0(iii) Based on (16) and (ii) we have 1198873(11988711198872 minus 1198873) minus 119887

211198874 gt 0

Because the expression of 1198873(11988711198872 minus 1198873) minus 119887211198874 is too

long we place it inAppendix B According toHurwitzcriterion all roots of 1198912(120582) have negative real partsHence endemic equilibrium 119875

lowast is local asymptoticalstability

4 Application of the Model

In this section model (1) will be applied to analyze thecharacteristics of EVD which is equivalent to system (3)In order to estimate its parameters and calculate the basicreproduced number 1198770 we use the onset and death data ofEVD to fit the observed variables by least square method

41 Data Sets The 2014 Ebola outbreak began in GuineaThepatient zero is a 2-year-old toddler named Emile Ouamounowho lived in the village ofMeliandou sitting close to Guinearsquosborders with Sierra Leone and Liberia The boy was infectedin December 2013 and it is not clear exactly how he gotinfected [8] In December Emile had a fever black stool andstarted vomiting Four days later on December 6 he wasdead Within a month so were his young sister his motherand his grandmother

In March 22 2014 the Ministry of health of Guinea hasreported the acute infectious disease named EVD whichbegan with fever severe diarrhea vomiting and high casefatality rate (59) WHO publicly announced outbreak ofEVD on its web site on March 23 49 cases and 29 deathswere officially reported in March 22 [21] By January 252015 2917 reported cases of Ebola in Guinea were identifiedof which 1910 individuals had died From the first case inDecember 1 2013 to January 18 2015 in Guinea the Ebolaepidemics lasted 400 days but the recorded data starting timeand evolution of the epidemic is March 22 2014 Thereforethe onset and death data were collected fromMarch 22 2014to January 25 2015 [22] see Figures 2 and 3

42 Least Square Method for Parameters Estimation In thissubsection we use the least square method to estimate theparameters of model (1) The spread of EVD started inDecember 2013 inwhich thewhole population ofGuineawas11745000 in this year [23] Based on the data sets (see Figures2 and 3) we know that 119868(0) = 49 119879(0) = 49 minus 29 = 20 and

Computational and Mathematical Methods in Medicine 5

020406080

100120140160180200

322

201

43

262

014

331

201

44

220

144

102

014

421

201

44

302

014

514

201

45

272

014

65

2014

618

201

47

102

014

720

201

47

272

014

81

2014

86

2014

811

201

48

162

014

820

201

48

312

014

97

2014

914

201

49

212

014

925

201

410

12

014

107

201

410

17

2014

102

420

1411

22

014

119

201

411

16

2014

112

320

1412

72

014

122

120

141

420

151

182

015N

umbe

r of i

nfec

tious

case

s

Figure 2 Number of recognized infectious cases

020406080

100120

322

201

43

262

014

331

201

44

220

144

102

014

421

201

44

302

014

514

201

45

272

014

65

2014

618

201

47

102

014

720

201

47

272

014

81

2014

86

2014

811

201

48

162

014

820

201

48

312

014

97

2014

914

201

49

212

014

925

201

410

12

014

107

201

410

17

2014

102

420

1411

22

014

119

201

411

16

2014

112

320

1412

72

014

122

120

141

420

151

182

015

Num

ber o

f dea

th ca

ses

Figure 3 Number of recognized death cases

00102030405060708

322

201

43

262

014

331

201

44

220

144

102

014

421

201

44

302

014

514

201

45

272

014

65

2014

618

201

47

102

014

720

201

47

272

014

81

2014

86

2014

811

201

48

162

014

820

201

48

312

014

97

2014

914

201

49

212

014

925

201

410

12

014

107

201

410

17

2014

102

420

1411

22

014

119

201

411

16

2014

112

320

1412

72

014

122

120

141

420

151

182

015

Dea

th ra

te o

f inf

ectio

us ca

ses

Figure 4 Ebola disease-caused death rate of recognized infectious cases

119878(0) = 11 745 000minus49 = 11 744 951Moreover since Ebolaincubation period ranges from 5 to 12 days in most cases[1] then we take 5 days as the incubation period Therefore119864(0) = 37 (that is the reported infected cases at the fifth dayminus the reported infected cases at the first day)

In model (1) there are ten parameters that correspond tothe recruitment rate Λ the natural and disease-caused deathrates 120583 12058310 12058320 and 12058330 the efficient contact rates in theinfected and the latent period 12057310 and 12057320 and the transferrates 1205760 V0 and 1205740 Among these parameters parts of themneed to be fixed and others will be estimated as follows

(P1) The recruitment rate Λ can be given by birth ratewhich is the total number of births per 1000 of apopulation in a year Based on [24ndash26] the 2014 birth

rate of Guinea is 003602 Thus Λ = (003602365) times

11 744 951 = 1 159

(P2) The natural death rate 120583 is typically expressed inunits of deaths per 1000 individuals per year Thuswe use the natural death rate 120583 = 00097365 =

2657 times 10minus5 of Guinea in 2013 [27] For two disease-caused death rates 12058310 in the latent period and 12058330 intreatedrecovery period their values are smaller thanthat of 12058320 in the infectious period Thus we assumethat 12058310 = 12058330 = 0 The rate 12058320 will be estimatedIn order to provide the value range of 12058320 we use themethod in which the accumulative death number isdivided by the accumulative casesThen it is obtainedthat 05 le 12058320 le 08 (see Figure 4)

6 Computational and Mathematical Methods in Medicine

0 50 100 150 200 250 300 3500

500

1000

1500

2000

2500

3000

3500

Day

Accu

mul

ated

infe

ctio

us p

opul

atio

n

Observed infectious valueEstimated value

0 50 100 150 200 250 300 3500

500

1000

1500

2000

2500

Day

Accu

mul

ated

dea

th p

opul

atio

n

Observed death valueEstimated value

Figure 5 Real values and fitting values of the accumulative infectious cases and death cases

(P3) In this paper we consider that Ebola virus during thelatent and exposed period is contagious In generalthe efficient contact rates 12057310 and 12057320 are differentIn order to reduce the risk of human-to-humantransmission WHO raises the awareness of the riskfactors of Ebola infection and the protectivemeasures[28 29]Thus we consider that 0 lt 12057310 le 12057320 lt 1 andall of them will be estimated

(P4) Since Ebola incubation period ranges from 2 to 21days (5ndash12 days in most cases) [1] the transfer rate 1205760is set following the constraint 2 lt 11205760 lt 21 that is004761 lt 1205760 lt 05 For the transfer rates V0 we willestimate it and let it satisfy 0 lt V0 lt 1

(P5) Based on [2 5] we know that the effective treated rate35 lt 11205740 lt 107 that is 0093 lt 1205740 lt 02857

Based on the conditions (P1)ndash(P5) parameters of model(1) to be estimated are 12057310 12057320 12058320 1205760 V0 and 1205740 We list theirestimated values in Table 1

43 Numerical Simulations Using the parameter values inTable 1 numerical simulations of model (1) respectively givethe comparison curves between the real values and fittingvalues of the accumulative infectious cases and accumulativedeath cases fromMarch 22 2014 to January 25 2015 [22] seeFigure 5

Through Figure 5 we observed that the EVD infectionstill does not get the effective control This point can beconfirmed by threshold value 1198770 Based on the estimatedvalues of parameters in Table 1 we have 1198770 = Λ(1205761205731 +

12059621205732)12058312059611205962 = 416 where 120573119894

= 1205731198940120583 119894 = 1 2 Since 1198770 gt 1

byTheorem 2 endemic equilibrium 119875lowast of model (1) is locally

asymptotically stable This indicates that EVD infection stillinfects humans and will be endemic in Guinea without theeffective control measures

Table 1 The estimated values of model parameters by least squaremethod (unit yearminus1)

Parameters Definition Initial values Estimates

12057310

Contact rate ininfectiousperiod

50138 times 10minus11 6474 times 10minus11

12057320

Contact rate inlatent andexposed period

93133 times 10minus8 5685 times 10minus9

12058320

Disease-causeddeath rate ininfectiousperiod

05061 06647

1205760

Transfer ratefrom latent andexposed to theinfectious

04 00596

V0

Transfer rateform theinfectious totreatment state

05140 08613

1205740Effective treatedrate 02 02999

44 Sensitivity and Uncertainty Analysis of 1198770 Because 1198770is an important threshed value for the spread of EVD weperform uncertainty and sensitivity analysis of 1198770 in model(1) using partial rank correlation coefficients (PRCCs) [30]Based on (1) (2) and (5) we get the expression of 1198770 asfollows

1198770 =Λ (120576012057310 + 12057320 (120583 + V0 + 12058320))

120583 (120583 + V0 + 12058320) (120583 + 1205760 + 12058310) (18)

Among all parameters of model (1) we only analyze theinfluence of seven parameters in determining the magnitude

Computational and Mathematical Methods in Medicine 7

0743408429

0090200172

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

minus04145

minus09510

12057310 12057320 12058320 1205760 0 1205740

Figure 6 PRCCs for the effect of six parameters on 1198770

of 1198770 except the fixed numbers Λ 120583 12058310 and 12058330 which havebeen discussed in Section 42 The ordering of these PRCCsdirectly corresponds to the level of statistical influence theimpact that uncertainty in the estimate of a parameter hason the variability of 1198770 A positive PRCC value indicatesthat an increase in that parameter leads to an increase in 1198770while a negative value shows that increasing that parameterdecreases 1198770 For the Ebola SEIT model seven parameterswere significantly different from 0 (119901 value lt 005) Amongthese parameters 12057310 12057320 12058320 and V0 have a positiveinfluence on 1198770 while 1205760 and 1205740 have a negative influence on1198770 see Figure 6

5 Conclusions

In this paper we establish the SEIT model to analyze thedynamical properties of EVD transmission in Guinea Oneof the reasons is that there is still no effective treatments forEVD and the current response is only support treatment [31]In particular if parameters 1205740 = 0 or 1205740 = 0 the SEIT modelwill become the SEIR or SEIRS types

The SEIT model is formed by four-dimensional differ-ential equations In order to understand the transmissionof Ebola virus we discuss the local stability of the disease-free equilibrium and the endemic equilibrium by basicreproduction number1198770 lt 1 and1198770 gt 1 Since there exist tenparameters in the expression of 1198770 it is important to estimatethe value of each parameter By the least square method andthe recorded data in Guinea we obtain the estimation ofseven parameters except for four fixed constants in Table 1In Table 1 we observe that the effective treated rate is only02999 and the disease-caused death rate in infectious periodis 06647Thus it is urgent to provide the effective vaccines tocure the infectious people and protect the susceptible

With suitable parameter values we obtain estimationvalue of 1198770 The result shows that the Ebola virus still infectspeople in Guinea and does not disappear in short timeWithout the effective control measures the EVD may be

endemic in Guinea By the PRCCs method we analyze thesensitivity and uncertainty of 1198770 The result shows that therates of the efficient contact especially in latent and exposedperiod lead to the significant increase in 1198770 (see Figure 6)

On the other hand it would be interesting to study moreproperties of the present model In particular a study involv-ing both stability properties of pulse vaccination strategy andglobal stability is worth pursuing We leave these for futureconsideration

Appendices

A Calculation of 1198871 1198872 1198873 1198874

By (6) (16) and the definition of 119860 we obtain

1198871 = 1205731119868lowast

+ 1205732119864lowast

minus 1205732119878lowast

+ 1205961 + 1205962 + 1205963 + 1

= 119860 minus 120573212059611205962

1205761205731 + 12059621205732+ 1205961 + 1205962 + 1205963 + 1

= 119860 + (1205961 minus120596112059621205732

1205761205731 + 12059621205732) + 1205962 + 1205963 + 1

= 119860 +12057612059611205731

1205761205731 + 12059621205732+ 1205962 + 1205963 + 1

(A1)

Calculating 1198872 we have

1198872 = (1205961 + 1205962 + 1205963) (119860 + 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 1205961 (1205962 + 1205963)

+ 12059621205963

= (1205961 + 1205962 + 1205963) (119860 + 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731)12059611205962

1205761205731 + 12059621205732

+ 1205961 (1205962 + 1205963) + 12059621205963

= (1205961 + 1205962 + 1205963) (119860 + 1) +120576120596112059631205731 minus 120596112059621205731

1205761205731 + 12059621205732

+ 12059621205963

= (1205961 + 1205962 + 1205963) 119860

+1205961 (1205761205731 + 12059621205732) + 120576120596112059631205731 minus 120596112059621205731

1205761205731 + 12059621205732+ 1205962

+ 1205963 + 12059621205963

= (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732+ 1205962

+ 1205963 + 12059621205963

(A2)

8 Computational and Mathematical Methods in Medicine

For 1198873 we have

1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast

+ 1205732119864lowast

)

minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast

+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963

= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)

minus (120596212059631205732 + 12057612059631205731 + (1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 120596112059621205963

= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)

minus ((1205962 + 1205963) 1205732 + 1205761205731)12059611205962

1205761205731 + 12059621205732

= (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731

1205761205731 + 12059621205732

(A3)

Lastly it is easy to get

1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast

+ 1205732119864lowast

)

= (120596112059621205963 minus 120576120574V) 119860

(A4)

B Calculation of 1198873(11988711198872

minus 1198873) minus 1198872

11198874

By calculating we have

1198873 (11988711198872 minus 1198873) = (1205961 + 1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 1198603

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 1198602

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961

+ 1205962 + 1205963) 1198602

+12057612059611205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+ (1205962 + 1205963)

sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963

+ 1) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961

+ 1205962 + 1205963) 120596212059631198602

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963

+ 12059621205963) (12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 11986211198622

(B1)

where

1198621 = 12059621205962 +120576120596112059631205731

1205761205731 + 12059621205732

1198622 =12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732+

1205761205961120596231205731

1205761205731 + 12059621205732

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732

+ (1205962 + 1205963) (1205962 + 1205963 + 1)

+ 12059621205963 (1205962 + 1205963)

(B2)

On the other hand

119887211198874 = 120596112059621205963119860

3+ 2

120576120596112057311205761205731 + 12059621205732

1205961120596212059631198602

+ 2 (1205962 + 1205963 + 1) 1205961120596212059631198602

+ (1205962 + 1205963 + 1)2

120596112059621205963119860

+ (12057612059611205731

1205761205731 + 12059621205732)

2120596112059621205963119860

Computational and Mathematical Methods in Medicine 9

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

minus 120576120574V1198603

minus 212057612059611205731

1205761205731 + 12059621205732120576120574V119860

2

minus 2 (1205962 + 1205963 + 1) 120576120574V1198602

minus (1205962 + 1205963 + 1)2

120576120574V119860

minus (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860

(B3)

Therefore we further have

1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 1198603

minus 1205961120596212059631198603

+12057612059611205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962

+ 1205963) 1198602

minus 212057612059611205731

1205761205731 + 12059621205732120596112059621205963119860

2+ (1205962 + 1205963)

sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963

+ 1) 1198602

minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2

120596112059621205963119860

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731

1205761205731 + 12059621205732)

2120596112059621205963119860

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))

sdot 119860 minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622

gt12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

10 Computational and Mathematical Methods in Medicine

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0

(B4)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)

References

[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013

[2] 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

[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014

[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014

[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006

[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014

[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014

[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1

[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015

[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014

[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014

[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1

[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015

[14] 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

[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012

[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012

[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010

[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014

[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006

[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml

[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen

[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news

Computational and Mathematical Methods in Medicine 11

[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen

[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml

[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates

[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate

[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate

[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen

[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf

[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994

[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377

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Page 3: Research Article Dynamical Analysis of an SEIT Epidemic ...downloads.hindawi.com/journals/cmmm/2015/582625.pdf · Research Article Dynamical Analysis of an SEIT Epidemic Model with

Computational and Mathematical Methods in Medicine 3

3 Equilibria and Local Stability

It is clear that model (3) always has a disease-free equilibrium1198750(1198780 0 0 0) with 1198780 = Λ120583 Let 119875

lowast

= (119878lowast

119864lowast

119868lowast

119879lowast

) be theendemic equilibrium then we have

Λ

120583minus 1205731119868lowast

119878lowast

minus 1205732119864lowast

119878lowast

minus 119878lowast

+ 120574119879lowast

= 0

1205731119868lowast

119878lowast

+ 1205732119864lowast

119878lowast

minus 1205961119864lowast

= 0

120576119864lowast

minus 1205962119868lowast

= 0

V119868lowast

minus 1205963119879lowast

= 0

(4)

Define the basic reproduction number as

1198770 =Λ (1205761205731 + 12059621205732)

12058312059611205962 (5)

Based on (2) we know that 1205961 1205962 1205963 gt 0 and 120596112059621205963 minus

120576120574V gt 0 When 1198770 gt 1 solving (4) we obtain the endemicequilibrium 119875

lowast

119878lowast

=12059611205962

1205761205731 + 12059621205732

119864lowast

=12059621205963 (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)

120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)

119868lowast

=1205761205963 (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)

120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)

119879lowast

=120576V (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)

120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)

(6)

Now we discuss the local stability of equilibria Firstlyon the stability of disease-free equilibrium 1198750 we have thefollowing result

Theorem 1 If 1198770 lt 1 then disease-free equilibrium 1198750 =

(Λ120583 0 0 0) is locally asymptotically stable

Proof The Jacobian of model (3) is

119869 = (

minus1205731119868 minus 1205732119864 minus 1 minus1205732119878 minus1205731119878 120574

1205731119868 + 1205732119864 1205732119878 minus 1205961 1205731119878 00 120576 minus1205962 00 0 V minus1205963

) (7)

Thus the Jacobian at point 1198750 is

119869 (1198750) =(((

(

minus1 minus1205732Λ

120583minus

1205731Λ

120583120574

01205732Λ

120583minus 1205961

1205731Λ

1205830

0 120576 minus1205962 00 0 V minus1205963

)))

)

(8)

and its characteristic equation is given by det(120582119868 minus 119869(1198750)) = 0where 119868 is the unit matrix since

det (120582119868 minus 119869 (1198750))

=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120582 + 11205732Λ

120583

1205731Λ

120583minus120574

0 120582 minus1205732Λ

120583+ 1205961 minus

1205731Λ

1205830

0 minus120576 120582 + 1205962 00 0 minusV 120582 + 1205963

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= (120582 + 1)

sdot (120582 + 1205963) [(120582 minus1205732Λ

120583+ 1205961) (120582 + 1205962) minus

1205761205731Λ

120583]

(9)

Clearly there exist two roots 1205821 = minus1 and 1205822 = minus1205963 andother roots satisfy

1198911 (120582) ≜ (120582 minus1205732Λ

120583+ 1205961) (120582 + 1205962) minus

1205761205731Λ

120583

= 1205822

+ 1198861120582 + 1198862

(10)

where

1198861 = 1205961 + 1205962 minus1205732Λ

120583

1198862 = 12059611205962 minusΛ

120583(12057321205962 + 1205761205731)

(11)

If 1198770 = Λ(1205761205731 + 12059621205732)12058312059611205962 lt 1 then 1205961 gt

Λ12057612057311205831205962 + Λ1205732120583 and 12059611205962 gt Λ1205761205731120583 + (Λ1205732120583)1205962 Clearly1198861 1198862 gt 0 and 11988611198862 gt 0 According to Hurwitz criterion allroots of 1198911(120582) have negative real parts Hence disease-freeequilibrium 1198750 is local asymptotical stability

Next on the stability of endemic equilibrium 119875lowast we have

the result as follows

Theorem 2 If 1198770 gt 1 then endemic equilibrium 119875lowast is locally

asymptotically stable

Proof By (7) the matrix of the linearization of model (3) atequilibrium 119875

lowast is

119869 (119875lowast

)

= (

minus1205731119868lowast

minus 1205732119864lowast

minus 1 minus1205732119878lowast

minus1205731119878lowast

120574

1205731119868lowast

+ 1205732119864lowast

1205732119878lowast

minus 1205961 1205731119878lowast 0

0 120576 minus1205962 00 0 V minus1205963

)

(12)

The corresponding characteristic equation is

1198912 (120582) ≜ det (120582119868 minus 119869 (119875lowast

))

= 1205824

+ 11988711205823

+ 11988721205822

+ 1198873120582 + 1198874

(13)

4 Computational and Mathematical Methods in Medicine

where

1198871 = 1205731119868lowast

+ 1205732119864lowast

minus 1205732119878lowast

+ 1205961 + 1205962 + 1205963 + 1

1198872 = (1205961 + 1205962 + 1205963) (1205731119868lowast

+ 1205732119864lowast

+ 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 1205961 (1205962 + 1205963)

+ 12059621205963

1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast

+ 1205732119864lowast

)

minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast

+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963

1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast

+ 1205732119864lowast

)

(14)

Denote119860 = (Λ(12057612059631205731+120596212059631205732)minus120583120596112059621205963)120583(120596112059621205963minus120576120574V)Based on (6) we have

1205731119868lowast

+ 1205732119864lowast

=1205761205731 + 12059621205732

120576119868lowast

= 119860 (15)

Thus

1198871 = 119860 +12057612059611205731

1205761205731 + 12059621205732+ 1205962 + 1205963 + 1

1198872 = (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732+ 1205962 + 1205963

+ 12059621205963

1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731

1205761205731 + 12059621205732

1198874 = (120596112059621205963 minus 120576120574V) 119860

(16)

The calculation of 119887119894 119894 = 1 2 3 4 is placed in Appendix A

In order to obtain the local asymptotical stability of119875lowast we

need to verify the following conditions according to Hurwitzcriterion (i) 119887

119894gt 0 for 119894 = 1 2 3 4 (ii) 11988711198872 minus 1198873 gt 0 and (iii)

1198873(11988711198872 minus 1198873) minus 119887211198874 gt 0

(i) Since 1205731 1205732 1205961 1205962 1205963 gt 0 and 120596112059621205963 minus 120576120574V gt 0 it iseasy to get 1198871 1198872 1198873 1198874 gt 0

(ii) For 11988711198872 minus 1198873 we have

11988711198872 minus 1198873 = (1205961 + 1205962 + 1205963) 1198602

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732119860

+ (1205961 + 1205962 + 1205963) 119860

+12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732+

1205761205961120596231205731

1205761205731 + 12059621205732

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732

+ (1205962 + 1205963) (1205962 + 1205963 + 1)

+ 12059621205963 (1205962 + 1205963)

(17)

Clearly 11988711198872 minus 1198873 gt 0(iii) Based on (16) and (ii) we have 1198873(11988711198872 minus 1198873) minus 119887

211198874 gt 0

Because the expression of 1198873(11988711198872 minus 1198873) minus 119887211198874 is too

long we place it inAppendix B According toHurwitzcriterion all roots of 1198912(120582) have negative real partsHence endemic equilibrium 119875

lowast is local asymptoticalstability

4 Application of the Model

In this section model (1) will be applied to analyze thecharacteristics of EVD which is equivalent to system (3)In order to estimate its parameters and calculate the basicreproduced number 1198770 we use the onset and death data ofEVD to fit the observed variables by least square method

41 Data Sets The 2014 Ebola outbreak began in GuineaThepatient zero is a 2-year-old toddler named Emile Ouamounowho lived in the village ofMeliandou sitting close to Guinearsquosborders with Sierra Leone and Liberia The boy was infectedin December 2013 and it is not clear exactly how he gotinfected [8] In December Emile had a fever black stool andstarted vomiting Four days later on December 6 he wasdead Within a month so were his young sister his motherand his grandmother

In March 22 2014 the Ministry of health of Guinea hasreported the acute infectious disease named EVD whichbegan with fever severe diarrhea vomiting and high casefatality rate (59) WHO publicly announced outbreak ofEVD on its web site on March 23 49 cases and 29 deathswere officially reported in March 22 [21] By January 252015 2917 reported cases of Ebola in Guinea were identifiedof which 1910 individuals had died From the first case inDecember 1 2013 to January 18 2015 in Guinea the Ebolaepidemics lasted 400 days but the recorded data starting timeand evolution of the epidemic is March 22 2014 Thereforethe onset and death data were collected fromMarch 22 2014to January 25 2015 [22] see Figures 2 and 3

42 Least Square Method for Parameters Estimation In thissubsection we use the least square method to estimate theparameters of model (1) The spread of EVD started inDecember 2013 inwhich thewhole population ofGuineawas11745000 in this year [23] Based on the data sets (see Figures2 and 3) we know that 119868(0) = 49 119879(0) = 49 minus 29 = 20 and

Computational and Mathematical Methods in Medicine 5

020406080

100120140160180200

322

201

43

262

014

331

201

44

220

144

102

014

421

201

44

302

014

514

201

45

272

014

65

2014

618

201

47

102

014

720

201

47

272

014

81

2014

86

2014

811

201

48

162

014

820

201

48

312

014

97

2014

914

201

49

212

014

925

201

410

12

014

107

201

410

17

2014

102

420

1411

22

014

119

201

411

16

2014

112

320

1412

72

014

122

120

141

420

151

182

015N

umbe

r of i

nfec

tious

case

s

Figure 2 Number of recognized infectious cases

020406080

100120

322

201

43

262

014

331

201

44

220

144

102

014

421

201

44

302

014

514

201

45

272

014

65

2014

618

201

47

102

014

720

201

47

272

014

81

2014

86

2014

811

201

48

162

014

820

201

48

312

014

97

2014

914

201

49

212

014

925

201

410

12

014

107

201

410

17

2014

102

420

1411

22

014

119

201

411

16

2014

112

320

1412

72

014

122

120

141

420

151

182

015

Num

ber o

f dea

th ca

ses

Figure 3 Number of recognized death cases

00102030405060708

322

201

43

262

014

331

201

44

220

144

102

014

421

201

44

302

014

514

201

45

272

014

65

2014

618

201

47

102

014

720

201

47

272

014

81

2014

86

2014

811

201

48

162

014

820

201

48

312

014

97

2014

914

201

49

212

014

925

201

410

12

014

107

201

410

17

2014

102

420

1411

22

014

119

201

411

16

2014

112

320

1412

72

014

122

120

141

420

151

182

015

Dea

th ra

te o

f inf

ectio

us ca

ses

Figure 4 Ebola disease-caused death rate of recognized infectious cases

119878(0) = 11 745 000minus49 = 11 744 951Moreover since Ebolaincubation period ranges from 5 to 12 days in most cases[1] then we take 5 days as the incubation period Therefore119864(0) = 37 (that is the reported infected cases at the fifth dayminus the reported infected cases at the first day)

In model (1) there are ten parameters that correspond tothe recruitment rate Λ the natural and disease-caused deathrates 120583 12058310 12058320 and 12058330 the efficient contact rates in theinfected and the latent period 12057310 and 12057320 and the transferrates 1205760 V0 and 1205740 Among these parameters parts of themneed to be fixed and others will be estimated as follows

(P1) The recruitment rate Λ can be given by birth ratewhich is the total number of births per 1000 of apopulation in a year Based on [24ndash26] the 2014 birth

rate of Guinea is 003602 Thus Λ = (003602365) times

11 744 951 = 1 159

(P2) The natural death rate 120583 is typically expressed inunits of deaths per 1000 individuals per year Thuswe use the natural death rate 120583 = 00097365 =

2657 times 10minus5 of Guinea in 2013 [27] For two disease-caused death rates 12058310 in the latent period and 12058330 intreatedrecovery period their values are smaller thanthat of 12058320 in the infectious period Thus we assumethat 12058310 = 12058330 = 0 The rate 12058320 will be estimatedIn order to provide the value range of 12058320 we use themethod in which the accumulative death number isdivided by the accumulative casesThen it is obtainedthat 05 le 12058320 le 08 (see Figure 4)

6 Computational and Mathematical Methods in Medicine

0 50 100 150 200 250 300 3500

500

1000

1500

2000

2500

3000

3500

Day

Accu

mul

ated

infe

ctio

us p

opul

atio

n

Observed infectious valueEstimated value

0 50 100 150 200 250 300 3500

500

1000

1500

2000

2500

Day

Accu

mul

ated

dea

th p

opul

atio

n

Observed death valueEstimated value

Figure 5 Real values and fitting values of the accumulative infectious cases and death cases

(P3) In this paper we consider that Ebola virus during thelatent and exposed period is contagious In generalthe efficient contact rates 12057310 and 12057320 are differentIn order to reduce the risk of human-to-humantransmission WHO raises the awareness of the riskfactors of Ebola infection and the protectivemeasures[28 29]Thus we consider that 0 lt 12057310 le 12057320 lt 1 andall of them will be estimated

(P4) Since Ebola incubation period ranges from 2 to 21days (5ndash12 days in most cases) [1] the transfer rate 1205760is set following the constraint 2 lt 11205760 lt 21 that is004761 lt 1205760 lt 05 For the transfer rates V0 we willestimate it and let it satisfy 0 lt V0 lt 1

(P5) Based on [2 5] we know that the effective treated rate35 lt 11205740 lt 107 that is 0093 lt 1205740 lt 02857

Based on the conditions (P1)ndash(P5) parameters of model(1) to be estimated are 12057310 12057320 12058320 1205760 V0 and 1205740 We list theirestimated values in Table 1

43 Numerical Simulations Using the parameter values inTable 1 numerical simulations of model (1) respectively givethe comparison curves between the real values and fittingvalues of the accumulative infectious cases and accumulativedeath cases fromMarch 22 2014 to January 25 2015 [22] seeFigure 5

Through Figure 5 we observed that the EVD infectionstill does not get the effective control This point can beconfirmed by threshold value 1198770 Based on the estimatedvalues of parameters in Table 1 we have 1198770 = Λ(1205761205731 +

12059621205732)12058312059611205962 = 416 where 120573119894

= 1205731198940120583 119894 = 1 2 Since 1198770 gt 1

byTheorem 2 endemic equilibrium 119875lowast of model (1) is locally

asymptotically stable This indicates that EVD infection stillinfects humans and will be endemic in Guinea without theeffective control measures

Table 1 The estimated values of model parameters by least squaremethod (unit yearminus1)

Parameters Definition Initial values Estimates

12057310

Contact rate ininfectiousperiod

50138 times 10minus11 6474 times 10minus11

12057320

Contact rate inlatent andexposed period

93133 times 10minus8 5685 times 10minus9

12058320

Disease-causeddeath rate ininfectiousperiod

05061 06647

1205760

Transfer ratefrom latent andexposed to theinfectious

04 00596

V0

Transfer rateform theinfectious totreatment state

05140 08613

1205740Effective treatedrate 02 02999

44 Sensitivity and Uncertainty Analysis of 1198770 Because 1198770is an important threshed value for the spread of EVD weperform uncertainty and sensitivity analysis of 1198770 in model(1) using partial rank correlation coefficients (PRCCs) [30]Based on (1) (2) and (5) we get the expression of 1198770 asfollows

1198770 =Λ (120576012057310 + 12057320 (120583 + V0 + 12058320))

120583 (120583 + V0 + 12058320) (120583 + 1205760 + 12058310) (18)

Among all parameters of model (1) we only analyze theinfluence of seven parameters in determining the magnitude

Computational and Mathematical Methods in Medicine 7

0743408429

0090200172

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

minus04145

minus09510

12057310 12057320 12058320 1205760 0 1205740

Figure 6 PRCCs for the effect of six parameters on 1198770

of 1198770 except the fixed numbers Λ 120583 12058310 and 12058330 which havebeen discussed in Section 42 The ordering of these PRCCsdirectly corresponds to the level of statistical influence theimpact that uncertainty in the estimate of a parameter hason the variability of 1198770 A positive PRCC value indicatesthat an increase in that parameter leads to an increase in 1198770while a negative value shows that increasing that parameterdecreases 1198770 For the Ebola SEIT model seven parameterswere significantly different from 0 (119901 value lt 005) Amongthese parameters 12057310 12057320 12058320 and V0 have a positiveinfluence on 1198770 while 1205760 and 1205740 have a negative influence on1198770 see Figure 6

5 Conclusions

In this paper we establish the SEIT model to analyze thedynamical properties of EVD transmission in Guinea Oneof the reasons is that there is still no effective treatments forEVD and the current response is only support treatment [31]In particular if parameters 1205740 = 0 or 1205740 = 0 the SEIT modelwill become the SEIR or SEIRS types

The SEIT model is formed by four-dimensional differ-ential equations In order to understand the transmissionof Ebola virus we discuss the local stability of the disease-free equilibrium and the endemic equilibrium by basicreproduction number1198770 lt 1 and1198770 gt 1 Since there exist tenparameters in the expression of 1198770 it is important to estimatethe value of each parameter By the least square method andthe recorded data in Guinea we obtain the estimation ofseven parameters except for four fixed constants in Table 1In Table 1 we observe that the effective treated rate is only02999 and the disease-caused death rate in infectious periodis 06647Thus it is urgent to provide the effective vaccines tocure the infectious people and protect the susceptible

With suitable parameter values we obtain estimationvalue of 1198770 The result shows that the Ebola virus still infectspeople in Guinea and does not disappear in short timeWithout the effective control measures the EVD may be

endemic in Guinea By the PRCCs method we analyze thesensitivity and uncertainty of 1198770 The result shows that therates of the efficient contact especially in latent and exposedperiod lead to the significant increase in 1198770 (see Figure 6)

On the other hand it would be interesting to study moreproperties of the present model In particular a study involv-ing both stability properties of pulse vaccination strategy andglobal stability is worth pursuing We leave these for futureconsideration

Appendices

A Calculation of 1198871 1198872 1198873 1198874

By (6) (16) and the definition of 119860 we obtain

1198871 = 1205731119868lowast

+ 1205732119864lowast

minus 1205732119878lowast

+ 1205961 + 1205962 + 1205963 + 1

= 119860 minus 120573212059611205962

1205761205731 + 12059621205732+ 1205961 + 1205962 + 1205963 + 1

= 119860 + (1205961 minus120596112059621205732

1205761205731 + 12059621205732) + 1205962 + 1205963 + 1

= 119860 +12057612059611205731

1205761205731 + 12059621205732+ 1205962 + 1205963 + 1

(A1)

Calculating 1198872 we have

1198872 = (1205961 + 1205962 + 1205963) (119860 + 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 1205961 (1205962 + 1205963)

+ 12059621205963

= (1205961 + 1205962 + 1205963) (119860 + 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731)12059611205962

1205761205731 + 12059621205732

+ 1205961 (1205962 + 1205963) + 12059621205963

= (1205961 + 1205962 + 1205963) (119860 + 1) +120576120596112059631205731 minus 120596112059621205731

1205761205731 + 12059621205732

+ 12059621205963

= (1205961 + 1205962 + 1205963) 119860

+1205961 (1205761205731 + 12059621205732) + 120576120596112059631205731 minus 120596112059621205731

1205761205731 + 12059621205732+ 1205962

+ 1205963 + 12059621205963

= (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732+ 1205962

+ 1205963 + 12059621205963

(A2)

8 Computational and Mathematical Methods in Medicine

For 1198873 we have

1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast

+ 1205732119864lowast

)

minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast

+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963

= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)

minus (120596212059631205732 + 12057612059631205731 + (1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 120596112059621205963

= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)

minus ((1205962 + 1205963) 1205732 + 1205761205731)12059611205962

1205761205731 + 12059621205732

= (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731

1205761205731 + 12059621205732

(A3)

Lastly it is easy to get

1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast

+ 1205732119864lowast

)

= (120596112059621205963 minus 120576120574V) 119860

(A4)

B Calculation of 1198873(11988711198872

minus 1198873) minus 1198872

11198874

By calculating we have

1198873 (11988711198872 minus 1198873) = (1205961 + 1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 1198603

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 1198602

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961

+ 1205962 + 1205963) 1198602

+12057612059611205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+ (1205962 + 1205963)

sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963

+ 1) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961

+ 1205962 + 1205963) 120596212059631198602

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963

+ 12059621205963) (12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 11986211198622

(B1)

where

1198621 = 12059621205962 +120576120596112059631205731

1205761205731 + 12059621205732

1198622 =12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732+

1205761205961120596231205731

1205761205731 + 12059621205732

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732

+ (1205962 + 1205963) (1205962 + 1205963 + 1)

+ 12059621205963 (1205962 + 1205963)

(B2)

On the other hand

119887211198874 = 120596112059621205963119860

3+ 2

120576120596112057311205761205731 + 12059621205732

1205961120596212059631198602

+ 2 (1205962 + 1205963 + 1) 1205961120596212059631198602

+ (1205962 + 1205963 + 1)2

120596112059621205963119860

+ (12057612059611205731

1205761205731 + 12059621205732)

2120596112059621205963119860

Computational and Mathematical Methods in Medicine 9

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

minus 120576120574V1198603

minus 212057612059611205731

1205761205731 + 12059621205732120576120574V119860

2

minus 2 (1205962 + 1205963 + 1) 120576120574V1198602

minus (1205962 + 1205963 + 1)2

120576120574V119860

minus (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860

(B3)

Therefore we further have

1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 1198603

minus 1205961120596212059631198603

+12057612059611205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962

+ 1205963) 1198602

minus 212057612059611205731

1205761205731 + 12059621205732120596112059621205963119860

2+ (1205962 + 1205963)

sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963

+ 1) 1198602

minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2

120596112059621205963119860

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731

1205761205731 + 12059621205732)

2120596112059621205963119860

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))

sdot 119860 minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622

gt12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

10 Computational and Mathematical Methods in Medicine

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0

(B4)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)

References

[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013

[2] 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

[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014

[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014

[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006

[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014

[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014

[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1

[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015

[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014

[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014

[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1

[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015

[14] 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

[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012

[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012

[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010

[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014

[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006

[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml

[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen

[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news

Computational and Mathematical Methods in Medicine 11

[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen

[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml

[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates

[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate

[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate

[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen

[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf

[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994

[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Page 4: Research Article Dynamical Analysis of an SEIT Epidemic ...downloads.hindawi.com/journals/cmmm/2015/582625.pdf · Research Article Dynamical Analysis of an SEIT Epidemic Model with

4 Computational and Mathematical Methods in Medicine

where

1198871 = 1205731119868lowast

+ 1205732119864lowast

minus 1205732119878lowast

+ 1205961 + 1205962 + 1205963 + 1

1198872 = (1205961 + 1205962 + 1205963) (1205731119868lowast

+ 1205732119864lowast

+ 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 1205961 (1205962 + 1205963)

+ 12059621205963

1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast

+ 1205732119864lowast

)

minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast

+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963

1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast

+ 1205732119864lowast

)

(14)

Denote119860 = (Λ(12057612059631205731+120596212059631205732)minus120583120596112059621205963)120583(120596112059621205963minus120576120574V)Based on (6) we have

1205731119868lowast

+ 1205732119864lowast

=1205761205731 + 12059621205732

120576119868lowast

= 119860 (15)

Thus

1198871 = 119860 +12057612059611205731

1205761205731 + 12059621205732+ 1205962 + 1205963 + 1

1198872 = (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732+ 1205962 + 1205963

+ 12059621205963

1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731

1205761205731 + 12059621205732

1198874 = (120596112059621205963 minus 120576120574V) 119860

(16)

The calculation of 119887119894 119894 = 1 2 3 4 is placed in Appendix A

In order to obtain the local asymptotical stability of119875lowast we

need to verify the following conditions according to Hurwitzcriterion (i) 119887

119894gt 0 for 119894 = 1 2 3 4 (ii) 11988711198872 minus 1198873 gt 0 and (iii)

1198873(11988711198872 minus 1198873) minus 119887211198874 gt 0

(i) Since 1205731 1205732 1205961 1205962 1205963 gt 0 and 120596112059621205963 minus 120576120574V gt 0 it iseasy to get 1198871 1198872 1198873 1198874 gt 0

(ii) For 11988711198872 minus 1198873 we have

11988711198872 minus 1198873 = (1205961 + 1205962 + 1205963) 1198602

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732119860

+ (1205961 + 1205962 + 1205963) 119860

+12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732+

1205761205961120596231205731

1205761205731 + 12059621205732

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732

+ (1205962 + 1205963) (1205962 + 1205963 + 1)

+ 12059621205963 (1205962 + 1205963)

(17)

Clearly 11988711198872 minus 1198873 gt 0(iii) Based on (16) and (ii) we have 1198873(11988711198872 minus 1198873) minus 119887

211198874 gt 0

Because the expression of 1198873(11988711198872 minus 1198873) minus 119887211198874 is too

long we place it inAppendix B According toHurwitzcriterion all roots of 1198912(120582) have negative real partsHence endemic equilibrium 119875

lowast is local asymptoticalstability

4 Application of the Model

In this section model (1) will be applied to analyze thecharacteristics of EVD which is equivalent to system (3)In order to estimate its parameters and calculate the basicreproduced number 1198770 we use the onset and death data ofEVD to fit the observed variables by least square method

41 Data Sets The 2014 Ebola outbreak began in GuineaThepatient zero is a 2-year-old toddler named Emile Ouamounowho lived in the village ofMeliandou sitting close to Guinearsquosborders with Sierra Leone and Liberia The boy was infectedin December 2013 and it is not clear exactly how he gotinfected [8] In December Emile had a fever black stool andstarted vomiting Four days later on December 6 he wasdead Within a month so were his young sister his motherand his grandmother

In March 22 2014 the Ministry of health of Guinea hasreported the acute infectious disease named EVD whichbegan with fever severe diarrhea vomiting and high casefatality rate (59) WHO publicly announced outbreak ofEVD on its web site on March 23 49 cases and 29 deathswere officially reported in March 22 [21] By January 252015 2917 reported cases of Ebola in Guinea were identifiedof which 1910 individuals had died From the first case inDecember 1 2013 to January 18 2015 in Guinea the Ebolaepidemics lasted 400 days but the recorded data starting timeand evolution of the epidemic is March 22 2014 Thereforethe onset and death data were collected fromMarch 22 2014to January 25 2015 [22] see Figures 2 and 3

42 Least Square Method for Parameters Estimation In thissubsection we use the least square method to estimate theparameters of model (1) The spread of EVD started inDecember 2013 inwhich thewhole population ofGuineawas11745000 in this year [23] Based on the data sets (see Figures2 and 3) we know that 119868(0) = 49 119879(0) = 49 minus 29 = 20 and

Computational and Mathematical Methods in Medicine 5

020406080

100120140160180200

322

201

43

262

014

331

201

44

220

144

102

014

421

201

44

302

014

514

201

45

272

014

65

2014

618

201

47

102

014

720

201

47

272

014

81

2014

86

2014

811

201

48

162

014

820

201

48

312

014

97

2014

914

201

49

212

014

925

201

410

12

014

107

201

410

17

2014

102

420

1411

22

014

119

201

411

16

2014

112

320

1412

72

014

122

120

141

420

151

182

015N

umbe

r of i

nfec

tious

case

s

Figure 2 Number of recognized infectious cases

020406080

100120

322

201

43

262

014

331

201

44

220

144

102

014

421

201

44

302

014

514

201

45

272

014

65

2014

618

201

47

102

014

720

201

47

272

014

81

2014

86

2014

811

201

48

162

014

820

201

48

312

014

97

2014

914

201

49

212

014

925

201

410

12

014

107

201

410

17

2014

102

420

1411

22

014

119

201

411

16

2014

112

320

1412

72

014

122

120

141

420

151

182

015

Num

ber o

f dea

th ca

ses

Figure 3 Number of recognized death cases

00102030405060708

322

201

43

262

014

331

201

44

220

144

102

014

421

201

44

302

014

514

201

45

272

014

65

2014

618

201

47

102

014

720

201

47

272

014

81

2014

86

2014

811

201

48

162

014

820

201

48

312

014

97

2014

914

201

49

212

014

925

201

410

12

014

107

201

410

17

2014

102

420

1411

22

014

119

201

411

16

2014

112

320

1412

72

014

122

120

141

420

151

182

015

Dea

th ra

te o

f inf

ectio

us ca

ses

Figure 4 Ebola disease-caused death rate of recognized infectious cases

119878(0) = 11 745 000minus49 = 11 744 951Moreover since Ebolaincubation period ranges from 5 to 12 days in most cases[1] then we take 5 days as the incubation period Therefore119864(0) = 37 (that is the reported infected cases at the fifth dayminus the reported infected cases at the first day)

In model (1) there are ten parameters that correspond tothe recruitment rate Λ the natural and disease-caused deathrates 120583 12058310 12058320 and 12058330 the efficient contact rates in theinfected and the latent period 12057310 and 12057320 and the transferrates 1205760 V0 and 1205740 Among these parameters parts of themneed to be fixed and others will be estimated as follows

(P1) The recruitment rate Λ can be given by birth ratewhich is the total number of births per 1000 of apopulation in a year Based on [24ndash26] the 2014 birth

rate of Guinea is 003602 Thus Λ = (003602365) times

11 744 951 = 1 159

(P2) The natural death rate 120583 is typically expressed inunits of deaths per 1000 individuals per year Thuswe use the natural death rate 120583 = 00097365 =

2657 times 10minus5 of Guinea in 2013 [27] For two disease-caused death rates 12058310 in the latent period and 12058330 intreatedrecovery period their values are smaller thanthat of 12058320 in the infectious period Thus we assumethat 12058310 = 12058330 = 0 The rate 12058320 will be estimatedIn order to provide the value range of 12058320 we use themethod in which the accumulative death number isdivided by the accumulative casesThen it is obtainedthat 05 le 12058320 le 08 (see Figure 4)

6 Computational and Mathematical Methods in Medicine

0 50 100 150 200 250 300 3500

500

1000

1500

2000

2500

3000

3500

Day

Accu

mul

ated

infe

ctio

us p

opul

atio

n

Observed infectious valueEstimated value

0 50 100 150 200 250 300 3500

500

1000

1500

2000

2500

Day

Accu

mul

ated

dea

th p

opul

atio

n

Observed death valueEstimated value

Figure 5 Real values and fitting values of the accumulative infectious cases and death cases

(P3) In this paper we consider that Ebola virus during thelatent and exposed period is contagious In generalthe efficient contact rates 12057310 and 12057320 are differentIn order to reduce the risk of human-to-humantransmission WHO raises the awareness of the riskfactors of Ebola infection and the protectivemeasures[28 29]Thus we consider that 0 lt 12057310 le 12057320 lt 1 andall of them will be estimated

(P4) Since Ebola incubation period ranges from 2 to 21days (5ndash12 days in most cases) [1] the transfer rate 1205760is set following the constraint 2 lt 11205760 lt 21 that is004761 lt 1205760 lt 05 For the transfer rates V0 we willestimate it and let it satisfy 0 lt V0 lt 1

(P5) Based on [2 5] we know that the effective treated rate35 lt 11205740 lt 107 that is 0093 lt 1205740 lt 02857

Based on the conditions (P1)ndash(P5) parameters of model(1) to be estimated are 12057310 12057320 12058320 1205760 V0 and 1205740 We list theirestimated values in Table 1

43 Numerical Simulations Using the parameter values inTable 1 numerical simulations of model (1) respectively givethe comparison curves between the real values and fittingvalues of the accumulative infectious cases and accumulativedeath cases fromMarch 22 2014 to January 25 2015 [22] seeFigure 5

Through Figure 5 we observed that the EVD infectionstill does not get the effective control This point can beconfirmed by threshold value 1198770 Based on the estimatedvalues of parameters in Table 1 we have 1198770 = Λ(1205761205731 +

12059621205732)12058312059611205962 = 416 where 120573119894

= 1205731198940120583 119894 = 1 2 Since 1198770 gt 1

byTheorem 2 endemic equilibrium 119875lowast of model (1) is locally

asymptotically stable This indicates that EVD infection stillinfects humans and will be endemic in Guinea without theeffective control measures

Table 1 The estimated values of model parameters by least squaremethod (unit yearminus1)

Parameters Definition Initial values Estimates

12057310

Contact rate ininfectiousperiod

50138 times 10minus11 6474 times 10minus11

12057320

Contact rate inlatent andexposed period

93133 times 10minus8 5685 times 10minus9

12058320

Disease-causeddeath rate ininfectiousperiod

05061 06647

1205760

Transfer ratefrom latent andexposed to theinfectious

04 00596

V0

Transfer rateform theinfectious totreatment state

05140 08613

1205740Effective treatedrate 02 02999

44 Sensitivity and Uncertainty Analysis of 1198770 Because 1198770is an important threshed value for the spread of EVD weperform uncertainty and sensitivity analysis of 1198770 in model(1) using partial rank correlation coefficients (PRCCs) [30]Based on (1) (2) and (5) we get the expression of 1198770 asfollows

1198770 =Λ (120576012057310 + 12057320 (120583 + V0 + 12058320))

120583 (120583 + V0 + 12058320) (120583 + 1205760 + 12058310) (18)

Among all parameters of model (1) we only analyze theinfluence of seven parameters in determining the magnitude

Computational and Mathematical Methods in Medicine 7

0743408429

0090200172

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

minus04145

minus09510

12057310 12057320 12058320 1205760 0 1205740

Figure 6 PRCCs for the effect of six parameters on 1198770

of 1198770 except the fixed numbers Λ 120583 12058310 and 12058330 which havebeen discussed in Section 42 The ordering of these PRCCsdirectly corresponds to the level of statistical influence theimpact that uncertainty in the estimate of a parameter hason the variability of 1198770 A positive PRCC value indicatesthat an increase in that parameter leads to an increase in 1198770while a negative value shows that increasing that parameterdecreases 1198770 For the Ebola SEIT model seven parameterswere significantly different from 0 (119901 value lt 005) Amongthese parameters 12057310 12057320 12058320 and V0 have a positiveinfluence on 1198770 while 1205760 and 1205740 have a negative influence on1198770 see Figure 6

5 Conclusions

In this paper we establish the SEIT model to analyze thedynamical properties of EVD transmission in Guinea Oneof the reasons is that there is still no effective treatments forEVD and the current response is only support treatment [31]In particular if parameters 1205740 = 0 or 1205740 = 0 the SEIT modelwill become the SEIR or SEIRS types

The SEIT model is formed by four-dimensional differ-ential equations In order to understand the transmissionof Ebola virus we discuss the local stability of the disease-free equilibrium and the endemic equilibrium by basicreproduction number1198770 lt 1 and1198770 gt 1 Since there exist tenparameters in the expression of 1198770 it is important to estimatethe value of each parameter By the least square method andthe recorded data in Guinea we obtain the estimation ofseven parameters except for four fixed constants in Table 1In Table 1 we observe that the effective treated rate is only02999 and the disease-caused death rate in infectious periodis 06647Thus it is urgent to provide the effective vaccines tocure the infectious people and protect the susceptible

With suitable parameter values we obtain estimationvalue of 1198770 The result shows that the Ebola virus still infectspeople in Guinea and does not disappear in short timeWithout the effective control measures the EVD may be

endemic in Guinea By the PRCCs method we analyze thesensitivity and uncertainty of 1198770 The result shows that therates of the efficient contact especially in latent and exposedperiod lead to the significant increase in 1198770 (see Figure 6)

On the other hand it would be interesting to study moreproperties of the present model In particular a study involv-ing both stability properties of pulse vaccination strategy andglobal stability is worth pursuing We leave these for futureconsideration

Appendices

A Calculation of 1198871 1198872 1198873 1198874

By (6) (16) and the definition of 119860 we obtain

1198871 = 1205731119868lowast

+ 1205732119864lowast

minus 1205732119878lowast

+ 1205961 + 1205962 + 1205963 + 1

= 119860 minus 120573212059611205962

1205761205731 + 12059621205732+ 1205961 + 1205962 + 1205963 + 1

= 119860 + (1205961 minus120596112059621205732

1205761205731 + 12059621205732) + 1205962 + 1205963 + 1

= 119860 +12057612059611205731

1205761205731 + 12059621205732+ 1205962 + 1205963 + 1

(A1)

Calculating 1198872 we have

1198872 = (1205961 + 1205962 + 1205963) (119860 + 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 1205961 (1205962 + 1205963)

+ 12059621205963

= (1205961 + 1205962 + 1205963) (119860 + 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731)12059611205962

1205761205731 + 12059621205732

+ 1205961 (1205962 + 1205963) + 12059621205963

= (1205961 + 1205962 + 1205963) (119860 + 1) +120576120596112059631205731 minus 120596112059621205731

1205761205731 + 12059621205732

+ 12059621205963

= (1205961 + 1205962 + 1205963) 119860

+1205961 (1205761205731 + 12059621205732) + 120576120596112059631205731 minus 120596112059621205731

1205761205731 + 12059621205732+ 1205962

+ 1205963 + 12059621205963

= (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732+ 1205962

+ 1205963 + 12059621205963

(A2)

8 Computational and Mathematical Methods in Medicine

For 1198873 we have

1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast

+ 1205732119864lowast

)

minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast

+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963

= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)

minus (120596212059631205732 + 12057612059631205731 + (1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 120596112059621205963

= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)

minus ((1205962 + 1205963) 1205732 + 1205761205731)12059611205962

1205761205731 + 12059621205732

= (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731

1205761205731 + 12059621205732

(A3)

Lastly it is easy to get

1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast

+ 1205732119864lowast

)

= (120596112059621205963 minus 120576120574V) 119860

(A4)

B Calculation of 1198873(11988711198872

minus 1198873) minus 1198872

11198874

By calculating we have

1198873 (11988711198872 minus 1198873) = (1205961 + 1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 1198603

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 1198602

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961

+ 1205962 + 1205963) 1198602

+12057612059611205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+ (1205962 + 1205963)

sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963

+ 1) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961

+ 1205962 + 1205963) 120596212059631198602

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963

+ 12059621205963) (12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 11986211198622

(B1)

where

1198621 = 12059621205962 +120576120596112059631205731

1205761205731 + 12059621205732

1198622 =12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732+

1205761205961120596231205731

1205761205731 + 12059621205732

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732

+ (1205962 + 1205963) (1205962 + 1205963 + 1)

+ 12059621205963 (1205962 + 1205963)

(B2)

On the other hand

119887211198874 = 120596112059621205963119860

3+ 2

120576120596112057311205761205731 + 12059621205732

1205961120596212059631198602

+ 2 (1205962 + 1205963 + 1) 1205961120596212059631198602

+ (1205962 + 1205963 + 1)2

120596112059621205963119860

+ (12057612059611205731

1205761205731 + 12059621205732)

2120596112059621205963119860

Computational and Mathematical Methods in Medicine 9

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

minus 120576120574V1198603

minus 212057612059611205731

1205761205731 + 12059621205732120576120574V119860

2

minus 2 (1205962 + 1205963 + 1) 120576120574V1198602

minus (1205962 + 1205963 + 1)2

120576120574V119860

minus (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860

(B3)

Therefore we further have

1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 1198603

minus 1205961120596212059631198603

+12057612059611205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962

+ 1205963) 1198602

minus 212057612059611205731

1205761205731 + 12059621205732120596112059621205963119860

2+ (1205962 + 1205963)

sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963

+ 1) 1198602

minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2

120596112059621205963119860

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731

1205761205731 + 12059621205732)

2120596112059621205963119860

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))

sdot 119860 minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622

gt12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

10 Computational and Mathematical Methods in Medicine

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0

(B4)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)

References

[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013

[2] 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

[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014

[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014

[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006

[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014

[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014

[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1

[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015

[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014

[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014

[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1

[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015

[14] 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

[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012

[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012

[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010

[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014

[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006

[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml

[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen

[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news

Computational and Mathematical Methods in Medicine 11

[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen

[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml

[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates

[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate

[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate

[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen

[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf

[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994

[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Page 5: Research Article Dynamical Analysis of an SEIT Epidemic ...downloads.hindawi.com/journals/cmmm/2015/582625.pdf · Research Article Dynamical Analysis of an SEIT Epidemic Model with

Computational and Mathematical Methods in Medicine 5

020406080

100120140160180200

322

201

43

262

014

331

201

44

220

144

102

014

421

201

44

302

014

514

201

45

272

014

65

2014

618

201

47

102

014

720

201

47

272

014

81

2014

86

2014

811

201

48

162

014

820

201

48

312

014

97

2014

914

201

49

212

014

925

201

410

12

014

107

201

410

17

2014

102

420

1411

22

014

119

201

411

16

2014

112

320

1412

72

014

122

120

141

420

151

182

015N

umbe

r of i

nfec

tious

case

s

Figure 2 Number of recognized infectious cases

020406080

100120

322

201

43

262

014

331

201

44

220

144

102

014

421

201

44

302

014

514

201

45

272

014

65

2014

618

201

47

102

014

720

201

47

272

014

81

2014

86

2014

811

201

48

162

014

820

201

48

312

014

97

2014

914

201

49

212

014

925

201

410

12

014

107

201

410

17

2014

102

420

1411

22

014

119

201

411

16

2014

112

320

1412

72

014

122

120

141

420

151

182

015

Num

ber o

f dea

th ca

ses

Figure 3 Number of recognized death cases

00102030405060708

322

201

43

262

014

331

201

44

220

144

102

014

421

201

44

302

014

514

201

45

272

014

65

2014

618

201

47

102

014

720

201

47

272

014

81

2014

86

2014

811

201

48

162

014

820

201

48

312

014

97

2014

914

201

49

212

014

925

201

410

12

014

107

201

410

17

2014

102

420

1411

22

014

119

201

411

16

2014

112

320

1412

72

014

122

120

141

420

151

182

015

Dea

th ra

te o

f inf

ectio

us ca

ses

Figure 4 Ebola disease-caused death rate of recognized infectious cases

119878(0) = 11 745 000minus49 = 11 744 951Moreover since Ebolaincubation period ranges from 5 to 12 days in most cases[1] then we take 5 days as the incubation period Therefore119864(0) = 37 (that is the reported infected cases at the fifth dayminus the reported infected cases at the first day)

In model (1) there are ten parameters that correspond tothe recruitment rate Λ the natural and disease-caused deathrates 120583 12058310 12058320 and 12058330 the efficient contact rates in theinfected and the latent period 12057310 and 12057320 and the transferrates 1205760 V0 and 1205740 Among these parameters parts of themneed to be fixed and others will be estimated as follows

(P1) The recruitment rate Λ can be given by birth ratewhich is the total number of births per 1000 of apopulation in a year Based on [24ndash26] the 2014 birth

rate of Guinea is 003602 Thus Λ = (003602365) times

11 744 951 = 1 159

(P2) The natural death rate 120583 is typically expressed inunits of deaths per 1000 individuals per year Thuswe use the natural death rate 120583 = 00097365 =

2657 times 10minus5 of Guinea in 2013 [27] For two disease-caused death rates 12058310 in the latent period and 12058330 intreatedrecovery period their values are smaller thanthat of 12058320 in the infectious period Thus we assumethat 12058310 = 12058330 = 0 The rate 12058320 will be estimatedIn order to provide the value range of 12058320 we use themethod in which the accumulative death number isdivided by the accumulative casesThen it is obtainedthat 05 le 12058320 le 08 (see Figure 4)

6 Computational and Mathematical Methods in Medicine

0 50 100 150 200 250 300 3500

500

1000

1500

2000

2500

3000

3500

Day

Accu

mul

ated

infe

ctio

us p

opul

atio

n

Observed infectious valueEstimated value

0 50 100 150 200 250 300 3500

500

1000

1500

2000

2500

Day

Accu

mul

ated

dea

th p

opul

atio

n

Observed death valueEstimated value

Figure 5 Real values and fitting values of the accumulative infectious cases and death cases

(P3) In this paper we consider that Ebola virus during thelatent and exposed period is contagious In generalthe efficient contact rates 12057310 and 12057320 are differentIn order to reduce the risk of human-to-humantransmission WHO raises the awareness of the riskfactors of Ebola infection and the protectivemeasures[28 29]Thus we consider that 0 lt 12057310 le 12057320 lt 1 andall of them will be estimated

(P4) Since Ebola incubation period ranges from 2 to 21days (5ndash12 days in most cases) [1] the transfer rate 1205760is set following the constraint 2 lt 11205760 lt 21 that is004761 lt 1205760 lt 05 For the transfer rates V0 we willestimate it and let it satisfy 0 lt V0 lt 1

(P5) Based on [2 5] we know that the effective treated rate35 lt 11205740 lt 107 that is 0093 lt 1205740 lt 02857

Based on the conditions (P1)ndash(P5) parameters of model(1) to be estimated are 12057310 12057320 12058320 1205760 V0 and 1205740 We list theirestimated values in Table 1

43 Numerical Simulations Using the parameter values inTable 1 numerical simulations of model (1) respectively givethe comparison curves between the real values and fittingvalues of the accumulative infectious cases and accumulativedeath cases fromMarch 22 2014 to January 25 2015 [22] seeFigure 5

Through Figure 5 we observed that the EVD infectionstill does not get the effective control This point can beconfirmed by threshold value 1198770 Based on the estimatedvalues of parameters in Table 1 we have 1198770 = Λ(1205761205731 +

12059621205732)12058312059611205962 = 416 where 120573119894

= 1205731198940120583 119894 = 1 2 Since 1198770 gt 1

byTheorem 2 endemic equilibrium 119875lowast of model (1) is locally

asymptotically stable This indicates that EVD infection stillinfects humans and will be endemic in Guinea without theeffective control measures

Table 1 The estimated values of model parameters by least squaremethod (unit yearminus1)

Parameters Definition Initial values Estimates

12057310

Contact rate ininfectiousperiod

50138 times 10minus11 6474 times 10minus11

12057320

Contact rate inlatent andexposed period

93133 times 10minus8 5685 times 10minus9

12058320

Disease-causeddeath rate ininfectiousperiod

05061 06647

1205760

Transfer ratefrom latent andexposed to theinfectious

04 00596

V0

Transfer rateform theinfectious totreatment state

05140 08613

1205740Effective treatedrate 02 02999

44 Sensitivity and Uncertainty Analysis of 1198770 Because 1198770is an important threshed value for the spread of EVD weperform uncertainty and sensitivity analysis of 1198770 in model(1) using partial rank correlation coefficients (PRCCs) [30]Based on (1) (2) and (5) we get the expression of 1198770 asfollows

1198770 =Λ (120576012057310 + 12057320 (120583 + V0 + 12058320))

120583 (120583 + V0 + 12058320) (120583 + 1205760 + 12058310) (18)

Among all parameters of model (1) we only analyze theinfluence of seven parameters in determining the magnitude

Computational and Mathematical Methods in Medicine 7

0743408429

0090200172

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

minus04145

minus09510

12057310 12057320 12058320 1205760 0 1205740

Figure 6 PRCCs for the effect of six parameters on 1198770

of 1198770 except the fixed numbers Λ 120583 12058310 and 12058330 which havebeen discussed in Section 42 The ordering of these PRCCsdirectly corresponds to the level of statistical influence theimpact that uncertainty in the estimate of a parameter hason the variability of 1198770 A positive PRCC value indicatesthat an increase in that parameter leads to an increase in 1198770while a negative value shows that increasing that parameterdecreases 1198770 For the Ebola SEIT model seven parameterswere significantly different from 0 (119901 value lt 005) Amongthese parameters 12057310 12057320 12058320 and V0 have a positiveinfluence on 1198770 while 1205760 and 1205740 have a negative influence on1198770 see Figure 6

5 Conclusions

In this paper we establish the SEIT model to analyze thedynamical properties of EVD transmission in Guinea Oneof the reasons is that there is still no effective treatments forEVD and the current response is only support treatment [31]In particular if parameters 1205740 = 0 or 1205740 = 0 the SEIT modelwill become the SEIR or SEIRS types

The SEIT model is formed by four-dimensional differ-ential equations In order to understand the transmissionof Ebola virus we discuss the local stability of the disease-free equilibrium and the endemic equilibrium by basicreproduction number1198770 lt 1 and1198770 gt 1 Since there exist tenparameters in the expression of 1198770 it is important to estimatethe value of each parameter By the least square method andthe recorded data in Guinea we obtain the estimation ofseven parameters except for four fixed constants in Table 1In Table 1 we observe that the effective treated rate is only02999 and the disease-caused death rate in infectious periodis 06647Thus it is urgent to provide the effective vaccines tocure the infectious people and protect the susceptible

With suitable parameter values we obtain estimationvalue of 1198770 The result shows that the Ebola virus still infectspeople in Guinea and does not disappear in short timeWithout the effective control measures the EVD may be

endemic in Guinea By the PRCCs method we analyze thesensitivity and uncertainty of 1198770 The result shows that therates of the efficient contact especially in latent and exposedperiod lead to the significant increase in 1198770 (see Figure 6)

On the other hand it would be interesting to study moreproperties of the present model In particular a study involv-ing both stability properties of pulse vaccination strategy andglobal stability is worth pursuing We leave these for futureconsideration

Appendices

A Calculation of 1198871 1198872 1198873 1198874

By (6) (16) and the definition of 119860 we obtain

1198871 = 1205731119868lowast

+ 1205732119864lowast

minus 1205732119878lowast

+ 1205961 + 1205962 + 1205963 + 1

= 119860 minus 120573212059611205962

1205761205731 + 12059621205732+ 1205961 + 1205962 + 1205963 + 1

= 119860 + (1205961 minus120596112059621205732

1205761205731 + 12059621205732) + 1205962 + 1205963 + 1

= 119860 +12057612059611205731

1205761205731 + 12059621205732+ 1205962 + 1205963 + 1

(A1)

Calculating 1198872 we have

1198872 = (1205961 + 1205962 + 1205963) (119860 + 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 1205961 (1205962 + 1205963)

+ 12059621205963

= (1205961 + 1205962 + 1205963) (119860 + 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731)12059611205962

1205761205731 + 12059621205732

+ 1205961 (1205962 + 1205963) + 12059621205963

= (1205961 + 1205962 + 1205963) (119860 + 1) +120576120596112059631205731 minus 120596112059621205731

1205761205731 + 12059621205732

+ 12059621205963

= (1205961 + 1205962 + 1205963) 119860

+1205961 (1205761205731 + 12059621205732) + 120576120596112059631205731 minus 120596112059621205731

1205761205731 + 12059621205732+ 1205962

+ 1205963 + 12059621205963

= (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732+ 1205962

+ 1205963 + 12059621205963

(A2)

8 Computational and Mathematical Methods in Medicine

For 1198873 we have

1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast

+ 1205732119864lowast

)

minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast

+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963

= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)

minus (120596212059631205732 + 12057612059631205731 + (1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 120596112059621205963

= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)

minus ((1205962 + 1205963) 1205732 + 1205761205731)12059611205962

1205761205731 + 12059621205732

= (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731

1205761205731 + 12059621205732

(A3)

Lastly it is easy to get

1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast

+ 1205732119864lowast

)

= (120596112059621205963 minus 120576120574V) 119860

(A4)

B Calculation of 1198873(11988711198872

minus 1198873) minus 1198872

11198874

By calculating we have

1198873 (11988711198872 minus 1198873) = (1205961 + 1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 1198603

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 1198602

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961

+ 1205962 + 1205963) 1198602

+12057612059611205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+ (1205962 + 1205963)

sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963

+ 1) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961

+ 1205962 + 1205963) 120596212059631198602

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963

+ 12059621205963) (12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 11986211198622

(B1)

where

1198621 = 12059621205962 +120576120596112059631205731

1205761205731 + 12059621205732

1198622 =12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732+

1205761205961120596231205731

1205761205731 + 12059621205732

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732

+ (1205962 + 1205963) (1205962 + 1205963 + 1)

+ 12059621205963 (1205962 + 1205963)

(B2)

On the other hand

119887211198874 = 120596112059621205963119860

3+ 2

120576120596112057311205761205731 + 12059621205732

1205961120596212059631198602

+ 2 (1205962 + 1205963 + 1) 1205961120596212059631198602

+ (1205962 + 1205963 + 1)2

120596112059621205963119860

+ (12057612059611205731

1205761205731 + 12059621205732)

2120596112059621205963119860

Computational and Mathematical Methods in Medicine 9

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

minus 120576120574V1198603

minus 212057612059611205731

1205761205731 + 12059621205732120576120574V119860

2

minus 2 (1205962 + 1205963 + 1) 120576120574V1198602

minus (1205962 + 1205963 + 1)2

120576120574V119860

minus (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860

(B3)

Therefore we further have

1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 1198603

minus 1205961120596212059631198603

+12057612059611205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962

+ 1205963) 1198602

minus 212057612059611205731

1205761205731 + 12059621205732120596112059621205963119860

2+ (1205962 + 1205963)

sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963

+ 1) 1198602

minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2

120596112059621205963119860

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731

1205761205731 + 12059621205732)

2120596112059621205963119860

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))

sdot 119860 minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622

gt12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

10 Computational and Mathematical Methods in Medicine

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0

(B4)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)

References

[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013

[2] 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

[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014

[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014

[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006

[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014

[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014

[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1

[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015

[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014

[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014

[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1

[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015

[14] 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

[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012

[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012

[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010

[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014

[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006

[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml

[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen

[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news

Computational and Mathematical Methods in Medicine 11

[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen

[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml

[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates

[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate

[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate

[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen

[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf

[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994

[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

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Diabetes ResearchJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Page 6: Research Article Dynamical Analysis of an SEIT Epidemic ...downloads.hindawi.com/journals/cmmm/2015/582625.pdf · Research Article Dynamical Analysis of an SEIT Epidemic Model with

6 Computational and Mathematical Methods in Medicine

0 50 100 150 200 250 300 3500

500

1000

1500

2000

2500

3000

3500

Day

Accu

mul

ated

infe

ctio

us p

opul

atio

n

Observed infectious valueEstimated value

0 50 100 150 200 250 300 3500

500

1000

1500

2000

2500

Day

Accu

mul

ated

dea

th p

opul

atio

n

Observed death valueEstimated value

Figure 5 Real values and fitting values of the accumulative infectious cases and death cases

(P3) In this paper we consider that Ebola virus during thelatent and exposed period is contagious In generalthe efficient contact rates 12057310 and 12057320 are differentIn order to reduce the risk of human-to-humantransmission WHO raises the awareness of the riskfactors of Ebola infection and the protectivemeasures[28 29]Thus we consider that 0 lt 12057310 le 12057320 lt 1 andall of them will be estimated

(P4) Since Ebola incubation period ranges from 2 to 21days (5ndash12 days in most cases) [1] the transfer rate 1205760is set following the constraint 2 lt 11205760 lt 21 that is004761 lt 1205760 lt 05 For the transfer rates V0 we willestimate it and let it satisfy 0 lt V0 lt 1

(P5) Based on [2 5] we know that the effective treated rate35 lt 11205740 lt 107 that is 0093 lt 1205740 lt 02857

Based on the conditions (P1)ndash(P5) parameters of model(1) to be estimated are 12057310 12057320 12058320 1205760 V0 and 1205740 We list theirestimated values in Table 1

43 Numerical Simulations Using the parameter values inTable 1 numerical simulations of model (1) respectively givethe comparison curves between the real values and fittingvalues of the accumulative infectious cases and accumulativedeath cases fromMarch 22 2014 to January 25 2015 [22] seeFigure 5

Through Figure 5 we observed that the EVD infectionstill does not get the effective control This point can beconfirmed by threshold value 1198770 Based on the estimatedvalues of parameters in Table 1 we have 1198770 = Λ(1205761205731 +

12059621205732)12058312059611205962 = 416 where 120573119894

= 1205731198940120583 119894 = 1 2 Since 1198770 gt 1

byTheorem 2 endemic equilibrium 119875lowast of model (1) is locally

asymptotically stable This indicates that EVD infection stillinfects humans and will be endemic in Guinea without theeffective control measures

Table 1 The estimated values of model parameters by least squaremethod (unit yearminus1)

Parameters Definition Initial values Estimates

12057310

Contact rate ininfectiousperiod

50138 times 10minus11 6474 times 10minus11

12057320

Contact rate inlatent andexposed period

93133 times 10minus8 5685 times 10minus9

12058320

Disease-causeddeath rate ininfectiousperiod

05061 06647

1205760

Transfer ratefrom latent andexposed to theinfectious

04 00596

V0

Transfer rateform theinfectious totreatment state

05140 08613

1205740Effective treatedrate 02 02999

44 Sensitivity and Uncertainty Analysis of 1198770 Because 1198770is an important threshed value for the spread of EVD weperform uncertainty and sensitivity analysis of 1198770 in model(1) using partial rank correlation coefficients (PRCCs) [30]Based on (1) (2) and (5) we get the expression of 1198770 asfollows

1198770 =Λ (120576012057310 + 12057320 (120583 + V0 + 12058320))

120583 (120583 + V0 + 12058320) (120583 + 1205760 + 12058310) (18)

Among all parameters of model (1) we only analyze theinfluence of seven parameters in determining the magnitude

Computational and Mathematical Methods in Medicine 7

0743408429

0090200172

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

minus04145

minus09510

12057310 12057320 12058320 1205760 0 1205740

Figure 6 PRCCs for the effect of six parameters on 1198770

of 1198770 except the fixed numbers Λ 120583 12058310 and 12058330 which havebeen discussed in Section 42 The ordering of these PRCCsdirectly corresponds to the level of statistical influence theimpact that uncertainty in the estimate of a parameter hason the variability of 1198770 A positive PRCC value indicatesthat an increase in that parameter leads to an increase in 1198770while a negative value shows that increasing that parameterdecreases 1198770 For the Ebola SEIT model seven parameterswere significantly different from 0 (119901 value lt 005) Amongthese parameters 12057310 12057320 12058320 and V0 have a positiveinfluence on 1198770 while 1205760 and 1205740 have a negative influence on1198770 see Figure 6

5 Conclusions

In this paper we establish the SEIT model to analyze thedynamical properties of EVD transmission in Guinea Oneof the reasons is that there is still no effective treatments forEVD and the current response is only support treatment [31]In particular if parameters 1205740 = 0 or 1205740 = 0 the SEIT modelwill become the SEIR or SEIRS types

The SEIT model is formed by four-dimensional differ-ential equations In order to understand the transmissionof Ebola virus we discuss the local stability of the disease-free equilibrium and the endemic equilibrium by basicreproduction number1198770 lt 1 and1198770 gt 1 Since there exist tenparameters in the expression of 1198770 it is important to estimatethe value of each parameter By the least square method andthe recorded data in Guinea we obtain the estimation ofseven parameters except for four fixed constants in Table 1In Table 1 we observe that the effective treated rate is only02999 and the disease-caused death rate in infectious periodis 06647Thus it is urgent to provide the effective vaccines tocure the infectious people and protect the susceptible

With suitable parameter values we obtain estimationvalue of 1198770 The result shows that the Ebola virus still infectspeople in Guinea and does not disappear in short timeWithout the effective control measures the EVD may be

endemic in Guinea By the PRCCs method we analyze thesensitivity and uncertainty of 1198770 The result shows that therates of the efficient contact especially in latent and exposedperiod lead to the significant increase in 1198770 (see Figure 6)

On the other hand it would be interesting to study moreproperties of the present model In particular a study involv-ing both stability properties of pulse vaccination strategy andglobal stability is worth pursuing We leave these for futureconsideration

Appendices

A Calculation of 1198871 1198872 1198873 1198874

By (6) (16) and the definition of 119860 we obtain

1198871 = 1205731119868lowast

+ 1205732119864lowast

minus 1205732119878lowast

+ 1205961 + 1205962 + 1205963 + 1

= 119860 minus 120573212059611205962

1205761205731 + 12059621205732+ 1205961 + 1205962 + 1205963 + 1

= 119860 + (1205961 minus120596112059621205732

1205761205731 + 12059621205732) + 1205962 + 1205963 + 1

= 119860 +12057612059611205731

1205761205731 + 12059621205732+ 1205962 + 1205963 + 1

(A1)

Calculating 1198872 we have

1198872 = (1205961 + 1205962 + 1205963) (119860 + 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 1205961 (1205962 + 1205963)

+ 12059621205963

= (1205961 + 1205962 + 1205963) (119860 + 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731)12059611205962

1205761205731 + 12059621205732

+ 1205961 (1205962 + 1205963) + 12059621205963

= (1205961 + 1205962 + 1205963) (119860 + 1) +120576120596112059631205731 minus 120596112059621205731

1205761205731 + 12059621205732

+ 12059621205963

= (1205961 + 1205962 + 1205963) 119860

+1205961 (1205761205731 + 12059621205732) + 120576120596112059631205731 minus 120596112059621205731

1205761205731 + 12059621205732+ 1205962

+ 1205963 + 12059621205963

= (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732+ 1205962

+ 1205963 + 12059621205963

(A2)

8 Computational and Mathematical Methods in Medicine

For 1198873 we have

1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast

+ 1205732119864lowast

)

minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast

+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963

= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)

minus (120596212059631205732 + 12057612059631205731 + (1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 120596112059621205963

= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)

minus ((1205962 + 1205963) 1205732 + 1205761205731)12059611205962

1205761205731 + 12059621205732

= (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731

1205761205731 + 12059621205732

(A3)

Lastly it is easy to get

1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast

+ 1205732119864lowast

)

= (120596112059621205963 minus 120576120574V) 119860

(A4)

B Calculation of 1198873(11988711198872

minus 1198873) minus 1198872

11198874

By calculating we have

1198873 (11988711198872 minus 1198873) = (1205961 + 1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 1198603

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 1198602

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961

+ 1205962 + 1205963) 1198602

+12057612059611205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+ (1205962 + 1205963)

sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963

+ 1) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961

+ 1205962 + 1205963) 120596212059631198602

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963

+ 12059621205963) (12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 11986211198622

(B1)

where

1198621 = 12059621205962 +120576120596112059631205731

1205761205731 + 12059621205732

1198622 =12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732+

1205761205961120596231205731

1205761205731 + 12059621205732

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732

+ (1205962 + 1205963) (1205962 + 1205963 + 1)

+ 12059621205963 (1205962 + 1205963)

(B2)

On the other hand

119887211198874 = 120596112059621205963119860

3+ 2

120576120596112057311205761205731 + 12059621205732

1205961120596212059631198602

+ 2 (1205962 + 1205963 + 1) 1205961120596212059631198602

+ (1205962 + 1205963 + 1)2

120596112059621205963119860

+ (12057612059611205731

1205761205731 + 12059621205732)

2120596112059621205963119860

Computational and Mathematical Methods in Medicine 9

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

minus 120576120574V1198603

minus 212057612059611205731

1205761205731 + 12059621205732120576120574V119860

2

minus 2 (1205962 + 1205963 + 1) 120576120574V1198602

minus (1205962 + 1205963 + 1)2

120576120574V119860

minus (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860

(B3)

Therefore we further have

1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 1198603

minus 1205961120596212059631198603

+12057612059611205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962

+ 1205963) 1198602

minus 212057612059611205731

1205761205731 + 12059621205732120596112059621205963119860

2+ (1205962 + 1205963)

sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963

+ 1) 1198602

minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2

120596112059621205963119860

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731

1205761205731 + 12059621205732)

2120596112059621205963119860

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))

sdot 119860 minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622

gt12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

10 Computational and Mathematical Methods in Medicine

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0

(B4)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)

References

[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013

[2] 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

[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014

[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014

[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006

[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014

[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014

[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1

[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015

[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014

[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014

[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1

[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015

[14] 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

[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012

[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012

[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010

[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014

[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006

[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml

[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen

[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news

Computational and Mathematical Methods in Medicine 11

[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen

[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml

[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates

[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate

[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate

[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen

[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf

[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994

[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Page 7: Research Article Dynamical Analysis of an SEIT Epidemic ...downloads.hindawi.com/journals/cmmm/2015/582625.pdf · Research Article Dynamical Analysis of an SEIT Epidemic Model with

Computational and Mathematical Methods in Medicine 7

0743408429

0090200172

minus12

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

minus04145

minus09510

12057310 12057320 12058320 1205760 0 1205740

Figure 6 PRCCs for the effect of six parameters on 1198770

of 1198770 except the fixed numbers Λ 120583 12058310 and 12058330 which havebeen discussed in Section 42 The ordering of these PRCCsdirectly corresponds to the level of statistical influence theimpact that uncertainty in the estimate of a parameter hason the variability of 1198770 A positive PRCC value indicatesthat an increase in that parameter leads to an increase in 1198770while a negative value shows that increasing that parameterdecreases 1198770 For the Ebola SEIT model seven parameterswere significantly different from 0 (119901 value lt 005) Amongthese parameters 12057310 12057320 12058320 and V0 have a positiveinfluence on 1198770 while 1205760 and 1205740 have a negative influence on1198770 see Figure 6

5 Conclusions

In this paper we establish the SEIT model to analyze thedynamical properties of EVD transmission in Guinea Oneof the reasons is that there is still no effective treatments forEVD and the current response is only support treatment [31]In particular if parameters 1205740 = 0 or 1205740 = 0 the SEIT modelwill become the SEIR or SEIRS types

The SEIT model is formed by four-dimensional differ-ential equations In order to understand the transmissionof Ebola virus we discuss the local stability of the disease-free equilibrium and the endemic equilibrium by basicreproduction number1198770 lt 1 and1198770 gt 1 Since there exist tenparameters in the expression of 1198770 it is important to estimatethe value of each parameter By the least square method andthe recorded data in Guinea we obtain the estimation ofseven parameters except for four fixed constants in Table 1In Table 1 we observe that the effective treated rate is only02999 and the disease-caused death rate in infectious periodis 06647Thus it is urgent to provide the effective vaccines tocure the infectious people and protect the susceptible

With suitable parameter values we obtain estimationvalue of 1198770 The result shows that the Ebola virus still infectspeople in Guinea and does not disappear in short timeWithout the effective control measures the EVD may be

endemic in Guinea By the PRCCs method we analyze thesensitivity and uncertainty of 1198770 The result shows that therates of the efficient contact especially in latent and exposedperiod lead to the significant increase in 1198770 (see Figure 6)

On the other hand it would be interesting to study moreproperties of the present model In particular a study involv-ing both stability properties of pulse vaccination strategy andglobal stability is worth pursuing We leave these for futureconsideration

Appendices

A Calculation of 1198871 1198872 1198873 1198874

By (6) (16) and the definition of 119860 we obtain

1198871 = 1205731119868lowast

+ 1205732119864lowast

minus 1205732119878lowast

+ 1205961 + 1205962 + 1205963 + 1

= 119860 minus 120573212059611205962

1205761205731 + 12059621205732+ 1205961 + 1205962 + 1205963 + 1

= 119860 + (1205961 minus120596112059621205732

1205761205731 + 12059621205732) + 1205962 + 1205963 + 1

= 119860 +12057612059611205731

1205761205731 + 12059621205732+ 1205962 + 1205963 + 1

(A1)

Calculating 1198872 we have

1198872 = (1205961 + 1205962 + 1205963) (119860 + 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 1205961 (1205962 + 1205963)

+ 12059621205963

= (1205961 + 1205962 + 1205963) (119860 + 1)

minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731)12059611205962

1205761205731 + 12059621205732

+ 1205961 (1205962 + 1205963) + 12059621205963

= (1205961 + 1205962 + 1205963) (119860 + 1) +120576120596112059631205731 minus 120596112059621205731

1205761205731 + 12059621205732

+ 12059621205963

= (1205961 + 1205962 + 1205963) 119860

+1205961 (1205761205731 + 12059621205732) + 120576120596112059631205731 minus 120596112059621205731

1205761205731 + 12059621205732+ 1205962

+ 1205963 + 12059621205963

= (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732+ 1205962

+ 1205963 + 12059621205963

(A2)

8 Computational and Mathematical Methods in Medicine

For 1198873 we have

1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast

+ 1205732119864lowast

)

minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast

+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963

= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)

minus (120596212059631205732 + 12057612059631205731 + (1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 120596112059621205963

= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)

minus ((1205962 + 1205963) 1205732 + 1205761205731)12059611205962

1205761205731 + 12059621205732

= (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731

1205761205731 + 12059621205732

(A3)

Lastly it is easy to get

1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast

+ 1205732119864lowast

)

= (120596112059621205963 minus 120576120574V) 119860

(A4)

B Calculation of 1198873(11988711198872

minus 1198873) minus 1198872

11198874

By calculating we have

1198873 (11988711198872 minus 1198873) = (1205961 + 1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 1198603

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 1198602

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961

+ 1205962 + 1205963) 1198602

+12057612059611205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+ (1205962 + 1205963)

sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963

+ 1) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961

+ 1205962 + 1205963) 120596212059631198602

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963

+ 12059621205963) (12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 11986211198622

(B1)

where

1198621 = 12059621205962 +120576120596112059631205731

1205761205731 + 12059621205732

1198622 =12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732+

1205761205961120596231205731

1205761205731 + 12059621205732

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732

+ (1205962 + 1205963) (1205962 + 1205963 + 1)

+ 12059621205963 (1205962 + 1205963)

(B2)

On the other hand

119887211198874 = 120596112059621205963119860

3+ 2

120576120596112057311205761205731 + 12059621205732

1205961120596212059631198602

+ 2 (1205962 + 1205963 + 1) 1205961120596212059631198602

+ (1205962 + 1205963 + 1)2

120596112059621205963119860

+ (12057612059611205731

1205761205731 + 12059621205732)

2120596112059621205963119860

Computational and Mathematical Methods in Medicine 9

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

minus 120576120574V1198603

minus 212057612059611205731

1205761205731 + 12059621205732120576120574V119860

2

minus 2 (1205962 + 1205963 + 1) 120576120574V1198602

minus (1205962 + 1205963 + 1)2

120576120574V119860

minus (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860

(B3)

Therefore we further have

1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 1198603

minus 1205961120596212059631198603

+12057612059611205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962

+ 1205963) 1198602

minus 212057612059611205731

1205761205731 + 12059621205732120596112059621205963119860

2+ (1205962 + 1205963)

sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963

+ 1) 1198602

minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2

120596112059621205963119860

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731

1205761205731 + 12059621205732)

2120596112059621205963119860

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))

sdot 119860 minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622

gt12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

10 Computational and Mathematical Methods in Medicine

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0

(B4)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)

References

[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013

[2] 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

[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014

[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014

[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006

[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014

[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014

[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1

[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015

[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014

[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014

[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1

[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015

[14] 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

[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012

[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012

[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010

[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014

[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006

[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml

[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen

[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news

Computational and Mathematical Methods in Medicine 11

[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen

[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml

[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates

[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate

[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate

[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen

[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf

[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994

[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Page 8: Research Article Dynamical Analysis of an SEIT Epidemic ...downloads.hindawi.com/journals/cmmm/2015/582625.pdf · Research Article Dynamical Analysis of an SEIT Epidemic Model with

8 Computational and Mathematical Methods in Medicine

For 1198873 we have

1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast

+ 1205732119864lowast

)

minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast

+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963

= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)

minus (120596212059631205732 + 12057612059631205731 + (1205962 + 1205963) 1205732 + 1205761205731) 119878lowast

+ 120596112059621205963

= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)

minus ((1205962 + 1205963) 1205732 + 1205761205731)12059611205962

1205761205731 + 12059621205732

= (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731

1205761205731 + 12059621205732

(A3)

Lastly it is easy to get

1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast

+ 1205732119864lowast

)

= (120596112059621205963 minus 120576120574V) 119860

(A4)

B Calculation of 1198873(11988711198872

minus 1198873) minus 1198872

11198874

By calculating we have

1198873 (11988711198872 minus 1198873) = (1205961 + 1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 1198603

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 1198602

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961

+ 1205962 + 1205963) 1198602

+12057612059611205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+ (1205962 + 1205963)

sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963

+ 1) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961

+ 1205962 + 1205963) 120596212059631198602

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963

+ 12059621205963) (12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 11986211198622

(B1)

where

1198621 = 12059621205962 +120576120596112059631205731

1205761205731 + 12059621205732

1198622 =12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732+

1205761205961120596231205731

1205761205731 + 12059621205732

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732

+ (1205962 + 1205963) (1205962 + 1205963 + 1)

+ 12059621205963 (1205962 + 1205963)

(B2)

On the other hand

119887211198874 = 120596112059621205963119860

3+ 2

120576120596112057311205761205731 + 12059621205732

1205961120596212059631198602

+ 2 (1205962 + 1205963 + 1) 1205961120596212059631198602

+ (1205962 + 1205963 + 1)2

120596112059621205963119860

+ (12057612059611205731

1205761205731 + 12059621205732)

2120596112059621205963119860

Computational and Mathematical Methods in Medicine 9

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

minus 120576120574V1198603

minus 212057612059611205731

1205761205731 + 12059621205732120576120574V119860

2

minus 2 (1205962 + 1205963 + 1) 120576120574V1198602

minus (1205962 + 1205963 + 1)2

120576120574V119860

minus (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860

(B3)

Therefore we further have

1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 1198603

minus 1205961120596212059631198603

+12057612059611205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962

+ 1205963) 1198602

minus 212057612059611205731

1205761205731 + 12059621205732120596112059621205963119860

2+ (1205962 + 1205963)

sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963

+ 1) 1198602

minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2

120596112059621205963119860

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731

1205761205731 + 12059621205732)

2120596112059621205963119860

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))

sdot 119860 minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622

gt12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

10 Computational and Mathematical Methods in Medicine

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0

(B4)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)

References

[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013

[2] 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

[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014

[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014

[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006

[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014

[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014

[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1

[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015

[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014

[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014

[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1

[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015

[14] 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

[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012

[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012

[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010

[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014

[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006

[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml

[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen

[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news

Computational and Mathematical Methods in Medicine 11

[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen

[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml

[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates

[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate

[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate

[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen

[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf

[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994

[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Page 9: Research Article Dynamical Analysis of an SEIT Epidemic ...downloads.hindawi.com/journals/cmmm/2015/582625.pdf · Research Article Dynamical Analysis of an SEIT Epidemic Model with

Computational and Mathematical Methods in Medicine 9

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

minus 120576120574V1198603

minus 212057612059611205731

1205761205731 + 12059621205732120576120574V119860

2

minus 2 (1205962 + 1205963 + 1) 120576120574V1198602

minus (1205962 + 1205963 + 1)2

120576120574V119860

minus (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860

(B3)

Therefore we further have

1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 1198603

minus 1205961120596212059631198603

+12057612059611205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962

+ 1205963) 1198602

minus 212057612059611205731

1205761205731 + 12059621205732120596112059621205963119860

2+ (1205962 + 1205963)

sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963

+ 1) 1198602

minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2

120596112059621205963119860

+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 (12059621205963

+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731

1205761205731 + 12059621205732)

2120596112059621205963119860

+12057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963

+ 1205961 (1205962 + 1205963)) 119860

+120576120596112059621205731 + 1205761205961120596212059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))

sdot 119860 minus 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622

gt12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

2

+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

2+ (1205961 + 1205962 + 1205963)

sdot 120596212059631198602

+1205761205961120596

231205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860

+12057612059611205731 + 120576120596112059631205731

1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)

sdot 12059621205963119860 +12057612059611205731

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860

10 Computational and Mathematical Methods in Medicine

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0

(B4)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)

References

[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013

[2] 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

[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014

[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014

[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006

[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014

[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014

[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1

[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015

[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014

[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014

[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1

[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015

[14] 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

[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012

[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012

[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010

[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014

[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006

[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml

[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen

[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news

Computational and Mathematical Methods in Medicine 11

[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen

[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml

[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates

[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate

[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate

[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen

[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf

[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994

[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Page 10: Research Article Dynamical Analysis of an SEIT Epidemic ...downloads.hindawi.com/journals/cmmm/2015/582625.pdf · Research Article Dynamical Analysis of an SEIT Epidemic Model with

10 Computational and Mathematical Methods in Medicine

+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860

+120576120596112059631205731

1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860

+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)

(1205761205731 + 12059621205732)2 119860

+1205762120596211205963120573

21

1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860

3+ 2

sdot12057612059611205731

1205761205731 + 12059621205732120576120574V119860

2+ 2 (1205962 + 1205963 + 1) 120576120574V119860

2

+ (1205962 + 1205963 + 1)2

120576120574V119860 + (12057612059611205731

1205761205731 + 12059621205732)

2120576120574V119860

+ 212057612059611205731

1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0

(B4)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)

References

[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013

[2] 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

[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014

[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014

[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006

[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014

[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014

[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1

[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015

[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014

[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014

[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1

[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015

[14] 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

[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012

[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012

[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010

[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014

[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006

[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml

[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen

[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news

Computational and Mathematical Methods in Medicine 11

[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen

[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml

[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates

[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate

[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate

[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen

[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf

[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994

[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Page 11: Research Article Dynamical Analysis of an SEIT Epidemic ...downloads.hindawi.com/journals/cmmm/2015/582625.pdf · Research Article Dynamical Analysis of an SEIT Epidemic Model with

Computational and Mathematical Methods in Medicine 11

[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen

[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml

[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates

[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate

[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate

[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen

[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf

[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994

[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Page 12: Research Article Dynamical Analysis of an SEIT Epidemic ...downloads.hindawi.com/journals/cmmm/2015/582625.pdf · Research Article Dynamical Analysis of an SEIT Epidemic Model with

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


Software Engineering and Internet Technologies Laboratory - SEIT lab

Software Engineering and Internet Technologies Laboratory - SEIT lab

INTEGRATIVE SYSTEMATICS (SEIT 2018)...INTEGRATIVE SYSTEMATICS (SEIT 2018) 2018 – Integrative Systematics 1 Schawaller, W. (2018): Taxonomy and distribution of the species of the

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Vergaberecht - SSF · „Vergaberecht“ geführt. Seit Jänner 1991 ist Dr. reitenfeld Partner bei Siemer – Siegl – Füreder & Partner, Rechtsanwälte, wobei er seit mehr als

Vergaberecht - SSF · „Vergaberecht“ geführt. Seit Jänner 1991 ist Dr. reitenfeld Partner bei Siemer – Siegl – Füreder & Partner, Rechtsanwälte, wobei er seit mehr als

Dynamical Performance Investigation of Dynamical Model ...Dynamical Performance Investigation of Dynamical Model Based Maximum Power Point Tracking Controller in Solar Photovoltaic

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SPECIALS 2020! - BEAUTY FORUM€¦ · Fr e Seit GW s Seit I s Seit Io Seit K Ethetika aris Seit Mar I Seit Oc Seit Orly Seit Raue Seit Y ac e/MakeUp 082 Happy New Year Aufmacher.indd

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SEP SEIT DIGIT -) INSTITUTO TECNOLIÓGICO DE TOLUCA …

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Obesit A Worldwide Epidemic Obesit A Worldwide Epidemic.

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Dynamical Dark Energy. What is dynamical dark energy ?

Curriculum vitae - uniklinikum-saarland.de · Journal of Refractive Surgery - seit 1999 Journal of Cataract and Refractive Surgery - seit 2000 British Journal of Ophthalmology - seit

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Willi Ross – Fleischerei seit 1899 › wp-content › uploads › 2019 › 01 › Stern_N… · seit 2008 stellt er in seiner Genuss-Manufaktur „hausge- machte Leckereien" her.

Willi Ross – Fleischerei seit 1899 › wp-content › uploads › 2019 › 01 › Stern_N… · seit 2008 stellt er in seiner Genuss-Manufaktur „hausge- machte Leckereien" her.

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Epidemic analysis of COVID-19 in China by dynamical modeling€¦ · 16/02/2020  · Epidemic analysis of COVID-19 in China by dynamical modeling Liangrong Peng1a), Wuyue Yang2a),

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Microsoft PowerPoint - Trainer.pptx · Web viewDan; Karate seit 19911. Kyu; Karate seit 20043. Kyu, Karate seit 1998 Foto Gesicht Foto Aktion Foto Gesicht Foto Aktion Foto Gesicht

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