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Dynamics of a Nonautonomous Stochastic SIS...
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Research Article Dynamics of a Nonautonomous Stochastic SIS Epidemic Model with Double Epidemic Hypothesis Haokun Qi, 1 Lidan Liu, 1 and Xinzhu Meng 1,2 1 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China 2 State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China Correspondence should be addressed to Xinzhu Meng; [email protected] Received 25 August 2017; Accepted 11 October 2017; Published 8 November 2017 Academic Editor: Benito M. Chen-Charpentier Copyright © 2017 Haokun Qi 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. We investigate the dynamics of a nonautonomous stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis. By constructing suitable stochastic Lyapunov functions and using Has’minskii theory, we prove that there exists at least one nontrivial positive periodic solution of the system. Moreover, the sufficient conditions for extinction of the disease are obtained by using the theory of nonautonomous stochastic differential equations. Finally, numerical simulations are utilized to illustrate our theoretical analysis. 1. Introduction e SIS (Susceptible-Infected-Susceptible) model is a basic biological mathematical model describing susceptible and infected epidemic process and is first introduced by Kermack and McKendrick [1]. e SIS model is defined in that indi- viduals start off susceptible, at some stage catch the disease, and aſter a short infectious period become susceptible again [2]. erefore, some deterministic SIS epidemic models have been studied by many authors [3–10]. Recently, the authors of [11–13] investigated the epidemic model with double epidemic hypothesis which has two epidemic diseases caused by two different viruses. For example, the deterministic SIS epidemic model with nonlinear saturated incidence rate and double epidemic hypothesis can be expressed as follows [11]: ̇ () = − () − 1 () 1 () 1 + 1 () − 2 () 2 () 2 + 2 () + 1 1 () + 2 2 () , ̇ 1 () = 1 () 1 () 1 + 1 () − ( + 1 + 1 ) 1 () , ̇ 2 () = 2 () 2 () 2 + 2 () − ( + 2 + 2 ) 2 () , (1) where (), 1 (), and 2 () represent the number of suscepti- bles and infected individuals with viruses and at time , respectively. e parameters in model (1) have the following meanings: is the total input susceptible population size, represents the natural death rate of , 1 , and 2 , represents the disease transmission coefficient between compartments and ( = 1, 2), 1 and 2 are the recovery rates of the two diseases, and 1 and 2 are mortality rates due to diseases, respectively. Functions 1 () 1 ()/( 1 + 2 ()) and 2 () 2 ()/( 2 + 2 ()) represent two different types of saturated incidence rates for the two epidemic diseases 1 () and 2 (). All parameter values are nonnegative. In the real world, population systems and epidemic sys- tems are inevitably infected by some uncertain environmental disturbances. Hence, many authors have introduced stochas- tic interferences into differential systems, and the stochastic dynamics of such systems were widely investigated (see [14– 28]). Moreover, numerous scholars have investigated some stochastic epidemic models (see [29–34]). For example, in [11, Hindawi Complexity Volume 2017, Article ID 4861391, 14 pages https://doi.org/10.1155/2017/4861391
Transcript of Dynamics of a Nonautonomous Stochastic SIS...
Research ArticleDynamics of a Nonautonomous Stochastic SISEpidemic Model with Double Epidemic Hypothesis
Haokun Qi1 Lidan Liu1 and Xinzhu Meng12
1College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao 266590 China2State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry ofScience and Technology Shandong University of Science and Technology Qingdao 266590 China
Correspondence should be addressed to Xinzhu Meng mxz721106sdusteducn
Received 25 August 2017 Accepted 11 October 2017 Published 8 November 2017
Academic Editor Benito M Chen-Charpentier
Copyright copy 2017 Haokun Qi 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
We investigate the dynamics of a nonautonomous stochastic SIS epidemicmodel with nonlinear incidence rate and double epidemichypothesis By constructing suitable stochastic Lyapunov functions and using Hasrsquominskii theory we prove that there exists at leastone nontrivial positive periodic solution of the systemMoreover the sufficient conditions for extinction of the disease are obtainedby using the theory of nonautonomous stochastic differential equations Finally numerical simulations are utilized to illustrate ourtheoretical analysis
1 Introduction
The SIS (Susceptible-Infected-Susceptible) model is a basicbiological mathematical model describing susceptible andinfected epidemic process and is first introduced by Kermackand McKendrick [1] The SIS model is defined in that indi-viduals start off susceptible at some stage catch the diseaseand after a short infectious period become susceptible again[2] Therefore some deterministic SIS epidemic models havebeen studied by many authors [3ndash10] Recently the authorsof [11ndash13] investigated the epidemic model with doubleepidemic hypothesis which has two epidemic diseases causedby two different viruses For example the deterministic SISepidemic model with nonlinear saturated incidence rate anddouble epidemic hypothesis can be expressed as follows [11]
(119905) = 119860 minus 119889119878 (119905) minus 1205731119878 (119905) 1198681 (119905)1198861 + 1198681 (119905) minus 1205732119878 (119905) 1198682 (119905)1198862 + 1198682 (119905)+ 11990311198681 (119905) + 11990321198682 (119905)
1 (119905) = 1205731119878 (119905) 1198681 (119905)1198861 + 1198681 (119905) minus (119889 + 1205721 + 1199031) 1198681 (119905)
2 (119905) = 1205732119878 (119905) 1198682 (119905)1198862 + 1198682 (119905) minus (119889 + 1205722 + 1199032) 1198682 (119905) (1)
where 119878(119905) 1198681(119905) and 1198682(119905) represent the number of suscepti-bles and infected individuals with viruses 119860 and 119861 at time 119905respectively The parameters in model (1) have the followingmeanings 119860 is the total input susceptible population size 119889represents the natural death rate of 119878 1198681 and 1198682 120573119894 representsthe disease transmission coefficient between compartments119878 and 119868119894 (119894 = 1 2) 1199031 and 1199032 are the recovery rates ofthe two diseases and 1205721 and 1205722 are mortality rates dueto diseases respectively Functions 1205731119878(119905)1198681(119905)(1198861 + 1198682(119905))and 1205732119878(119905)1198682(119905)(1198862 + 1198682(119905)) represent two different types ofsaturated incidence rates for the two epidemic diseases 1198681(119905)and 1198682(119905) All parameter values are nonnegative
In the real world population systems and epidemic sys-tems are inevitably infected by someuncertain environmentaldisturbances Hence many authors have introduced stochas-tic interferences into differential systems and the stochasticdynamics of such systems were widely investigated (see [14ndash28]) Moreover numerous scholars have investigated somestochastic epidemicmodels (see [29ndash34]) For example in [11
HindawiComplexityVolume 2017 Article ID 4861391 14 pageshttpsdoiorg10115520174861391
2 Complexity
30] they obtained thresholds of the stochastic system whichdetermines the extinction and persistence of the epidemicZhang et al [29] proved that there is a unique ergodic station-ary distribution of his model We assume that environmentfluctuations will manifest themselves mainly as fluctuationsin the saturated response rate so that 120573119894119878(119905)119868119894(119905)(119886119894+119868119894(119905)) rarr120573119894119878(119905)119868119894(119905)(119886119894 + 119868119894(119905)) + (120590119894119878(119905)119868119894(119905)(119886119894 + 119868119894(119905)))119894(119905) (119894 = 1 2)where 119861(119905) = (1198611(119905) 1198612(119905)) is a standard Brownian motionwith intensity 120590119894 gt 0 (119894 = 1 2) Therefore a stochastic modelis described by [11]
d119878 (119905) = (119860 minus 119889119878 (119905) minus 1205731119878 (119905) 1198681 (119905)1198861 + 1198681 (119905) minus 1205732119878 (119905) 1198682 (119905)1198862 + 1198682 (119905)+ 11990311198681 (119905) + 11990321198682 (119905)) d119905 minus 1205901119878 (119905) 1198681 (119905)1198861 + 1198681 (119905) d1198611 (119905)minus 1205902119878 (119905) 1198682 (119905)1198862 + 1198682 (119905) d1198612 (119905)
d1198681 (119905) = (1205731119878 (119905) 1198681 (119905)1198861 + 1198681 (119905) minus (119889 + 1205721 + 1199031) 1198681 (119905)) d119905+ 1205901119878 (119905) 1198681 (119905)1198861 + 1198681 (119905) d1198611 (119905)
d1198682 (119905) = (1205732119878 (119905) 1198682 (119905)1198862 + 1198682 (119905) minus (119889 + 1205722 + 1199032) 1198682 (119905)) d119905+ 1205902119878 (119905) 1198682 (119905)1198862 + 1198682 (119905) d1198612 (119905)
(2)
However many infectious diseases of human fluctuateover time and often show the seasonal morbidity Thereforethe existence of periodic solutions of some nonautonomousepidemic models was explored [35ndash37] Recently manyscholars focused on nonautonomous stochastic periodicsystems With the development of stochastic differentialequations and application ofHasrsquominskii theory the existenceof stochastic periodic solution has been studied [23 3839] In [23] Zhang et al considered a nonautonomousstochastic Lotka-Volterra predator-prey model with impul-sive effects they got thresholds for stochastic persistence andextinction of the system Authors of [38ndash40] investigatedperiodic solution of a stochastic nonautonomous epidemicmodel
Based on the discussion above in this paper we considera nonautonomous stochastic SIS model with periodic coeffi-cients
d119878 (119905) = (119860 (119905) minus 119889 (119905) 119878 (119905) minus 1205731 (119905) 119878 (119905) 1198681 (119905)1198861 (119905) + 1198681 (119905)minus 1205732 (119905) 119878 (119905) 1198682 (119905)1198862 (119905) + 1198682 (119905) + 1199031 (119905) 1198681 (119905) + 1199032 (119905) 1198682 (119905)) d119905minus 1205901 (119905) 119878 (119905) 1198681 (119905)1198861 (119905) + 1198681 (119905) d1198611 (119905) minus 1205902 (119905) 119878 (119905) 1198682 (119905)1198862 (119905) + 1198682 (119905) d1198612 (119905)
d1198681 (119905) = (1205731 (119905) 119878 (119905) 1198681 (119905)1198861 (119905) + 1198681 (119905)minus (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681 (119905)) d119905+ 1205901 (119905) 119878 (119905) 1198681 (119905)1198861 (119905) + 1198681 (119905) d1198611 (119905)
d1198682 (119905) = (1205732 (119905) 119878 (119905) 1198682 (119905)1198862 (119905) + 1198682 (119905)minus (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682 (119905)) d119905+ 1205902 (119905) 119878 (119905) 1198682 (119905)1198862 (119905) + 1198682 (119905) d1198612 (119905)
(3)
where the parameter functions 119860(119905) 119889(119905) 120573119894(119905) 119886119894(119905) 119903119894(119905)120572119894(119905) 120590119894(119905) (119894 = 1 2) are positive nonconstant and contin-uous periodic functions with positive period 119879
To the best of our knowledge there are only fewworks on research of nonautonomous stochastic epidemicmodels with nonlinear saturated incidence rate and doubleepidemic hypothesis Therefore based on an autonomousstochastic epidemic model we propose a nonautonomousstochastic model and investigate the existence of stochasticperiodic solution and the extinction of the two epidemicdiseases
This paper is organized as follows In Section 2 we givesome definitions and known results In Section 3 we provethat system (3) has a unique global positive solution InSection 4 we present sufficient condition for the existencea nontrivial positive periodic solution of system (3) InSection 5 we obtain the sufficient conditions of system (3)for extinction of the two epidemic diseases In Section 6 wecarry out a series of numerical simulations to illustrate ourtheoretical findings
2 Preliminaries
Throughout this paper let (ΩFP) be a complete prob-ability space with a filtration F119905119905ge0 satisfying the usualconditions (ie it is increasing and right continuouswhileF0contains all P-null sets) The function 119861119894(119905) (119894 = 1 2 3 4) isdefined on this complete probability space
For simplicity some notations are given first If 119891(119905) isan integrable function defined on [0infin) define ⟨119891⟩119905 =(1119905) int119905
0119891(119904)119889119904 119905 gt 0 If 119891(119905) is a bounded function on
[0infin) define119891119897 = inf 119905isin[0infin)119891(119905) and 119891119906 = sup119905isin[0infin)119891(119905)Here we present some basic theory in stochastic differen-
tial equations which are introduced in [41]In general consider the 119897-dimensional stochastic differ-
ential equation
d119883 (119905) = 119891 (119883 (119905) 119905) d119905 + 119892 (119883 (119905) 119905) 119889119861 (119905) 119905 ge 1199050 (4)
with initial value 119909(1199050) = 1199090 isin R119897 119861(119905) stands for a119897-dimensional standard Brownian motion defined on the
Complexity 3
complete probability space (ΩF F119905119905ge0P) Denote by11986221(R119897 times [1199050infin]R+) the family of all nonnegative functions119881(119883 119905)defined onR119897times[1199050infin] such that they are continuouslytwice differentiable in 119883 and once in 119905 The differentialoperator 119871 of (4) is defined by [41]
119871 = 120597120597119905 +sum119891119894 (119883 119905) 120597120597119883119894+ 12119897sum119894119895=1
[119892119879 (119883 119905) 119892 (119883 119905)]119894119895
1205972120597119883119894120597119883119895 (5)
If 119871 acts on a function 119881 isin 11986221(R119897 times [1199050infin]R+) then119871119881 (119883 119905) = 119881119905 (119883 119905) + 119881119883 (119883 119905) 119891 (119883 119905)
+ 12 trace [119892119879 (119883 119905) 119881119883119883 (119883 119905) 119892 (119883 119905)] (6)
where 119881119905 = 120597119881120597119905 119881119883 = (1205971198811205971199091 120597119881120597119909119897) and 119881119883119883 =(1205972119881120597119883119894120597119883119895)119897times119897 In view of Itorsquos formula if119883(119905) isin R119897 then
d119881 (119883 (119905) 119905) = 119871119881 (119909 (119905) 119905) d119905+ 119881119883 (119883 (119905) 119905) 119892 (119883 (119905) 119905) 119889119861 (119905) (7)
Definition 1 (see [42]) A stochastic process 120585(119905) =120585(119905 120596) (minusinfin lt 119905 lt +infin) is said to be periodic with period119879 if for every finite sequence of numbers 1199051 1199052 119905119899 the jointdistribution of random variables 120585(1199051 +ℎ) 120585(1199052 +ℎ) 120585(119905119899 +ℎ) is independent of ℎ where ℎ = 119896119879 119896 = plusmn1 plusmn2
It is shown in [42] that a Markov process 119909(119905) is 119879-periodic if and only if its transition probability function is 119879-periodic and the functionP0(119905 119860) = P119909(119905) isin 119860 satisfies theequation
P0 (119904 119860) = intR119897P0 (119904 119889119909)P (119904 119909 119904 + 119879 119860)
= P0 (119904 + 119879 119860) (8)
Consider the following equation
119883(119905) = 119883 (1199050) + int1199051199050
119887 (119904 119883 (119904)) 119889119904
+ 119896sum119903=1
int1199051199050
120590119903 (119904 119883 (119904)) 119889119861119903 (119904) 119883 isin R119897
(9)
Lemma 2 (see [42]) Suppose that coefficients of (9) are 119879-periodic in 119905 and satisfy the condition
1003816100381610038161003816119887 (119904 119909) minus 119887 (119904 119910)1003816100381610038161003816 +119896sum119903=1
1003816100381610038161003816120590119903 (119904 119909) minus 120590119903 (119904 119910)1003816100381610038161003816le 119875 1003816100381610038161003816119909 minus 1199101003816100381610038161003816
|119887 (119904 119909)| + 119896sum119903=1
1003816100381610038161003816120590119903 (119904 119909)1003816100381610038161003816 le 119875 (1 + |119909|) (10)
in every cylinder 119868 times 119880 where 119875 is a constant And supposefurther that there exists a function 119881(119905 119909) isin 1198622 inR119889 which is119879-periodic in 119905 and satisfies the following conditions(1198601) inf |119909|geR119881(119905 119909) rarr infin as R rarrinfin(1198602) 119871119881(119905 119909) le minus1 outside some compact set where the
operator 119871 is given by
119871 = 120597120597119905 +119897sum119894=1
119887119894 (119905 119909) 120597120597119909119894 +12119897sum119894119895=1
119886119894119895 (119905 119909) 1205972120597119909119894120597119909119895
119886119894119895 = 119896sum119903=1
120590119894119903 (119905 119909) 120590119895119903 (119905 119909) (11)
Then there exists a solution of (9) which is a119879-periodicMarkovprocess
Lemma 3 (see [2] strong law of large numbers) Let 119872 =119872119905119905 ge 0 be a real-valued continuous local martingalevanishing at 119905 = 0
Thenlim119905rarrinfin
⟨119872119872⟩119905 = infin 119886119904 997904rArrlim119905rarrinfin
119872119905⟨119872119872⟩119905 = 0 119886119904 (12)
and also
lim sup119905rarrinfin
⟨119872119872⟩119905119905 lt infin 119886119904 997904rArrlim119905rarrinfin
119872119905119905 = 0 119886119904(13)
3 Existence and Uniqueness of theGlobal Positive Solution
In this section we prove that system (3) has a unique globalpositive solution
Theorem 4 For any initial value (119878(0) 1198681(0) 1198682(0)) isin 1198773+there is a unique positive solution (119878(119905) 1198681(119905) 1198682(119905)) of (3) on119905 ge 0 and the solution will remain in 1198773+ with probability oneProof From system (3) we can get
d (119878 (119905) + 1198681 (119905) + 1198682 (119905))d119905
= 119860 (119905) minus 119889 (119905) (119878 (119905) + 1198681 (119905) + 1198682 (119905))minus (1205721 (119905) 1198681 (119905) + 1205722 (119905) 1198682 (119905))
le 119860 (119905) minus 119889 (119905) (119878 (119905) + 1198681 (119905) + 1198682 (119905))le 119860119906 minus 119889119897 (119878 (119905) + 1198681 (119905) + 1198682 (119905))
(14)
Then
lim119905rarr+infin
(119878 (119905) + 1198681 (119905) + 1198682 (119905)) le 119860119906119889119897 (15)
4 Complexity
obviously we have
lim sup119905rarr+infin
119878 (119905) le 119860119906119889119897 lim sup119905rarr+infin
1198681 (119905) le 119860119906119889119897 lim sup119905rarr+infin
1198682 (119905) le 119860119906119889119897 (16)
Since the coefficients of system (3) satisfy the localLipschitz conditions then for any given initial value(119878(0) 1198681(0) 1198682(0)) isin 1198773+ there is a unique local solution(119878(119905) 1198681(119905) 1198682(119905)) on 119905 isin [0 120591119890) where 120591119890 is the explosion timeTo demonstrate that this solution is global we only need toprove that 120591119890 = infin as
Let 1198960 gt 0 be sufficiently large for any initial value119878(0) 1198681(0) and 1198682(0) lying within the interval [11198960 119896] Foreach integer 119896 ge 1198960 define the following stopping time
120591119896 = inf 119905 isin [0 120591119890) min 119878 (119905) 1198681 (119905) 1198682 (119905)le 1119896 ormax 119878 (119905) 1198681 (119905) 1198682 (119905) ge 119896
(17)
where we set inf 0 = infin (as usual 0 denotes the empty set)Clearly 120591119896 is increasing as 119896 rarr infin Let 120591infin = lim119896rarrinfin120591119896 hence120591infin le 120591119896 as Next we only need to verify 120591infin = infin as If thisstatement is false then there exist two constants 119879 gt 0 and120598 isin (0 1) such that
P 120591infin le 119879 gt 120598 (18)
Thus there is an integer 1198961 ge 1198960 such that
P 120591119896 le 119879 ge 120598 119896 ge 1198961 (19)
Define a 1198622-function 119881 R3+ rarr R+ as follows
119881 (119878 1198681 1198682) = 119878 minus 1 minus ln 119878 + 1198681 minus 1 minus ln 1198681 + 1198682 minus 1minus ln 1198682 (20)
the nonnegativity of this function can be obtained from
119906 minus 1 minus ln 119906 ge 0 119906 gt 0 (21)
Applying Itorsquos formula yields
d119881 (119878 1198681 1198682) = 119871119881 (119878 1198681 1198682) d119905+ (21198781198681 minus 119878 minus 1198681) 1205901 (119905)1198861 (119905) + 1198681 1198891198611 (119905)+ (21198781198682 minus 119878 minus 1198682) 1205902 (119905)1198862 (119905) + 1198682 1198891198612 (119905)
(22)
where
119871119881 = (1 minus 1119878)(119860 (119905) minus 119889 (119905) 119878 minus 1205731 (119905) 11987811986811198861 (119905) + 1198681minus 1205732 (119905) 11987811986821198862 (119905) + 1198682 + 1199031 (119905) 1198681 + 1199032 (119905) 1198682) + 12
sdot 12059021 (119905) 11986821(1198861 (119905) + 1198681)2 +12
12059022 (119905) 11986822(1198862 (119905) + 1198682)2 + (1 minus11198681)
sdot ( 1205731 (119905) 11987811986811198861 (119905) + 1198681 minus (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681) + 12sdot 12059021 (119905) 1198782(1198861 (119905) + 1198681)2 + (1 minus
11198682)(1205732 (119905) 11987811986821198862 (119905) + 1198682
minus (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682) + 1212059022 (119905) 1198782(1198862 (119905) + 1198682)2
= 119860 (119905) minus 119889 (119905) 119878 minus 119860 (119905)119878 + 119889 (119905) + 1205731 (119905) 11986811198861 (119905) + 1198681+ 1205732 (119905) 11986821198862 (119905) + 1198682 minus
1199031 (119905) 1198681119878 minus 1199032 (119905) 1198682119878+ 12059021 (119905) 119868212 (1198861 (119905) + 1198681)2 +
12059022 (119905) 119868222 (1198862 (119905) + 1198682)2 minus (119889 (119905)+ 1205721 (119905)) 1198681 minus 1205731 (119905) 1198781198861 (119905) + 1198681 + 119889 (119905) + 1205721 (119905) + 1199031 (119905)+ 12059021 (119905) 11987822 (1198861 (119905) + 1198681)2 minus (119889 (119905) + 1205722 (119905)) 1198682 minus
1205732 (119905) 1198781198862 (119905) + 1198682+ 119889 (119905) + 1205722 (119905) + 1199032 (119905) + 12059022 (119905) 11987822 (1198862 (119905) + 1198682)2 le 119860119906
+ 3119889119906 + 1205721199061 + 1205721199062 + 1199031199061 + 1199031199062 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622+ 12059021199061 1198602119906211988621198971 1198892119897 +
12059021199062 1198602119906211988621198972 1198892119897 fl 119870(23)
where 119870 is a positive constantSo we have
d119881 (119878 1198681 1198682) le 119870d119905+ (21198781198681 minus 119878 minus 1198681) 1205901 (119905)1198861 (119905) + 1198681 1198891198611 (119905)+ (21198781198682 minus 119878 minus 1198682) 1205902 (119905)1198862 (119905) + 1198682 1198891198612 (119905)
(24)
Integrating (24) from 0 to 120591119896 and 119879 and taking expectations onboth sides yield
E119881 (119878 (120591119896 and 119879) 1198681 (120591119896 and 119879) 1198682 (120591119896 and 119879))le 119881 (119878 (0) 1198681 (0) 1198682 (0)) + 119870119879 (25)
Let Ω119896 = 120591119896 le 119879 from inequality (25) we can see that119875(Ω119896) ge 120598We have
119881 (119878 (120591119896 and 119879) 1198681 (120591119896 and 119879) 1198682 (120591119896 and 119879))ge (119896 minus 1 minus ln 119896) and (1119896 minus 1 minus ln 1119896)
(26)
Complexity 5
By (25) and (26) one has
119881 (119878 (0) 1198681 (0) 1198682 (0)) + 119870119879 ge E [1Ω119896 (120596)sdot 119881 (119878 (120591119896 and 119879) 1198681 (120591119896 and 119879) 1198682 (120591119896 and 119879))] ge 120598 (119896minus 1 minus ln 119896) and (1119896 minus 1 minus ln 1119896)
(27)
where 1Ω119896 is the indicator function ofΩ119896
Let 119896 rarr infin we have
infin gt 119881(119878 (0) 1198681 (0) 1198682 (0)) + 119870119879 = infin (28)
So we obtain 120591infin = infinThe proof is completed
4 Existence of Nontrivial 119879-Periodic SolutionIn this section we verify that system (3) admits at least onenontrivial positive 119879-periodic solution Define
R = 2sum119894=1
⟨119860120572119894120573119894⟩119879⟨119889 + 120572119894 + 119903119894 + 1205902119894 119860211990621198862119894 1198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩119879 ⟨119860 + 11988611205721 + 11988621205722⟩119879 (29)
Theorem 5 When (1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)))119897 gt 119860119906119889119897and (1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)))119897 gt 119860119906119889119897 hold if R gt 1then there exists a nontrivial positive T-periodic solution ofsystem (3)
Proof Define a 1198622-function 119881 [0infin) timesR3+ rarr R+
119881 (119905 119878 1198681 1198682) = 119872(minus (1198901 + 1198902) ln 119878 minus 1198881 ln 1198681 minus 1198882 ln 1198682+ 119878 + 1198681 + 1198682 + 120596 (119905)) minus ln 119878 minus 1198681 minus 1198682 + 1120579 + 1 (119878+ 1198681 + 1198682)120579+1 = 119872(1198811 + 120596 (119905)) + 1198812 + 1198813 + 1198814+ 1198815
(30)
where1198881= ⟨11986012057211205731⟩119879⟨119889 + 1205721 + 1199031 + 1205902111986021199062119886211198892119897⟩2119879 ⟨119889 + 120590212 + 120590222⟩119879 1198901= ⟨11986012057211205731⟩119879⟨119889 + 1205721 + 1199031 + 1205902111986021199062119886211198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩2119879 1198882= ⟨11986012057221205732⟩119879⟨119889 + 1205722 + 1199032 + 1205902211986021199062119886221198892119897⟩2119879 ⟨119889 + 120590212 + 120590222⟩119879 1198902= ⟨11986012057221205732⟩119879⟨119889 + 1205722 + 1199032 + 1205902211986021199062119886221198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩2119879
(31)
and 0 lt 120579 lt min1 119889119897(12059021199061 + 12059021199062 )119872 is a sufficiently largepositive constant and satisfies the following conditions
119889119897 minus 120579 (12059021199061 + 12059021199062 ) gt 0minus119872120582 +max 119863 119864 le minus2 (32)
where
120582 = ⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1) 119863 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
119864 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(33)
where
119874 = sup(1198781198681 1198682)isinR
3+
119860119906 (119878 + 1198681 + 1198682)120579
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 lt infin(34)
Next we prove that condition (1198601) in Lemma 2 holds It iseasy to check that 119881(119905 119878 1198681 1198682) is a 119879-periodic function in 119905and satisfies
lim inf119870rarrinfin(1198781198681 1198682)isinR
3+U119896
119881 (119905 119878 1198681 1198682) = infin (35)
where U119896 = (1119896 119896) times (1119896 119896) times (1119896 119896) and 119896 gt 1 is asufficiently large number
6 Complexity
By Itorsquos formula we obtain
1198711198811 = minus1198901 + 1198902119878 (119860 (119905) minus 119889 (119905) 119878 minus 1205731 (119905) 11987811986811198861 (119905) + 1198681minus 1205732 (119905) 11987811986821198862 (119905) + 1198682 + 1199031 (119905) 1198681 + 1199032 (119905) 1198682)minus 11988811198681 (
1205731 (119905) 11987811986811198861 (119905) + I1minus (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681)
+ 1198901 + 1198902212059021 (119905) 11986821(1198861 (119905) + 1198681)2 minus
11988821198682 (1205732 (119905) 11987811986821198862 (119905) + 1198682
minus (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682) + 1198901 + 11989022sdot 12059022 (119905) 11986822(1198862 (119905) + 1198682)2 +
1198881212059021 (119905) 1198782(1198861 (119905) + 1198681)2 +
11988822sdot 12059022 (119905) 1198782(1198862 (119905) + 1198682)2 + 119860 (119905) minus 119889 (119905) (119878 + 1198681 + 1198682) minus 1205721 (119905)sdot 1198681 minus 1205722 (119905) 1198682 le minus1198901119860 (119905)119878 minus 11988811205731 (119905) 1198781198861 (119905) + 1198681 minus 1205721 (119905)sdot (1198861 (119905) + 1198681) minus 1198902119860 (119905)119878 minus 11988821205732 (119905) 1198781198862 (119905) + 1198682 minus 1205722 (119905)sdot (1198862 (119905) + 1198682) + 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905)
+ 12059021 (119905) 1198602119906211988621 (119905) 1198892119897) + 1198901 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )
+ 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905) + 12059022 (119905) 1198602119906211988622 (119905) 1198892119897)
+ 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 ) minus (1198901 + 1198902) (1199031 (119905)119878+ 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878 + 119889 (119905)minus 1205732 (119905)1198862 (119905) ) 1198682 + 119860 (119905) + 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905)le minus3 (119860 (119905) 1205721 (119905) 1205731 (119905) 11988811198901)13minus 3 (119860 (119905) 1205722 (119905) 1205732 (119905) 11988821198902)13 + 1198901 (119889 (119905) + 12059021 (119905)2+ 12059022 (119905)2 ) + 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905)
+ 12059021 (119905) 1198602119906211988621 (119905) 1198892119897) + 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905)
+ 12059022 (119905) 1198602119906211988622 (119905) 1198892119897) + 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )
+ 119860 (119905) + 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905) minus (1198901 + 1198902)sdot (1199031 (119905)119878 + 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878+ 119889 (119905) minus 1205732 (119905)1198862 (119905) ) 1198682 fl R0 (119905) minus (1198901 + 1198902) (1199031 (119905)119878+ 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878 + 119889 (119905)minus 1205732 (119905)1198862 (119905) ) 1198682
(36)
where
R0 (119905) = 3 (119860 (119905) 1205721 (119905) 1205731 (119905) 11988811198901)13+ 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905) + 12059021 (119905) 1198602119906211988621 (119905) 1198892119897)
+ 1198901 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )minus 3 (119860 (119905) 1205722 (119905) 1205732 (119905) 11988821198902)13+ 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905) + 12059022 (119905) 1198602119906211988622 (119905) 1198892119897)
+ 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 ) + 119860 (119905)+ 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905)
(37)
Note that (1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)))119897 gt 119860119906119889119897 and(1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)))119897 gt 119860119906119889119897 hold then119871 (1198811) le R0 (119905) (38)
Define the 119879-periodic function 120596(119905) satisfying1205961015840 (119905) = ⟨R0⟩119879 minusR0 (119905) (39)
So
119871 (1198811 + 120596 (119905)) le ⟨R0⟩119879= minus ⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1)fl minus120582
(40)
Applying Itorsquos formula we can also have
Complexity 7
1198711198812 = minus119860 (119905)119878 + 119889 (119905) + 1205731 (119905) 11986811198861 (119905) + 1198681 +1205732 (119905) 11986821198862 (119905) + 1198682
minus 1199031 (119905) 1198681119878 minus 1199032 (119905) 1198682119878 + 12059021 (119905) 119868212 (1198861 (119905) + 1198681)2+ 12059022 (119905) 119868222 (1198862 (119905) + 1198682)2 le minus119860119897119878 + 119889119906 + 1205731199061 + 1205731199062 + 120590211990612+ 120590211990622
1198711198813 = minus 1205731 (119905) 11987811986811198861 (119905) + 1198681 + (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681 le (119889119906+ 1205721199061 + 1199031199061 ) 1198681
1198711198814 = minus 1205732 (119905) 11987811986821198862 (119905) + 1198682 + (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682 le (119889119906+ 1205721199062 + 1199031199062 ) 1198682
1198711198815 = (119878 + 1198681 + 1198682)120579 (119860 (119905) minus 119889 (119905) 119878 minus 119889 (119905) 1198681 minus 119889 (119905) 1198682minus 1205721 (119905) 1198681 minus 1205722 (119905) 1198682) + 120579 (119878 + 1198681 + 1198682)120579minus1sdot ( 12059021 (119905) 119878211986821(1198861 (119905) + 1198681)2 +
12059022 (119905) 119878211986822(1198862 (119905) + 1198682)2) le 119860119906 (119878 + 1198681+ 1198682)120579 minus 119889119897 (119878 + 1198681 + 1198682)120579+1 + 120579 (119878 + 1198681 + 1198682)120579minus1 (12059021199061+ 12059021199062 ) 1198782 le 119860119906 (119878 + 1198681 + 1198682)120579 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 le 119874 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12
(41)
where
119874 = sup(1198781198681 1198682)isinR
3+
119860119906 (119878 + 1198681 + 1198682)120579
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 lt infin(42)
Therefore119871119881 (119905 119878 1198681 1198682) le minus119872⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1)
+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682 minus 119860119897119878minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874+ 119889119906 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622
= minus119872120582 minus 119860119897119878 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874+ 119889119906 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622
(43)
Now we are in the position to construct a compact subset119880 such that 1198602 in Lemma 2 holds Define the followingbounded closed set
119880 = (119878 1198681 1198682) isin R3+ 120598 le 119878 le 1120598 120598 le 1198681 le 1120598 120598 le 1198682
le 1120598 (44)
where 120598 gt 0 is a sufficiently small number In the set R3+119880we can choose 120598 sufficiently small such that
minus119872120582 minus 119860119897120598 + 119862 le minus1 (45)
minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 120598 + 119863 le minus1 (46)
minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 120598 + 119864 le minus1 (47)
minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119862 le minus1 (48)
minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119866 le minus1 (49)
minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119867 le minus1 (50)
where 119862 119863 119864 119866 and119867 are positive constants which can befound from the following inequations (52) (54) (56) (59)and (61) respectively For the sake of convenience we divideinto six domains
8 Complexity
1198801 = (119878 1198681 1198682) isin R3+ 0 lt 119878 lt 120598
1198802 = (119878 1198681 1198682) isin R3+ 0 lt 1198681 lt 120598
1198803 = (119878 1198681 1198682) isin R3+ 0 lt 1198682 lt 120598
1198804 = (119878 1198681 1198682) isin R3+ 119878 gt 1120598
1198805 = (119878 1198681 1198682) isin R3+ 1198681 gt 1120598
1198806 = (119878 1198681 1198682) isin R3+ 1198682 gt 1120598
(51)
Nextwewill prove that119871119881(119878 1198681 1198682) le minus1 onR3+119880 whichis equivalent to proving it on the above six domains
Case 1 If (119878 1198681 1198682) isin 1198801 one can see that
119871119881 le minus119872120582 minus 119860119897119878 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622 le minus119872120582 minus 119860119897119878 + 119862
le minus119872120582 minus 119860119897120598 + 119862
(52)
where
119862 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(53)
According to (45) we have 119871119881 le minus1 for all (119878 1198681 1198682) isin 1198801Case 2 If (119878 1198681 1198682) isin 1198802 one can get that
119871119881 le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681 + 119863le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 120598 + 119863
(54)
where
119863 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(55)
In view of (46) we can obtain that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198802Case 3 If (119878 1198681 1198682) isin 1198803 we have
119871119881 le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682 + 119864le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 120598 + 119864
(56)
where
119864 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(57)
By (47) we can conclude that 119871119881 le minus1 for all (119878 1198681 1198682) isin1198803Case 4 If (119878 1198681 1198682) isin 1198804 one can derive that
119871119881 le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1 + 119862le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119862
(58)
Complexity 9
Together with (48) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198804Case 5 If (119878 1198681 1198682) isin 1198805 it follows that
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119866le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119866
(59)
where
119866 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(60)
By virtue of (49) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198805Case 6 If (119878 1198681 1198682) isin 1198806 we obtain
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119867le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119867
(61)
where
119867 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(62)
It follows from (50) that 119871119881 le minus1 for all (119878 1198681 1198682) isin 1198806Clearly one can see from (52) (54) (56) (58) (59) and
(61) that for a sufficiently small 120598119871119881 (119878 1198681 1198682) le minus1
(119878 1198681 1198682) isin R3+119880 (63)
Hence 1198602 in Lemma 2 is satisfied This completes the proofof Theorem 5
5 Extinction
In this section we investigate the conditions for the extinc-tion of the two infectious diseases of system (3)
Let
R1 = 119860119906 ⟨1205731⟩119879119889119897 ⟨1198861⟩119879 ⟨119889 + 1205721 + 1199031⟩119879minus 1198602119906 ⟨12059021⟩11987921198892119897 ⟨11988621⟩119879 ⟨119889 + 1205721 + 1199031⟩119879
R2 = 119860119906 ⟨1205732⟩119879119889119897 ⟨1198862⟩119879 ⟨119889 + 1205722 + 1199032⟩119879minus 1198602119906 ⟨12059022⟩11987921198892119897 ⟨11988622⟩119879 ⟨119889 + 1205722 + 1199032⟩119879
(64)
Theorem 6 Let (119878(119905) 1198681(119905) 1198682(119905)) be a solution of system (3)with initial value (119878(0) 1198681(0) 1198682(0)) isin 1198773+
Then if
(i) ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792 ⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 119894 = 1 2 (65)
or(ii) R119894 lt 1⟨1205902119894 ⟩119879 le 119889119897 ⟨1198862119894 ⟩119879 ⟨120573119894⟩119879119860119906 ⟨119886119894⟩119879
119894 = 1 2(66)
hold the two infectious diseases of system (3) go to extinctionas that is
lim119905rarr+infin
119868119894 (119905) = 0 119894 = 1 2 (67)
10 Complexity
Proof Applying Itorsquos formula to system (3) we have
d ln 119868119894 (119905) = ( 120573119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905) minus (119889 (119905) + 120572119894 (119905) + 119903119894 (119905))minus 1205902119894 (119905) 1198782 (119905)2 (119886119894 (119905) + 119868119894 (119905))2) d119905 + 120590119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905)d119861119894 (119905)
119894 = 1 2
(68)
Case (i) Integrating (68) from 0 to 119905 and dividing 119905 on bothsides we obtain
ln 119868119894 (119905)119905 = minus1205902119894 (119905)2119905 int1199050( 119878 (120591)119886119894 (120591) + 119868119894 (120591) minus
120573119894 (120591)1205902119894 (120591))2
d120591minus 1119905 int
119905
0(119889 (120591) + 120572119894 (120591) + 119903119894 (120591)) d120591
+ 1119905 int119905
0
1205732119894 (120591)21205902119894 (120591)d120591 +119872119894 (119905)119905 + ln 119868119894 (0)119905
le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905
(69)
Case (ii) Integrating (68) from 0 to 119905 first and then dividingby 119905 on both sides yield
ln 119868119894 (119905)119905 = 1119905 int119905
0( 120573119894 (120591) 119878 (120591)119886119894 (120591) + 119868119894 (120591)
minus (119889 (120591) + 120572119894 (120591) + 119903119894 (120591))minus 1205902119894 (120591) 1198782 (120591)2 (119886119894 (120591) + 119868119894 (120591))2) d120591 + 119872119894 (119905)119905 + ln 119868119894 (0)119905le ⟨120573119894119860119906119886119894119889119897 minus (119889 + 120572119894 + 119903119894) minus
1205902119894 119860211990621198862119894 1198892119897⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) + 119872119894 (119905)119905+ ln 119868119894 (0)119905
(70)
where 119872119894(119905) = int1199050120590119894(120591)119878(120591)(119886119894(120591) + 119868119894(120591))d119861119894(120591) 119894 = 1 2
which is a local continuous martingale with 119872119894(0) = 0 ByLemma 3 we have
lim119905rarr+infin
119872119894 (119905)119905 = 0 119894 = 1 2 (71)
Taking the limit superior of both sides of (69) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119879 lt 0 (72)
which implies lim119905rarr+infin119868119894(119905) = 0
Taking the superior limit of both sides of (70) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) lt 0 (73)
which implies lim119905rarr+infin119868119894(119905) = 0 119894 = 1 2This completes theproof
Remark 7 Theorem 6 shows that the two diseases will dieout if the white noise disturbance is large or the white noisedisturbance is not large and R119894 lt 1 When ⟨1205902119894 ⟩119879 gt⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectious diseases of system(3) die out almost surely that is to say large white noisestochastic disturbance can lead to the two epidemics beingextinct
6 Numerical Simulations
Now we introduce some numerical simulations exampleswhich illustrate our theoretical results
Example 8 In model (3) let
119860 (119905) = 05 + 01 sin1205871199051198861 (119905) = 03 + 01 sin1205871199051198862 (119905) = 031 + 01 sin1205871199051205731 (119905) = 062 + 01 sin1205871199051205732 (119905) = 062 + 01 sin1205871199051199031 (119905) = 02 + 01 sin1205871199051199032 (119905) = 035 + 01 sin120587119905119889 (119905) = 02 + 01 sin1205871199051205721 (119905) = 02 + 01 sin1205871199051205722 (119905) = 025 + 01 sin1205871199051205901 (119905) = 01 + 005 sin1205871199051205902 (119905) = 01 + 005 sin120587119905
(74)
Note that R gt 1 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897and 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 hold that isthe conditions of Theorem 5 hold Hence system (3) has apositive periodic solution with 119879 = 1 Figure 1(a) shows theperiodicity of the nonautonomous stochastic model (3) with1205901 = 0 and 1205902 = 0 Figure 1(b) shows that solution of thenonautonomous stochastic model (3) with the initial value(119878(119905) 1198681(119905) 1198682(119905)) = (03 015 015) tends to a periodic orbitin the sense of joint distribution
Example 9 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin1205871199051198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin120587119905
Complexity 11
S(t)
I1(t)
I2(t)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
(a)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 1 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (1 02 02) to the nonautonomous stochastic model (3)
0 25 50 75 100t
59 60 61
078
08
082
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
59 60 61
078
08
082
0
1
2
14
14
S(t)I 1(t)I 2(t)
25 50 75 1000t
S(t)
I1(t)
I2(t)
(b)
Figure 2 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 04 + 02 sin1205871199051205902 (119905) = 05 + 02 sin120587119905
(75)
Note that ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897+120572119897119894+119903119897119894⟩119879Therefore conditions(i) of Theorem 6 hold Then the two infectious diseases willgo to extinction Figure 2(a) shows that one of two diseases inthe deterministic SIS epidemic model is extinct and the otheris persistent Figure 2(b) shows that the two diseases will dieout under the large white noise disturbance of model (3)
Example 10 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin120587119905
12 Complexity
25 50 75 1000t
0
05
1
15
2S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
0 25 50 75 100t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 3 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin1205871199051205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 02 + 02 sin1205871199051205902 (119905) = 02 + 02 sin120587119905
(76)
Note that R1 lt 1 R2 lt 1 and ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 That is conditions (ii) of Theorem 6 hold Thenthe two infectious diseases will go to extinction Figure 3(a)
shows that one of two diseases in the deterministic SISepidemic model is extinct and the other is persistent withoutthe white noises Figure 3(b) shows that the two diseases willdie out under a small white noise disturbance of model (3)
7 Discussion and Conclusions
This paper explores the existence of nontrivial positiveT-periodic solution of a nonautonomous stochastic SISepidemic model with nonlinear growth rate and doubleepidemic hypothesis By constructing a suitable stochasticLyapunov function we establish sufficient conditions for theexistence of nontrivial positive 119879-periodic solution of system(3) Furthermore the sufficient conditions for the extinctionof the two diseases are obtained Our results are given asfollows
(1) If 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 and R gt 1 hold the SISmodel has at least one nontrivial positive 119879-periodicsolution where
R = 2sum119894=1
⟨119860120572119894120573119894⟩119879⟨119889 + 120572119894 + 119903119894 + 1205902119894 119860211990621198862119894 1198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩119879 ⟨119860 + 11988611205721 + 11988621205722⟩119879 (77)
(2) If ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectiousdiseases go extinct
(3) If ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 and R119894 lt 1 the two
infectious diseases also go extinct where
R119894 = 119860119906 ⟨120573119894⟩119879119889119897 ⟨119886119894⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879minus 1198602119906 ⟨1205902119894 ⟩11987921198892119897 ⟨1198862119894 ⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879
(78)
Complexity 13
Some interesting questions deserve further investigationOn the one hand we may explore some realistic but complexmodels such as considering the effects of impulsive ordelay perturbations on system (3) On the other hand wecan concern the dynamics of a nonautonomous stochasticSIS epidemic model with two infectious diseases driven byLevy jumps What is more we can also investigate thenonautonomous stochastic SIS epidemic model with twoinfectious diseases by a continuous time Markov chain Wewill investigate these cases in our future work
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (11371230) Joint Innovative Center forSafe andEffectiveMiningTechnology andEquipment of CoalResources Shandong Province the SDUST Research Fund(2014TDJH102) and Shandong Provincial Natural ScienceFoundation China (ZR2015AQ001)
References
[1] W O Kermack and A G McKendrick ldquoA contributions to themathematical theory of epidemics (Part I)rdquo Proceedings of theRoyal Society of London A vol 115 pp 700ndash721 1927
[2] A Gray D Greenhalgh L Hu X Mao and J Pan ldquoA stochasticdifferential equation SIS epidemic modelrdquo SIAM Journal onApplied Mathematics vol 71 no 3 pp 876ndash902 2011
[3] A Korobeinikov and G C Wake ldquoLyapunov functions andglobal stability for SIR SIRS and SIS epidemiological modelsrdquoApplied Mathematics Letters vol 15 no 8 pp 955ndash960 2002
[4] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995
[5] T Zhang X Meng Y Song and T Zhang ldquoA stage-structuredpredator-prey SI model with disease in the prey and impulsiveeffectsrdquoMathematical Modelling and Analysis vol 18 no 4 pp505ndash528 2013
[6] J Li andZMa ldquoQualitative analyses of SIS epidemicmodelwithvaccination and varying total population sizerdquo Mathematicaland Computer Modelling vol 35 no 11-12 pp 1235ndash1243 2002
[7] A drsquoOnofrio ldquoA note on the global behaviour of the network-based SIS epidemic modelrdquo Nonlinear Analysis Real WorldApplications vol 9 no 4 pp 1567ndash1572 2008
[8] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[9] T Zhang X Meng T Zhang and Y Song ldquoGlobal dynamicsfor a new high-dimensional SIR model with distributed delayrdquoApplied Mathematics and Computation vol 218 no 24 pp11806ndash11819 2012
[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006
[11] X Meng S Zhao T Feng and T Zhang ldquoDynamics of a novelnonlinear stochastic SIS epidemic model with double epidemichypothesisrdquo Journal of Mathematical Analysis and Applicationsvol 433 no 1 pp 227ndash242 2016
[12] A Miao X Wang T Zhang W Wang and B Sampath ArunaPradeep ldquoDynamical analysis of a stochastic SIS epidemicmodel with nonlinear incidence rate and double epidemichypothesisrdquo Advances in Difference Equations 2017226 pages2017
[13] X-Z Meng ldquoStability of a novel stochastic epidemic modelwith double epidemic hypothesisrdquo Applied Mathematics andComputation vol 217 no 2 pp 506ndash515 2010
[14] X P Li X Y Lin and Y Q Lin ldquoLyapunov-type conditions andstochastic differential equations driven byG-BrownianmotionrdquoJournal of Mathematical Analysis and Applications vol 439 no1 pp 235ndash255 2016
[15] M Liu and M Fan ldquoPermanence of stochastic Lotka-Volterrasystemsrdquo Journal of Nonlinear Science vol 27 no 2 pp 425ndash452 2017
[16] L D Liu and X Z Meng ldquoOptimal harvesting controland dynamics of two-species stochastic model with delaysrdquoAdvances in Difference Equations vol 2017 18 pages 2017
[17] H Ma and Y Jia ldquoStability analysis for stochastic differentialequations with infiniteMarkovian switchingsrdquo Journal of Math-ematical Analysis and Applications vol 435 no 1 pp 593ndash6052016
[18] G D Liu X H Wang X Z Meng and S J Gao ldquoExtinctionand persistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexity vol 3pp 1ndash15 2017
[19] C Tan and W H Zhang ldquoOn observability and detectabilityof continuous-time stochastic Markov jump systemsrdquo Journalof Systems Science and Complexity vol 28 no 4 pp 830ndash8472015
[20] X Leng T Feng and X Meng ldquoStochastic inequalities andapplications to dynamics analysis of a novel SIVS epidemicmodel with jumpsrdquo Journal of Inequalities and ApplicationsPaper No 138 25 pages 2017
[21] G Li and M Chen ldquoInfinite horizon linear quadratic optimalcontrol for stochastic difference time-delay systemsrdquo Advancesin Difference Equations vol 2015 14 pages 2015
[22] Y Zhao and W Zhang ldquoObserver-based controller design forsingular stochastic Markov jump systems with state dependentnoiserdquo Journal of Systems Science and Complexity vol 29 no 4pp 946ndash958 2016
[23] S Zhang X Meng T Feng and T Zhang ldquoDynamics analysisand numerical simulations of a stochastic non-autonomouspredatorndashprey system with impulsive effectsrdquo Nonlinear Anal-ysis Hybrid Systems vol 26 pp 19ndash37 2017
[24] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013
[25] X Meng L Wang and T Zhang ldquoGlobal dynamics analysis ofa nonlinear impulsive stochastic chemostat system in a pollutedenvironmentrdquo Journal of AppliedAnalysis andComputation vol6 no 3 pp 865ndash875 2016
[26] X Liu Y Li andW Zhang ldquoStochastic linear quadratic optimalcontrol with constraint for discrete-time systemsrdquo AppliedMathematics and Computation vol 228 pp 264ndash270 2014
14 Complexity
[27] X Z Meng and X H Wang ldquoStochastic predator-prey systemsubject to levy jumprdquo Discrete Dynamics in Nature and Societyvol 2016 Article ID 5749892 13 pages 2016
[28] X Lv L Wang and X Meng ldquoGlobal analysis of a new non-linear stochastic differential competition systemwith impulsiveeffectrdquo Advances in Difference Equations vol 2017 296 pages2017
[29] X Zhang D Jiang A Alsaedi and T Hayat ldquoStationarydistribution of stochastic SIS epidemic model with vaccinationunder regime switchingrdquo Applied Mathematics Letters vol 59pp 87ndash93 2016
[30] F Li X Meng and Y Cui ldquoNonlinear stochastic analysis fora stochastic SIS epidemic modelrdquo Journal of Nonlinear Sciencesand Applications vol 10 no 09 pp 5116ndash5124 2017
[31] A Q Miao J Zhang T Q Zhang and B G Pradeep ldquoSampathAruna Pradeep Threshold dynamics of a stochastic SIR modelwith vertical transmission and vaccinationrdquo Computationaland Mathematical Methods in Medicine vol 2017 Article ID4820183 10 pages 2017
[32] Q Liu D Jiang N Shi T Hayat and A Alsaedi ldquoNontrivialperiodic solution of a stochastic non-autonomous SISV epi-demic modelrdquo Physica A Statistical Mechanics and its Applica-tions vol 462 pp 837ndash845 2016
[33] Y L Zhou S L Yuan and D L Zhao ldquoThreshold behavior ofa stochastic SIS model with levy jumpsrdquo Applied Mathematicsand Computation vol 275 pp 255ndash267 2016
[34] T Feng X Meng L Liu and S Gao ldquoApplication of inequal-ities technique to dynamics analysis of a stochastic eco-epidemiology modelrdquo Journal of Inequalities and ApplicationsPaper No 327 29 pages 2016
[35] M Liu and K Wang ldquoPersistence and extinction in stochasticnon-autonomous logistic systemsrdquo Journal of MathematicalAnalysis and Applications vol 375 no 2 pp 443ndash457 2011
[36] Z Bai and Y Zhou ldquoExistence of two periodic solutions for anon-autonomous SIR epidemic modelrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 1 pp 382ndash391 2011
[37] S Gao F Zhang and Y He ldquoThe effects of migratory birdpopulation in a nonautonomous eco-epidemiological modelrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 37 no 6 pp3903ndash3916 2013
[38] T Kuniya ldquoExistence of a nontrivial periodic solution in an age-structured SIR epidemicmodel with time periodic coefficientsrdquoApplied Mathematics Letters vol 27 pp 15ndash20 2014
[39] L Zu D Jiang D OrsquoRegan and B Ge ldquoPeriodic solution fora non-autonomous Lotka-Volterra predator-prey model withrandom perturbationrdquo Journal of Mathematical Analysis andApplications vol 430 no 1 pp 428ndash437 2015
[40] Y Lin D Jiang and T Liu ldquoNontrivial periodic solution ofa stochastic epidemic model with seasonal variationrdquo AppliedMathematics Letters vol 45 pp 103ndash107 2015
[41] X R Mao Stochastic Differential Equations and Their Applica-tions Horwood Chichester UK 1997
[42] R Khasminskii Stochastic Stability of Differential EquationsSpringer Berlin Germany 2011
Submit your manuscripts athttpswwwhindawicom
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Differential EquationsInternational Journal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Complexity
30] they obtained thresholds of the stochastic system whichdetermines the extinction and persistence of the epidemicZhang et al [29] proved that there is a unique ergodic station-ary distribution of his model We assume that environmentfluctuations will manifest themselves mainly as fluctuationsin the saturated response rate so that 120573119894119878(119905)119868119894(119905)(119886119894+119868119894(119905)) rarr120573119894119878(119905)119868119894(119905)(119886119894 + 119868119894(119905)) + (120590119894119878(119905)119868119894(119905)(119886119894 + 119868119894(119905)))119894(119905) (119894 = 1 2)where 119861(119905) = (1198611(119905) 1198612(119905)) is a standard Brownian motionwith intensity 120590119894 gt 0 (119894 = 1 2) Therefore a stochastic modelis described by [11]
d119878 (119905) = (119860 minus 119889119878 (119905) minus 1205731119878 (119905) 1198681 (119905)1198861 + 1198681 (119905) minus 1205732119878 (119905) 1198682 (119905)1198862 + 1198682 (119905)+ 11990311198681 (119905) + 11990321198682 (119905)) d119905 minus 1205901119878 (119905) 1198681 (119905)1198861 + 1198681 (119905) d1198611 (119905)minus 1205902119878 (119905) 1198682 (119905)1198862 + 1198682 (119905) d1198612 (119905)
d1198681 (119905) = (1205731119878 (119905) 1198681 (119905)1198861 + 1198681 (119905) minus (119889 + 1205721 + 1199031) 1198681 (119905)) d119905+ 1205901119878 (119905) 1198681 (119905)1198861 + 1198681 (119905) d1198611 (119905)
d1198682 (119905) = (1205732119878 (119905) 1198682 (119905)1198862 + 1198682 (119905) minus (119889 + 1205722 + 1199032) 1198682 (119905)) d119905+ 1205902119878 (119905) 1198682 (119905)1198862 + 1198682 (119905) d1198612 (119905)
(2)
However many infectious diseases of human fluctuateover time and often show the seasonal morbidity Thereforethe existence of periodic solutions of some nonautonomousepidemic models was explored [35ndash37] Recently manyscholars focused on nonautonomous stochastic periodicsystems With the development of stochastic differentialequations and application ofHasrsquominskii theory the existenceof stochastic periodic solution has been studied [23 3839] In [23] Zhang et al considered a nonautonomousstochastic Lotka-Volterra predator-prey model with impul-sive effects they got thresholds for stochastic persistence andextinction of the system Authors of [38ndash40] investigatedperiodic solution of a stochastic nonautonomous epidemicmodel
Based on the discussion above in this paper we considera nonautonomous stochastic SIS model with periodic coeffi-cients
d119878 (119905) = (119860 (119905) minus 119889 (119905) 119878 (119905) minus 1205731 (119905) 119878 (119905) 1198681 (119905)1198861 (119905) + 1198681 (119905)minus 1205732 (119905) 119878 (119905) 1198682 (119905)1198862 (119905) + 1198682 (119905) + 1199031 (119905) 1198681 (119905) + 1199032 (119905) 1198682 (119905)) d119905minus 1205901 (119905) 119878 (119905) 1198681 (119905)1198861 (119905) + 1198681 (119905) d1198611 (119905) minus 1205902 (119905) 119878 (119905) 1198682 (119905)1198862 (119905) + 1198682 (119905) d1198612 (119905)
d1198681 (119905) = (1205731 (119905) 119878 (119905) 1198681 (119905)1198861 (119905) + 1198681 (119905)minus (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681 (119905)) d119905+ 1205901 (119905) 119878 (119905) 1198681 (119905)1198861 (119905) + 1198681 (119905) d1198611 (119905)
d1198682 (119905) = (1205732 (119905) 119878 (119905) 1198682 (119905)1198862 (119905) + 1198682 (119905)minus (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682 (119905)) d119905+ 1205902 (119905) 119878 (119905) 1198682 (119905)1198862 (119905) + 1198682 (119905) d1198612 (119905)
(3)
where the parameter functions 119860(119905) 119889(119905) 120573119894(119905) 119886119894(119905) 119903119894(119905)120572119894(119905) 120590119894(119905) (119894 = 1 2) are positive nonconstant and contin-uous periodic functions with positive period 119879
To the best of our knowledge there are only fewworks on research of nonautonomous stochastic epidemicmodels with nonlinear saturated incidence rate and doubleepidemic hypothesis Therefore based on an autonomousstochastic epidemic model we propose a nonautonomousstochastic model and investigate the existence of stochasticperiodic solution and the extinction of the two epidemicdiseases
This paper is organized as follows In Section 2 we givesome definitions and known results In Section 3 we provethat system (3) has a unique global positive solution InSection 4 we present sufficient condition for the existencea nontrivial positive periodic solution of system (3) InSection 5 we obtain the sufficient conditions of system (3)for extinction of the two epidemic diseases In Section 6 wecarry out a series of numerical simulations to illustrate ourtheoretical findings
2 Preliminaries
Throughout this paper let (ΩFP) be a complete prob-ability space with a filtration F119905119905ge0 satisfying the usualconditions (ie it is increasing and right continuouswhileF0contains all P-null sets) The function 119861119894(119905) (119894 = 1 2 3 4) isdefined on this complete probability space
For simplicity some notations are given first If 119891(119905) isan integrable function defined on [0infin) define ⟨119891⟩119905 =(1119905) int119905
0119891(119904)119889119904 119905 gt 0 If 119891(119905) is a bounded function on
[0infin) define119891119897 = inf 119905isin[0infin)119891(119905) and 119891119906 = sup119905isin[0infin)119891(119905)Here we present some basic theory in stochastic differen-
tial equations which are introduced in [41]In general consider the 119897-dimensional stochastic differ-
ential equation
d119883 (119905) = 119891 (119883 (119905) 119905) d119905 + 119892 (119883 (119905) 119905) 119889119861 (119905) 119905 ge 1199050 (4)
with initial value 119909(1199050) = 1199090 isin R119897 119861(119905) stands for a119897-dimensional standard Brownian motion defined on the
Complexity 3
complete probability space (ΩF F119905119905ge0P) Denote by11986221(R119897 times [1199050infin]R+) the family of all nonnegative functions119881(119883 119905)defined onR119897times[1199050infin] such that they are continuouslytwice differentiable in 119883 and once in 119905 The differentialoperator 119871 of (4) is defined by [41]
119871 = 120597120597119905 +sum119891119894 (119883 119905) 120597120597119883119894+ 12119897sum119894119895=1
[119892119879 (119883 119905) 119892 (119883 119905)]119894119895
1205972120597119883119894120597119883119895 (5)
If 119871 acts on a function 119881 isin 11986221(R119897 times [1199050infin]R+) then119871119881 (119883 119905) = 119881119905 (119883 119905) + 119881119883 (119883 119905) 119891 (119883 119905)
+ 12 trace [119892119879 (119883 119905) 119881119883119883 (119883 119905) 119892 (119883 119905)] (6)
where 119881119905 = 120597119881120597119905 119881119883 = (1205971198811205971199091 120597119881120597119909119897) and 119881119883119883 =(1205972119881120597119883119894120597119883119895)119897times119897 In view of Itorsquos formula if119883(119905) isin R119897 then
d119881 (119883 (119905) 119905) = 119871119881 (119909 (119905) 119905) d119905+ 119881119883 (119883 (119905) 119905) 119892 (119883 (119905) 119905) 119889119861 (119905) (7)
Definition 1 (see [42]) A stochastic process 120585(119905) =120585(119905 120596) (minusinfin lt 119905 lt +infin) is said to be periodic with period119879 if for every finite sequence of numbers 1199051 1199052 119905119899 the jointdistribution of random variables 120585(1199051 +ℎ) 120585(1199052 +ℎ) 120585(119905119899 +ℎ) is independent of ℎ where ℎ = 119896119879 119896 = plusmn1 plusmn2
It is shown in [42] that a Markov process 119909(119905) is 119879-periodic if and only if its transition probability function is 119879-periodic and the functionP0(119905 119860) = P119909(119905) isin 119860 satisfies theequation
P0 (119904 119860) = intR119897P0 (119904 119889119909)P (119904 119909 119904 + 119879 119860)
= P0 (119904 + 119879 119860) (8)
Consider the following equation
119883(119905) = 119883 (1199050) + int1199051199050
119887 (119904 119883 (119904)) 119889119904
+ 119896sum119903=1
int1199051199050
120590119903 (119904 119883 (119904)) 119889119861119903 (119904) 119883 isin R119897
(9)
Lemma 2 (see [42]) Suppose that coefficients of (9) are 119879-periodic in 119905 and satisfy the condition
1003816100381610038161003816119887 (119904 119909) minus 119887 (119904 119910)1003816100381610038161003816 +119896sum119903=1
1003816100381610038161003816120590119903 (119904 119909) minus 120590119903 (119904 119910)1003816100381610038161003816le 119875 1003816100381610038161003816119909 minus 1199101003816100381610038161003816
|119887 (119904 119909)| + 119896sum119903=1
1003816100381610038161003816120590119903 (119904 119909)1003816100381610038161003816 le 119875 (1 + |119909|) (10)
in every cylinder 119868 times 119880 where 119875 is a constant And supposefurther that there exists a function 119881(119905 119909) isin 1198622 inR119889 which is119879-periodic in 119905 and satisfies the following conditions(1198601) inf |119909|geR119881(119905 119909) rarr infin as R rarrinfin(1198602) 119871119881(119905 119909) le minus1 outside some compact set where the
operator 119871 is given by
119871 = 120597120597119905 +119897sum119894=1
119887119894 (119905 119909) 120597120597119909119894 +12119897sum119894119895=1
119886119894119895 (119905 119909) 1205972120597119909119894120597119909119895
119886119894119895 = 119896sum119903=1
120590119894119903 (119905 119909) 120590119895119903 (119905 119909) (11)
Then there exists a solution of (9) which is a119879-periodicMarkovprocess
Lemma 3 (see [2] strong law of large numbers) Let 119872 =119872119905119905 ge 0 be a real-valued continuous local martingalevanishing at 119905 = 0
Thenlim119905rarrinfin
⟨119872119872⟩119905 = infin 119886119904 997904rArrlim119905rarrinfin
119872119905⟨119872119872⟩119905 = 0 119886119904 (12)
and also
lim sup119905rarrinfin
⟨119872119872⟩119905119905 lt infin 119886119904 997904rArrlim119905rarrinfin
119872119905119905 = 0 119886119904(13)
3 Existence and Uniqueness of theGlobal Positive Solution
In this section we prove that system (3) has a unique globalpositive solution
Theorem 4 For any initial value (119878(0) 1198681(0) 1198682(0)) isin 1198773+there is a unique positive solution (119878(119905) 1198681(119905) 1198682(119905)) of (3) on119905 ge 0 and the solution will remain in 1198773+ with probability oneProof From system (3) we can get
d (119878 (119905) + 1198681 (119905) + 1198682 (119905))d119905
= 119860 (119905) minus 119889 (119905) (119878 (119905) + 1198681 (119905) + 1198682 (119905))minus (1205721 (119905) 1198681 (119905) + 1205722 (119905) 1198682 (119905))
le 119860 (119905) minus 119889 (119905) (119878 (119905) + 1198681 (119905) + 1198682 (119905))le 119860119906 minus 119889119897 (119878 (119905) + 1198681 (119905) + 1198682 (119905))
(14)
Then
lim119905rarr+infin
(119878 (119905) + 1198681 (119905) + 1198682 (119905)) le 119860119906119889119897 (15)
4 Complexity
obviously we have
lim sup119905rarr+infin
119878 (119905) le 119860119906119889119897 lim sup119905rarr+infin
1198681 (119905) le 119860119906119889119897 lim sup119905rarr+infin
1198682 (119905) le 119860119906119889119897 (16)
Since the coefficients of system (3) satisfy the localLipschitz conditions then for any given initial value(119878(0) 1198681(0) 1198682(0)) isin 1198773+ there is a unique local solution(119878(119905) 1198681(119905) 1198682(119905)) on 119905 isin [0 120591119890) where 120591119890 is the explosion timeTo demonstrate that this solution is global we only need toprove that 120591119890 = infin as
Let 1198960 gt 0 be sufficiently large for any initial value119878(0) 1198681(0) and 1198682(0) lying within the interval [11198960 119896] Foreach integer 119896 ge 1198960 define the following stopping time
120591119896 = inf 119905 isin [0 120591119890) min 119878 (119905) 1198681 (119905) 1198682 (119905)le 1119896 ormax 119878 (119905) 1198681 (119905) 1198682 (119905) ge 119896
(17)
where we set inf 0 = infin (as usual 0 denotes the empty set)Clearly 120591119896 is increasing as 119896 rarr infin Let 120591infin = lim119896rarrinfin120591119896 hence120591infin le 120591119896 as Next we only need to verify 120591infin = infin as If thisstatement is false then there exist two constants 119879 gt 0 and120598 isin (0 1) such that
P 120591infin le 119879 gt 120598 (18)
Thus there is an integer 1198961 ge 1198960 such that
P 120591119896 le 119879 ge 120598 119896 ge 1198961 (19)
Define a 1198622-function 119881 R3+ rarr R+ as follows
119881 (119878 1198681 1198682) = 119878 minus 1 minus ln 119878 + 1198681 minus 1 minus ln 1198681 + 1198682 minus 1minus ln 1198682 (20)
the nonnegativity of this function can be obtained from
119906 minus 1 minus ln 119906 ge 0 119906 gt 0 (21)
Applying Itorsquos formula yields
d119881 (119878 1198681 1198682) = 119871119881 (119878 1198681 1198682) d119905+ (21198781198681 minus 119878 minus 1198681) 1205901 (119905)1198861 (119905) + 1198681 1198891198611 (119905)+ (21198781198682 minus 119878 minus 1198682) 1205902 (119905)1198862 (119905) + 1198682 1198891198612 (119905)
(22)
where
119871119881 = (1 minus 1119878)(119860 (119905) minus 119889 (119905) 119878 minus 1205731 (119905) 11987811986811198861 (119905) + 1198681minus 1205732 (119905) 11987811986821198862 (119905) + 1198682 + 1199031 (119905) 1198681 + 1199032 (119905) 1198682) + 12
sdot 12059021 (119905) 11986821(1198861 (119905) + 1198681)2 +12
12059022 (119905) 11986822(1198862 (119905) + 1198682)2 + (1 minus11198681)
sdot ( 1205731 (119905) 11987811986811198861 (119905) + 1198681 minus (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681) + 12sdot 12059021 (119905) 1198782(1198861 (119905) + 1198681)2 + (1 minus
11198682)(1205732 (119905) 11987811986821198862 (119905) + 1198682
minus (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682) + 1212059022 (119905) 1198782(1198862 (119905) + 1198682)2
= 119860 (119905) minus 119889 (119905) 119878 minus 119860 (119905)119878 + 119889 (119905) + 1205731 (119905) 11986811198861 (119905) + 1198681+ 1205732 (119905) 11986821198862 (119905) + 1198682 minus
1199031 (119905) 1198681119878 minus 1199032 (119905) 1198682119878+ 12059021 (119905) 119868212 (1198861 (119905) + 1198681)2 +
12059022 (119905) 119868222 (1198862 (119905) + 1198682)2 minus (119889 (119905)+ 1205721 (119905)) 1198681 minus 1205731 (119905) 1198781198861 (119905) + 1198681 + 119889 (119905) + 1205721 (119905) + 1199031 (119905)+ 12059021 (119905) 11987822 (1198861 (119905) + 1198681)2 minus (119889 (119905) + 1205722 (119905)) 1198682 minus
1205732 (119905) 1198781198862 (119905) + 1198682+ 119889 (119905) + 1205722 (119905) + 1199032 (119905) + 12059022 (119905) 11987822 (1198862 (119905) + 1198682)2 le 119860119906
+ 3119889119906 + 1205721199061 + 1205721199062 + 1199031199061 + 1199031199062 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622+ 12059021199061 1198602119906211988621198971 1198892119897 +
12059021199062 1198602119906211988621198972 1198892119897 fl 119870(23)
where 119870 is a positive constantSo we have
d119881 (119878 1198681 1198682) le 119870d119905+ (21198781198681 minus 119878 minus 1198681) 1205901 (119905)1198861 (119905) + 1198681 1198891198611 (119905)+ (21198781198682 minus 119878 minus 1198682) 1205902 (119905)1198862 (119905) + 1198682 1198891198612 (119905)
(24)
Integrating (24) from 0 to 120591119896 and 119879 and taking expectations onboth sides yield
E119881 (119878 (120591119896 and 119879) 1198681 (120591119896 and 119879) 1198682 (120591119896 and 119879))le 119881 (119878 (0) 1198681 (0) 1198682 (0)) + 119870119879 (25)
Let Ω119896 = 120591119896 le 119879 from inequality (25) we can see that119875(Ω119896) ge 120598We have
119881 (119878 (120591119896 and 119879) 1198681 (120591119896 and 119879) 1198682 (120591119896 and 119879))ge (119896 minus 1 minus ln 119896) and (1119896 minus 1 minus ln 1119896)
(26)
Complexity 5
By (25) and (26) one has
119881 (119878 (0) 1198681 (0) 1198682 (0)) + 119870119879 ge E [1Ω119896 (120596)sdot 119881 (119878 (120591119896 and 119879) 1198681 (120591119896 and 119879) 1198682 (120591119896 and 119879))] ge 120598 (119896minus 1 minus ln 119896) and (1119896 minus 1 minus ln 1119896)
(27)
where 1Ω119896 is the indicator function ofΩ119896
Let 119896 rarr infin we have
infin gt 119881(119878 (0) 1198681 (0) 1198682 (0)) + 119870119879 = infin (28)
So we obtain 120591infin = infinThe proof is completed
4 Existence of Nontrivial 119879-Periodic SolutionIn this section we verify that system (3) admits at least onenontrivial positive 119879-periodic solution Define
R = 2sum119894=1
⟨119860120572119894120573119894⟩119879⟨119889 + 120572119894 + 119903119894 + 1205902119894 119860211990621198862119894 1198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩119879 ⟨119860 + 11988611205721 + 11988621205722⟩119879 (29)
Theorem 5 When (1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)))119897 gt 119860119906119889119897and (1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)))119897 gt 119860119906119889119897 hold if R gt 1then there exists a nontrivial positive T-periodic solution ofsystem (3)
Proof Define a 1198622-function 119881 [0infin) timesR3+ rarr R+
119881 (119905 119878 1198681 1198682) = 119872(minus (1198901 + 1198902) ln 119878 minus 1198881 ln 1198681 minus 1198882 ln 1198682+ 119878 + 1198681 + 1198682 + 120596 (119905)) minus ln 119878 minus 1198681 minus 1198682 + 1120579 + 1 (119878+ 1198681 + 1198682)120579+1 = 119872(1198811 + 120596 (119905)) + 1198812 + 1198813 + 1198814+ 1198815
(30)
where1198881= ⟨11986012057211205731⟩119879⟨119889 + 1205721 + 1199031 + 1205902111986021199062119886211198892119897⟩2119879 ⟨119889 + 120590212 + 120590222⟩119879 1198901= ⟨11986012057211205731⟩119879⟨119889 + 1205721 + 1199031 + 1205902111986021199062119886211198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩2119879 1198882= ⟨11986012057221205732⟩119879⟨119889 + 1205722 + 1199032 + 1205902211986021199062119886221198892119897⟩2119879 ⟨119889 + 120590212 + 120590222⟩119879 1198902= ⟨11986012057221205732⟩119879⟨119889 + 1205722 + 1199032 + 1205902211986021199062119886221198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩2119879
(31)
and 0 lt 120579 lt min1 119889119897(12059021199061 + 12059021199062 )119872 is a sufficiently largepositive constant and satisfies the following conditions
119889119897 minus 120579 (12059021199061 + 12059021199062 ) gt 0minus119872120582 +max 119863 119864 le minus2 (32)
where
120582 = ⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1) 119863 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
119864 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(33)
where
119874 = sup(1198781198681 1198682)isinR
3+
119860119906 (119878 + 1198681 + 1198682)120579
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 lt infin(34)
Next we prove that condition (1198601) in Lemma 2 holds It iseasy to check that 119881(119905 119878 1198681 1198682) is a 119879-periodic function in 119905and satisfies
lim inf119870rarrinfin(1198781198681 1198682)isinR
3+U119896
119881 (119905 119878 1198681 1198682) = infin (35)
where U119896 = (1119896 119896) times (1119896 119896) times (1119896 119896) and 119896 gt 1 is asufficiently large number
6 Complexity
By Itorsquos formula we obtain
1198711198811 = minus1198901 + 1198902119878 (119860 (119905) minus 119889 (119905) 119878 minus 1205731 (119905) 11987811986811198861 (119905) + 1198681minus 1205732 (119905) 11987811986821198862 (119905) + 1198682 + 1199031 (119905) 1198681 + 1199032 (119905) 1198682)minus 11988811198681 (
1205731 (119905) 11987811986811198861 (119905) + I1minus (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681)
+ 1198901 + 1198902212059021 (119905) 11986821(1198861 (119905) + 1198681)2 minus
11988821198682 (1205732 (119905) 11987811986821198862 (119905) + 1198682
minus (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682) + 1198901 + 11989022sdot 12059022 (119905) 11986822(1198862 (119905) + 1198682)2 +
1198881212059021 (119905) 1198782(1198861 (119905) + 1198681)2 +
11988822sdot 12059022 (119905) 1198782(1198862 (119905) + 1198682)2 + 119860 (119905) minus 119889 (119905) (119878 + 1198681 + 1198682) minus 1205721 (119905)sdot 1198681 minus 1205722 (119905) 1198682 le minus1198901119860 (119905)119878 minus 11988811205731 (119905) 1198781198861 (119905) + 1198681 minus 1205721 (119905)sdot (1198861 (119905) + 1198681) minus 1198902119860 (119905)119878 minus 11988821205732 (119905) 1198781198862 (119905) + 1198682 minus 1205722 (119905)sdot (1198862 (119905) + 1198682) + 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905)
+ 12059021 (119905) 1198602119906211988621 (119905) 1198892119897) + 1198901 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )
+ 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905) + 12059022 (119905) 1198602119906211988622 (119905) 1198892119897)
+ 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 ) minus (1198901 + 1198902) (1199031 (119905)119878+ 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878 + 119889 (119905)minus 1205732 (119905)1198862 (119905) ) 1198682 + 119860 (119905) + 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905)le minus3 (119860 (119905) 1205721 (119905) 1205731 (119905) 11988811198901)13minus 3 (119860 (119905) 1205722 (119905) 1205732 (119905) 11988821198902)13 + 1198901 (119889 (119905) + 12059021 (119905)2+ 12059022 (119905)2 ) + 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905)
+ 12059021 (119905) 1198602119906211988621 (119905) 1198892119897) + 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905)
+ 12059022 (119905) 1198602119906211988622 (119905) 1198892119897) + 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )
+ 119860 (119905) + 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905) minus (1198901 + 1198902)sdot (1199031 (119905)119878 + 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878+ 119889 (119905) minus 1205732 (119905)1198862 (119905) ) 1198682 fl R0 (119905) minus (1198901 + 1198902) (1199031 (119905)119878+ 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878 + 119889 (119905)minus 1205732 (119905)1198862 (119905) ) 1198682
(36)
where
R0 (119905) = 3 (119860 (119905) 1205721 (119905) 1205731 (119905) 11988811198901)13+ 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905) + 12059021 (119905) 1198602119906211988621 (119905) 1198892119897)
+ 1198901 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )minus 3 (119860 (119905) 1205722 (119905) 1205732 (119905) 11988821198902)13+ 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905) + 12059022 (119905) 1198602119906211988622 (119905) 1198892119897)
+ 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 ) + 119860 (119905)+ 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905)
(37)
Note that (1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)))119897 gt 119860119906119889119897 and(1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)))119897 gt 119860119906119889119897 hold then119871 (1198811) le R0 (119905) (38)
Define the 119879-periodic function 120596(119905) satisfying1205961015840 (119905) = ⟨R0⟩119879 minusR0 (119905) (39)
So
119871 (1198811 + 120596 (119905)) le ⟨R0⟩119879= minus ⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1)fl minus120582
(40)
Applying Itorsquos formula we can also have
Complexity 7
1198711198812 = minus119860 (119905)119878 + 119889 (119905) + 1205731 (119905) 11986811198861 (119905) + 1198681 +1205732 (119905) 11986821198862 (119905) + 1198682
minus 1199031 (119905) 1198681119878 minus 1199032 (119905) 1198682119878 + 12059021 (119905) 119868212 (1198861 (119905) + 1198681)2+ 12059022 (119905) 119868222 (1198862 (119905) + 1198682)2 le minus119860119897119878 + 119889119906 + 1205731199061 + 1205731199062 + 120590211990612+ 120590211990622
1198711198813 = minus 1205731 (119905) 11987811986811198861 (119905) + 1198681 + (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681 le (119889119906+ 1205721199061 + 1199031199061 ) 1198681
1198711198814 = minus 1205732 (119905) 11987811986821198862 (119905) + 1198682 + (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682 le (119889119906+ 1205721199062 + 1199031199062 ) 1198682
1198711198815 = (119878 + 1198681 + 1198682)120579 (119860 (119905) minus 119889 (119905) 119878 minus 119889 (119905) 1198681 minus 119889 (119905) 1198682minus 1205721 (119905) 1198681 minus 1205722 (119905) 1198682) + 120579 (119878 + 1198681 + 1198682)120579minus1sdot ( 12059021 (119905) 119878211986821(1198861 (119905) + 1198681)2 +
12059022 (119905) 119878211986822(1198862 (119905) + 1198682)2) le 119860119906 (119878 + 1198681+ 1198682)120579 minus 119889119897 (119878 + 1198681 + 1198682)120579+1 + 120579 (119878 + 1198681 + 1198682)120579minus1 (12059021199061+ 12059021199062 ) 1198782 le 119860119906 (119878 + 1198681 + 1198682)120579 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 le 119874 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12
(41)
where
119874 = sup(1198781198681 1198682)isinR
3+
119860119906 (119878 + 1198681 + 1198682)120579
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 lt infin(42)
Therefore119871119881 (119905 119878 1198681 1198682) le minus119872⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1)
+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682 minus 119860119897119878minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874+ 119889119906 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622
= minus119872120582 minus 119860119897119878 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874+ 119889119906 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622
(43)
Now we are in the position to construct a compact subset119880 such that 1198602 in Lemma 2 holds Define the followingbounded closed set
119880 = (119878 1198681 1198682) isin R3+ 120598 le 119878 le 1120598 120598 le 1198681 le 1120598 120598 le 1198682
le 1120598 (44)
where 120598 gt 0 is a sufficiently small number In the set R3+119880we can choose 120598 sufficiently small such that
minus119872120582 minus 119860119897120598 + 119862 le minus1 (45)
minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 120598 + 119863 le minus1 (46)
minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 120598 + 119864 le minus1 (47)
minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119862 le minus1 (48)
minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119866 le minus1 (49)
minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119867 le minus1 (50)
where 119862 119863 119864 119866 and119867 are positive constants which can befound from the following inequations (52) (54) (56) (59)and (61) respectively For the sake of convenience we divideinto six domains
8 Complexity
1198801 = (119878 1198681 1198682) isin R3+ 0 lt 119878 lt 120598
1198802 = (119878 1198681 1198682) isin R3+ 0 lt 1198681 lt 120598
1198803 = (119878 1198681 1198682) isin R3+ 0 lt 1198682 lt 120598
1198804 = (119878 1198681 1198682) isin R3+ 119878 gt 1120598
1198805 = (119878 1198681 1198682) isin R3+ 1198681 gt 1120598
1198806 = (119878 1198681 1198682) isin R3+ 1198682 gt 1120598
(51)
Nextwewill prove that119871119881(119878 1198681 1198682) le minus1 onR3+119880 whichis equivalent to proving it on the above six domains
Case 1 If (119878 1198681 1198682) isin 1198801 one can see that
119871119881 le minus119872120582 minus 119860119897119878 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622 le minus119872120582 minus 119860119897119878 + 119862
le minus119872120582 minus 119860119897120598 + 119862
(52)
where
119862 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(53)
According to (45) we have 119871119881 le minus1 for all (119878 1198681 1198682) isin 1198801Case 2 If (119878 1198681 1198682) isin 1198802 one can get that
119871119881 le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681 + 119863le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 120598 + 119863
(54)
where
119863 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(55)
In view of (46) we can obtain that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198802Case 3 If (119878 1198681 1198682) isin 1198803 we have
119871119881 le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682 + 119864le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 120598 + 119864
(56)
where
119864 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(57)
By (47) we can conclude that 119871119881 le minus1 for all (119878 1198681 1198682) isin1198803Case 4 If (119878 1198681 1198682) isin 1198804 one can derive that
119871119881 le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1 + 119862le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119862
(58)
Complexity 9
Together with (48) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198804Case 5 If (119878 1198681 1198682) isin 1198805 it follows that
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119866le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119866
(59)
where
119866 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(60)
By virtue of (49) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198805Case 6 If (119878 1198681 1198682) isin 1198806 we obtain
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119867le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119867
(61)
where
119867 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(62)
It follows from (50) that 119871119881 le minus1 for all (119878 1198681 1198682) isin 1198806Clearly one can see from (52) (54) (56) (58) (59) and
(61) that for a sufficiently small 120598119871119881 (119878 1198681 1198682) le minus1
(119878 1198681 1198682) isin R3+119880 (63)
Hence 1198602 in Lemma 2 is satisfied This completes the proofof Theorem 5
5 Extinction
In this section we investigate the conditions for the extinc-tion of the two infectious diseases of system (3)
Let
R1 = 119860119906 ⟨1205731⟩119879119889119897 ⟨1198861⟩119879 ⟨119889 + 1205721 + 1199031⟩119879minus 1198602119906 ⟨12059021⟩11987921198892119897 ⟨11988621⟩119879 ⟨119889 + 1205721 + 1199031⟩119879
R2 = 119860119906 ⟨1205732⟩119879119889119897 ⟨1198862⟩119879 ⟨119889 + 1205722 + 1199032⟩119879minus 1198602119906 ⟨12059022⟩11987921198892119897 ⟨11988622⟩119879 ⟨119889 + 1205722 + 1199032⟩119879
(64)
Theorem 6 Let (119878(119905) 1198681(119905) 1198682(119905)) be a solution of system (3)with initial value (119878(0) 1198681(0) 1198682(0)) isin 1198773+
Then if
(i) ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792 ⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 119894 = 1 2 (65)
or(ii) R119894 lt 1⟨1205902119894 ⟩119879 le 119889119897 ⟨1198862119894 ⟩119879 ⟨120573119894⟩119879119860119906 ⟨119886119894⟩119879
119894 = 1 2(66)
hold the two infectious diseases of system (3) go to extinctionas that is
lim119905rarr+infin
119868119894 (119905) = 0 119894 = 1 2 (67)
10 Complexity
Proof Applying Itorsquos formula to system (3) we have
d ln 119868119894 (119905) = ( 120573119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905) minus (119889 (119905) + 120572119894 (119905) + 119903119894 (119905))minus 1205902119894 (119905) 1198782 (119905)2 (119886119894 (119905) + 119868119894 (119905))2) d119905 + 120590119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905)d119861119894 (119905)
119894 = 1 2
(68)
Case (i) Integrating (68) from 0 to 119905 and dividing 119905 on bothsides we obtain
ln 119868119894 (119905)119905 = minus1205902119894 (119905)2119905 int1199050( 119878 (120591)119886119894 (120591) + 119868119894 (120591) minus
120573119894 (120591)1205902119894 (120591))2
d120591minus 1119905 int
119905
0(119889 (120591) + 120572119894 (120591) + 119903119894 (120591)) d120591
+ 1119905 int119905
0
1205732119894 (120591)21205902119894 (120591)d120591 +119872119894 (119905)119905 + ln 119868119894 (0)119905
le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905
(69)
Case (ii) Integrating (68) from 0 to 119905 first and then dividingby 119905 on both sides yield
ln 119868119894 (119905)119905 = 1119905 int119905
0( 120573119894 (120591) 119878 (120591)119886119894 (120591) + 119868119894 (120591)
minus (119889 (120591) + 120572119894 (120591) + 119903119894 (120591))minus 1205902119894 (120591) 1198782 (120591)2 (119886119894 (120591) + 119868119894 (120591))2) d120591 + 119872119894 (119905)119905 + ln 119868119894 (0)119905le ⟨120573119894119860119906119886119894119889119897 minus (119889 + 120572119894 + 119903119894) minus
1205902119894 119860211990621198862119894 1198892119897⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) + 119872119894 (119905)119905+ ln 119868119894 (0)119905
(70)
where 119872119894(119905) = int1199050120590119894(120591)119878(120591)(119886119894(120591) + 119868119894(120591))d119861119894(120591) 119894 = 1 2
which is a local continuous martingale with 119872119894(0) = 0 ByLemma 3 we have
lim119905rarr+infin
119872119894 (119905)119905 = 0 119894 = 1 2 (71)
Taking the limit superior of both sides of (69) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119879 lt 0 (72)
which implies lim119905rarr+infin119868119894(119905) = 0
Taking the superior limit of both sides of (70) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) lt 0 (73)
which implies lim119905rarr+infin119868119894(119905) = 0 119894 = 1 2This completes theproof
Remark 7 Theorem 6 shows that the two diseases will dieout if the white noise disturbance is large or the white noisedisturbance is not large and R119894 lt 1 When ⟨1205902119894 ⟩119879 gt⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectious diseases of system(3) die out almost surely that is to say large white noisestochastic disturbance can lead to the two epidemics beingextinct
6 Numerical Simulations
Now we introduce some numerical simulations exampleswhich illustrate our theoretical results
Example 8 In model (3) let
119860 (119905) = 05 + 01 sin1205871199051198861 (119905) = 03 + 01 sin1205871199051198862 (119905) = 031 + 01 sin1205871199051205731 (119905) = 062 + 01 sin1205871199051205732 (119905) = 062 + 01 sin1205871199051199031 (119905) = 02 + 01 sin1205871199051199032 (119905) = 035 + 01 sin120587119905119889 (119905) = 02 + 01 sin1205871199051205721 (119905) = 02 + 01 sin1205871199051205722 (119905) = 025 + 01 sin1205871199051205901 (119905) = 01 + 005 sin1205871199051205902 (119905) = 01 + 005 sin120587119905
(74)
Note that R gt 1 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897and 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 hold that isthe conditions of Theorem 5 hold Hence system (3) has apositive periodic solution with 119879 = 1 Figure 1(a) shows theperiodicity of the nonautonomous stochastic model (3) with1205901 = 0 and 1205902 = 0 Figure 1(b) shows that solution of thenonautonomous stochastic model (3) with the initial value(119878(119905) 1198681(119905) 1198682(119905)) = (03 015 015) tends to a periodic orbitin the sense of joint distribution
Example 9 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin1205871199051198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin120587119905
Complexity 11
S(t)
I1(t)
I2(t)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
(a)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 1 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (1 02 02) to the nonautonomous stochastic model (3)
0 25 50 75 100t
59 60 61
078
08
082
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
59 60 61
078
08
082
0
1
2
14
14
S(t)I 1(t)I 2(t)
25 50 75 1000t
S(t)
I1(t)
I2(t)
(b)
Figure 2 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 04 + 02 sin1205871199051205902 (119905) = 05 + 02 sin120587119905
(75)
Note that ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897+120572119897119894+119903119897119894⟩119879Therefore conditions(i) of Theorem 6 hold Then the two infectious diseases willgo to extinction Figure 2(a) shows that one of two diseases inthe deterministic SIS epidemic model is extinct and the otheris persistent Figure 2(b) shows that the two diseases will dieout under the large white noise disturbance of model (3)
Example 10 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin120587119905
12 Complexity
25 50 75 1000t
0
05
1
15
2S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
0 25 50 75 100t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 3 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin1205871199051205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 02 + 02 sin1205871199051205902 (119905) = 02 + 02 sin120587119905
(76)
Note that R1 lt 1 R2 lt 1 and ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 That is conditions (ii) of Theorem 6 hold Thenthe two infectious diseases will go to extinction Figure 3(a)
shows that one of two diseases in the deterministic SISepidemic model is extinct and the other is persistent withoutthe white noises Figure 3(b) shows that the two diseases willdie out under a small white noise disturbance of model (3)
7 Discussion and Conclusions
This paper explores the existence of nontrivial positiveT-periodic solution of a nonautonomous stochastic SISepidemic model with nonlinear growth rate and doubleepidemic hypothesis By constructing a suitable stochasticLyapunov function we establish sufficient conditions for theexistence of nontrivial positive 119879-periodic solution of system(3) Furthermore the sufficient conditions for the extinctionof the two diseases are obtained Our results are given asfollows
(1) If 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 and R gt 1 hold the SISmodel has at least one nontrivial positive 119879-periodicsolution where
R = 2sum119894=1
⟨119860120572119894120573119894⟩119879⟨119889 + 120572119894 + 119903119894 + 1205902119894 119860211990621198862119894 1198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩119879 ⟨119860 + 11988611205721 + 11988621205722⟩119879 (77)
(2) If ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectiousdiseases go extinct
(3) If ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 and R119894 lt 1 the two
infectious diseases also go extinct where
R119894 = 119860119906 ⟨120573119894⟩119879119889119897 ⟨119886119894⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879minus 1198602119906 ⟨1205902119894 ⟩11987921198892119897 ⟨1198862119894 ⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879
(78)
Complexity 13
Some interesting questions deserve further investigationOn the one hand we may explore some realistic but complexmodels such as considering the effects of impulsive ordelay perturbations on system (3) On the other hand wecan concern the dynamics of a nonautonomous stochasticSIS epidemic model with two infectious diseases driven byLevy jumps What is more we can also investigate thenonautonomous stochastic SIS epidemic model with twoinfectious diseases by a continuous time Markov chain Wewill investigate these cases in our future work
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (11371230) Joint Innovative Center forSafe andEffectiveMiningTechnology andEquipment of CoalResources Shandong Province the SDUST Research Fund(2014TDJH102) and Shandong Provincial Natural ScienceFoundation China (ZR2015AQ001)
References
[1] W O Kermack and A G McKendrick ldquoA contributions to themathematical theory of epidemics (Part I)rdquo Proceedings of theRoyal Society of London A vol 115 pp 700ndash721 1927
[2] A Gray D Greenhalgh L Hu X Mao and J Pan ldquoA stochasticdifferential equation SIS epidemic modelrdquo SIAM Journal onApplied Mathematics vol 71 no 3 pp 876ndash902 2011
[3] A Korobeinikov and G C Wake ldquoLyapunov functions andglobal stability for SIR SIRS and SIS epidemiological modelsrdquoApplied Mathematics Letters vol 15 no 8 pp 955ndash960 2002
[4] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995
[5] T Zhang X Meng Y Song and T Zhang ldquoA stage-structuredpredator-prey SI model with disease in the prey and impulsiveeffectsrdquoMathematical Modelling and Analysis vol 18 no 4 pp505ndash528 2013
[6] J Li andZMa ldquoQualitative analyses of SIS epidemicmodelwithvaccination and varying total population sizerdquo Mathematicaland Computer Modelling vol 35 no 11-12 pp 1235ndash1243 2002
[7] A drsquoOnofrio ldquoA note on the global behaviour of the network-based SIS epidemic modelrdquo Nonlinear Analysis Real WorldApplications vol 9 no 4 pp 1567ndash1572 2008
[8] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[9] T Zhang X Meng T Zhang and Y Song ldquoGlobal dynamicsfor a new high-dimensional SIR model with distributed delayrdquoApplied Mathematics and Computation vol 218 no 24 pp11806ndash11819 2012
[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006
[11] X Meng S Zhao T Feng and T Zhang ldquoDynamics of a novelnonlinear stochastic SIS epidemic model with double epidemichypothesisrdquo Journal of Mathematical Analysis and Applicationsvol 433 no 1 pp 227ndash242 2016
[12] A Miao X Wang T Zhang W Wang and B Sampath ArunaPradeep ldquoDynamical analysis of a stochastic SIS epidemicmodel with nonlinear incidence rate and double epidemichypothesisrdquo Advances in Difference Equations 2017226 pages2017
[13] X-Z Meng ldquoStability of a novel stochastic epidemic modelwith double epidemic hypothesisrdquo Applied Mathematics andComputation vol 217 no 2 pp 506ndash515 2010
[14] X P Li X Y Lin and Y Q Lin ldquoLyapunov-type conditions andstochastic differential equations driven byG-BrownianmotionrdquoJournal of Mathematical Analysis and Applications vol 439 no1 pp 235ndash255 2016
[15] M Liu and M Fan ldquoPermanence of stochastic Lotka-Volterrasystemsrdquo Journal of Nonlinear Science vol 27 no 2 pp 425ndash452 2017
[16] L D Liu and X Z Meng ldquoOptimal harvesting controland dynamics of two-species stochastic model with delaysrdquoAdvances in Difference Equations vol 2017 18 pages 2017
[17] H Ma and Y Jia ldquoStability analysis for stochastic differentialequations with infiniteMarkovian switchingsrdquo Journal of Math-ematical Analysis and Applications vol 435 no 1 pp 593ndash6052016
[18] G D Liu X H Wang X Z Meng and S J Gao ldquoExtinctionand persistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexity vol 3pp 1ndash15 2017
[19] C Tan and W H Zhang ldquoOn observability and detectabilityof continuous-time stochastic Markov jump systemsrdquo Journalof Systems Science and Complexity vol 28 no 4 pp 830ndash8472015
[20] X Leng T Feng and X Meng ldquoStochastic inequalities andapplications to dynamics analysis of a novel SIVS epidemicmodel with jumpsrdquo Journal of Inequalities and ApplicationsPaper No 138 25 pages 2017
[21] G Li and M Chen ldquoInfinite horizon linear quadratic optimalcontrol for stochastic difference time-delay systemsrdquo Advancesin Difference Equations vol 2015 14 pages 2015
[22] Y Zhao and W Zhang ldquoObserver-based controller design forsingular stochastic Markov jump systems with state dependentnoiserdquo Journal of Systems Science and Complexity vol 29 no 4pp 946ndash958 2016
[23] S Zhang X Meng T Feng and T Zhang ldquoDynamics analysisand numerical simulations of a stochastic non-autonomouspredatorndashprey system with impulsive effectsrdquo Nonlinear Anal-ysis Hybrid Systems vol 26 pp 19ndash37 2017
[24] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013
[25] X Meng L Wang and T Zhang ldquoGlobal dynamics analysis ofa nonlinear impulsive stochastic chemostat system in a pollutedenvironmentrdquo Journal of AppliedAnalysis andComputation vol6 no 3 pp 865ndash875 2016
[26] X Liu Y Li andW Zhang ldquoStochastic linear quadratic optimalcontrol with constraint for discrete-time systemsrdquo AppliedMathematics and Computation vol 228 pp 264ndash270 2014
14 Complexity
[27] X Z Meng and X H Wang ldquoStochastic predator-prey systemsubject to levy jumprdquo Discrete Dynamics in Nature and Societyvol 2016 Article ID 5749892 13 pages 2016
[28] X Lv L Wang and X Meng ldquoGlobal analysis of a new non-linear stochastic differential competition systemwith impulsiveeffectrdquo Advances in Difference Equations vol 2017 296 pages2017
[29] X Zhang D Jiang A Alsaedi and T Hayat ldquoStationarydistribution of stochastic SIS epidemic model with vaccinationunder regime switchingrdquo Applied Mathematics Letters vol 59pp 87ndash93 2016
[30] F Li X Meng and Y Cui ldquoNonlinear stochastic analysis fora stochastic SIS epidemic modelrdquo Journal of Nonlinear Sciencesand Applications vol 10 no 09 pp 5116ndash5124 2017
[31] A Q Miao J Zhang T Q Zhang and B G Pradeep ldquoSampathAruna Pradeep Threshold dynamics of a stochastic SIR modelwith vertical transmission and vaccinationrdquo Computationaland Mathematical Methods in Medicine vol 2017 Article ID4820183 10 pages 2017
[32] Q Liu D Jiang N Shi T Hayat and A Alsaedi ldquoNontrivialperiodic solution of a stochastic non-autonomous SISV epi-demic modelrdquo Physica A Statistical Mechanics and its Applica-tions vol 462 pp 837ndash845 2016
[33] Y L Zhou S L Yuan and D L Zhao ldquoThreshold behavior ofa stochastic SIS model with levy jumpsrdquo Applied Mathematicsand Computation vol 275 pp 255ndash267 2016
[34] T Feng X Meng L Liu and S Gao ldquoApplication of inequal-ities technique to dynamics analysis of a stochastic eco-epidemiology modelrdquo Journal of Inequalities and ApplicationsPaper No 327 29 pages 2016
[35] M Liu and K Wang ldquoPersistence and extinction in stochasticnon-autonomous logistic systemsrdquo Journal of MathematicalAnalysis and Applications vol 375 no 2 pp 443ndash457 2011
[36] Z Bai and Y Zhou ldquoExistence of two periodic solutions for anon-autonomous SIR epidemic modelrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 1 pp 382ndash391 2011
[37] S Gao F Zhang and Y He ldquoThe effects of migratory birdpopulation in a nonautonomous eco-epidemiological modelrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 37 no 6 pp3903ndash3916 2013
[38] T Kuniya ldquoExistence of a nontrivial periodic solution in an age-structured SIR epidemicmodel with time periodic coefficientsrdquoApplied Mathematics Letters vol 27 pp 15ndash20 2014
[39] L Zu D Jiang D OrsquoRegan and B Ge ldquoPeriodic solution fora non-autonomous Lotka-Volterra predator-prey model withrandom perturbationrdquo Journal of Mathematical Analysis andApplications vol 430 no 1 pp 428ndash437 2015
[40] Y Lin D Jiang and T Liu ldquoNontrivial periodic solution ofa stochastic epidemic model with seasonal variationrdquo AppliedMathematics Letters vol 45 pp 103ndash107 2015
[41] X R Mao Stochastic Differential Equations and Their Applica-tions Horwood Chichester UK 1997
[42] R Khasminskii Stochastic Stability of Differential EquationsSpringer Berlin Germany 2011
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Complexity 3
complete probability space (ΩF F119905119905ge0P) Denote by11986221(R119897 times [1199050infin]R+) the family of all nonnegative functions119881(119883 119905)defined onR119897times[1199050infin] such that they are continuouslytwice differentiable in 119883 and once in 119905 The differentialoperator 119871 of (4) is defined by [41]
119871 = 120597120597119905 +sum119891119894 (119883 119905) 120597120597119883119894+ 12119897sum119894119895=1
[119892119879 (119883 119905) 119892 (119883 119905)]119894119895
1205972120597119883119894120597119883119895 (5)
If 119871 acts on a function 119881 isin 11986221(R119897 times [1199050infin]R+) then119871119881 (119883 119905) = 119881119905 (119883 119905) + 119881119883 (119883 119905) 119891 (119883 119905)
+ 12 trace [119892119879 (119883 119905) 119881119883119883 (119883 119905) 119892 (119883 119905)] (6)
where 119881119905 = 120597119881120597119905 119881119883 = (1205971198811205971199091 120597119881120597119909119897) and 119881119883119883 =(1205972119881120597119883119894120597119883119895)119897times119897 In view of Itorsquos formula if119883(119905) isin R119897 then
d119881 (119883 (119905) 119905) = 119871119881 (119909 (119905) 119905) d119905+ 119881119883 (119883 (119905) 119905) 119892 (119883 (119905) 119905) 119889119861 (119905) (7)
Definition 1 (see [42]) A stochastic process 120585(119905) =120585(119905 120596) (minusinfin lt 119905 lt +infin) is said to be periodic with period119879 if for every finite sequence of numbers 1199051 1199052 119905119899 the jointdistribution of random variables 120585(1199051 +ℎ) 120585(1199052 +ℎ) 120585(119905119899 +ℎ) is independent of ℎ where ℎ = 119896119879 119896 = plusmn1 plusmn2
It is shown in [42] that a Markov process 119909(119905) is 119879-periodic if and only if its transition probability function is 119879-periodic and the functionP0(119905 119860) = P119909(119905) isin 119860 satisfies theequation
P0 (119904 119860) = intR119897P0 (119904 119889119909)P (119904 119909 119904 + 119879 119860)
= P0 (119904 + 119879 119860) (8)
Consider the following equation
119883(119905) = 119883 (1199050) + int1199051199050
119887 (119904 119883 (119904)) 119889119904
+ 119896sum119903=1
int1199051199050
120590119903 (119904 119883 (119904)) 119889119861119903 (119904) 119883 isin R119897
(9)
Lemma 2 (see [42]) Suppose that coefficients of (9) are 119879-periodic in 119905 and satisfy the condition
1003816100381610038161003816119887 (119904 119909) minus 119887 (119904 119910)1003816100381610038161003816 +119896sum119903=1
1003816100381610038161003816120590119903 (119904 119909) minus 120590119903 (119904 119910)1003816100381610038161003816le 119875 1003816100381610038161003816119909 minus 1199101003816100381610038161003816
|119887 (119904 119909)| + 119896sum119903=1
1003816100381610038161003816120590119903 (119904 119909)1003816100381610038161003816 le 119875 (1 + |119909|) (10)
in every cylinder 119868 times 119880 where 119875 is a constant And supposefurther that there exists a function 119881(119905 119909) isin 1198622 inR119889 which is119879-periodic in 119905 and satisfies the following conditions(1198601) inf |119909|geR119881(119905 119909) rarr infin as R rarrinfin(1198602) 119871119881(119905 119909) le minus1 outside some compact set where the
operator 119871 is given by
119871 = 120597120597119905 +119897sum119894=1
119887119894 (119905 119909) 120597120597119909119894 +12119897sum119894119895=1
119886119894119895 (119905 119909) 1205972120597119909119894120597119909119895
119886119894119895 = 119896sum119903=1
120590119894119903 (119905 119909) 120590119895119903 (119905 119909) (11)
Then there exists a solution of (9) which is a119879-periodicMarkovprocess
Lemma 3 (see [2] strong law of large numbers) Let 119872 =119872119905119905 ge 0 be a real-valued continuous local martingalevanishing at 119905 = 0
Thenlim119905rarrinfin
⟨119872119872⟩119905 = infin 119886119904 997904rArrlim119905rarrinfin
119872119905⟨119872119872⟩119905 = 0 119886119904 (12)
and also
lim sup119905rarrinfin
⟨119872119872⟩119905119905 lt infin 119886119904 997904rArrlim119905rarrinfin
119872119905119905 = 0 119886119904(13)
3 Existence and Uniqueness of theGlobal Positive Solution
In this section we prove that system (3) has a unique globalpositive solution
Theorem 4 For any initial value (119878(0) 1198681(0) 1198682(0)) isin 1198773+there is a unique positive solution (119878(119905) 1198681(119905) 1198682(119905)) of (3) on119905 ge 0 and the solution will remain in 1198773+ with probability oneProof From system (3) we can get
d (119878 (119905) + 1198681 (119905) + 1198682 (119905))d119905
= 119860 (119905) minus 119889 (119905) (119878 (119905) + 1198681 (119905) + 1198682 (119905))minus (1205721 (119905) 1198681 (119905) + 1205722 (119905) 1198682 (119905))
le 119860 (119905) minus 119889 (119905) (119878 (119905) + 1198681 (119905) + 1198682 (119905))le 119860119906 minus 119889119897 (119878 (119905) + 1198681 (119905) + 1198682 (119905))
(14)
Then
lim119905rarr+infin
(119878 (119905) + 1198681 (119905) + 1198682 (119905)) le 119860119906119889119897 (15)
4 Complexity
obviously we have
lim sup119905rarr+infin
119878 (119905) le 119860119906119889119897 lim sup119905rarr+infin
1198681 (119905) le 119860119906119889119897 lim sup119905rarr+infin
1198682 (119905) le 119860119906119889119897 (16)
Since the coefficients of system (3) satisfy the localLipschitz conditions then for any given initial value(119878(0) 1198681(0) 1198682(0)) isin 1198773+ there is a unique local solution(119878(119905) 1198681(119905) 1198682(119905)) on 119905 isin [0 120591119890) where 120591119890 is the explosion timeTo demonstrate that this solution is global we only need toprove that 120591119890 = infin as
Let 1198960 gt 0 be sufficiently large for any initial value119878(0) 1198681(0) and 1198682(0) lying within the interval [11198960 119896] Foreach integer 119896 ge 1198960 define the following stopping time
120591119896 = inf 119905 isin [0 120591119890) min 119878 (119905) 1198681 (119905) 1198682 (119905)le 1119896 ormax 119878 (119905) 1198681 (119905) 1198682 (119905) ge 119896
(17)
where we set inf 0 = infin (as usual 0 denotes the empty set)Clearly 120591119896 is increasing as 119896 rarr infin Let 120591infin = lim119896rarrinfin120591119896 hence120591infin le 120591119896 as Next we only need to verify 120591infin = infin as If thisstatement is false then there exist two constants 119879 gt 0 and120598 isin (0 1) such that
P 120591infin le 119879 gt 120598 (18)
Thus there is an integer 1198961 ge 1198960 such that
P 120591119896 le 119879 ge 120598 119896 ge 1198961 (19)
Define a 1198622-function 119881 R3+ rarr R+ as follows
119881 (119878 1198681 1198682) = 119878 minus 1 minus ln 119878 + 1198681 minus 1 minus ln 1198681 + 1198682 minus 1minus ln 1198682 (20)
the nonnegativity of this function can be obtained from
119906 minus 1 minus ln 119906 ge 0 119906 gt 0 (21)
Applying Itorsquos formula yields
d119881 (119878 1198681 1198682) = 119871119881 (119878 1198681 1198682) d119905+ (21198781198681 minus 119878 minus 1198681) 1205901 (119905)1198861 (119905) + 1198681 1198891198611 (119905)+ (21198781198682 minus 119878 minus 1198682) 1205902 (119905)1198862 (119905) + 1198682 1198891198612 (119905)
(22)
where
119871119881 = (1 minus 1119878)(119860 (119905) minus 119889 (119905) 119878 minus 1205731 (119905) 11987811986811198861 (119905) + 1198681minus 1205732 (119905) 11987811986821198862 (119905) + 1198682 + 1199031 (119905) 1198681 + 1199032 (119905) 1198682) + 12
sdot 12059021 (119905) 11986821(1198861 (119905) + 1198681)2 +12
12059022 (119905) 11986822(1198862 (119905) + 1198682)2 + (1 minus11198681)
sdot ( 1205731 (119905) 11987811986811198861 (119905) + 1198681 minus (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681) + 12sdot 12059021 (119905) 1198782(1198861 (119905) + 1198681)2 + (1 minus
11198682)(1205732 (119905) 11987811986821198862 (119905) + 1198682
minus (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682) + 1212059022 (119905) 1198782(1198862 (119905) + 1198682)2
= 119860 (119905) minus 119889 (119905) 119878 minus 119860 (119905)119878 + 119889 (119905) + 1205731 (119905) 11986811198861 (119905) + 1198681+ 1205732 (119905) 11986821198862 (119905) + 1198682 minus
1199031 (119905) 1198681119878 minus 1199032 (119905) 1198682119878+ 12059021 (119905) 119868212 (1198861 (119905) + 1198681)2 +
12059022 (119905) 119868222 (1198862 (119905) + 1198682)2 minus (119889 (119905)+ 1205721 (119905)) 1198681 minus 1205731 (119905) 1198781198861 (119905) + 1198681 + 119889 (119905) + 1205721 (119905) + 1199031 (119905)+ 12059021 (119905) 11987822 (1198861 (119905) + 1198681)2 minus (119889 (119905) + 1205722 (119905)) 1198682 minus
1205732 (119905) 1198781198862 (119905) + 1198682+ 119889 (119905) + 1205722 (119905) + 1199032 (119905) + 12059022 (119905) 11987822 (1198862 (119905) + 1198682)2 le 119860119906
+ 3119889119906 + 1205721199061 + 1205721199062 + 1199031199061 + 1199031199062 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622+ 12059021199061 1198602119906211988621198971 1198892119897 +
12059021199062 1198602119906211988621198972 1198892119897 fl 119870(23)
where 119870 is a positive constantSo we have
d119881 (119878 1198681 1198682) le 119870d119905+ (21198781198681 minus 119878 minus 1198681) 1205901 (119905)1198861 (119905) + 1198681 1198891198611 (119905)+ (21198781198682 minus 119878 minus 1198682) 1205902 (119905)1198862 (119905) + 1198682 1198891198612 (119905)
(24)
Integrating (24) from 0 to 120591119896 and 119879 and taking expectations onboth sides yield
E119881 (119878 (120591119896 and 119879) 1198681 (120591119896 and 119879) 1198682 (120591119896 and 119879))le 119881 (119878 (0) 1198681 (0) 1198682 (0)) + 119870119879 (25)
Let Ω119896 = 120591119896 le 119879 from inequality (25) we can see that119875(Ω119896) ge 120598We have
119881 (119878 (120591119896 and 119879) 1198681 (120591119896 and 119879) 1198682 (120591119896 and 119879))ge (119896 minus 1 minus ln 119896) and (1119896 minus 1 minus ln 1119896)
(26)
Complexity 5
By (25) and (26) one has
119881 (119878 (0) 1198681 (0) 1198682 (0)) + 119870119879 ge E [1Ω119896 (120596)sdot 119881 (119878 (120591119896 and 119879) 1198681 (120591119896 and 119879) 1198682 (120591119896 and 119879))] ge 120598 (119896minus 1 minus ln 119896) and (1119896 minus 1 minus ln 1119896)
(27)
where 1Ω119896 is the indicator function ofΩ119896
Let 119896 rarr infin we have
infin gt 119881(119878 (0) 1198681 (0) 1198682 (0)) + 119870119879 = infin (28)
So we obtain 120591infin = infinThe proof is completed
4 Existence of Nontrivial 119879-Periodic SolutionIn this section we verify that system (3) admits at least onenontrivial positive 119879-periodic solution Define
R = 2sum119894=1
⟨119860120572119894120573119894⟩119879⟨119889 + 120572119894 + 119903119894 + 1205902119894 119860211990621198862119894 1198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩119879 ⟨119860 + 11988611205721 + 11988621205722⟩119879 (29)
Theorem 5 When (1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)))119897 gt 119860119906119889119897and (1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)))119897 gt 119860119906119889119897 hold if R gt 1then there exists a nontrivial positive T-periodic solution ofsystem (3)
Proof Define a 1198622-function 119881 [0infin) timesR3+ rarr R+
119881 (119905 119878 1198681 1198682) = 119872(minus (1198901 + 1198902) ln 119878 minus 1198881 ln 1198681 minus 1198882 ln 1198682+ 119878 + 1198681 + 1198682 + 120596 (119905)) minus ln 119878 minus 1198681 minus 1198682 + 1120579 + 1 (119878+ 1198681 + 1198682)120579+1 = 119872(1198811 + 120596 (119905)) + 1198812 + 1198813 + 1198814+ 1198815
(30)
where1198881= ⟨11986012057211205731⟩119879⟨119889 + 1205721 + 1199031 + 1205902111986021199062119886211198892119897⟩2119879 ⟨119889 + 120590212 + 120590222⟩119879 1198901= ⟨11986012057211205731⟩119879⟨119889 + 1205721 + 1199031 + 1205902111986021199062119886211198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩2119879 1198882= ⟨11986012057221205732⟩119879⟨119889 + 1205722 + 1199032 + 1205902211986021199062119886221198892119897⟩2119879 ⟨119889 + 120590212 + 120590222⟩119879 1198902= ⟨11986012057221205732⟩119879⟨119889 + 1205722 + 1199032 + 1205902211986021199062119886221198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩2119879
(31)
and 0 lt 120579 lt min1 119889119897(12059021199061 + 12059021199062 )119872 is a sufficiently largepositive constant and satisfies the following conditions
119889119897 minus 120579 (12059021199061 + 12059021199062 ) gt 0minus119872120582 +max 119863 119864 le minus2 (32)
where
120582 = ⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1) 119863 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
119864 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(33)
where
119874 = sup(1198781198681 1198682)isinR
3+
119860119906 (119878 + 1198681 + 1198682)120579
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 lt infin(34)
Next we prove that condition (1198601) in Lemma 2 holds It iseasy to check that 119881(119905 119878 1198681 1198682) is a 119879-periodic function in 119905and satisfies
lim inf119870rarrinfin(1198781198681 1198682)isinR
3+U119896
119881 (119905 119878 1198681 1198682) = infin (35)
where U119896 = (1119896 119896) times (1119896 119896) times (1119896 119896) and 119896 gt 1 is asufficiently large number
6 Complexity
By Itorsquos formula we obtain
1198711198811 = minus1198901 + 1198902119878 (119860 (119905) minus 119889 (119905) 119878 minus 1205731 (119905) 11987811986811198861 (119905) + 1198681minus 1205732 (119905) 11987811986821198862 (119905) + 1198682 + 1199031 (119905) 1198681 + 1199032 (119905) 1198682)minus 11988811198681 (
1205731 (119905) 11987811986811198861 (119905) + I1minus (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681)
+ 1198901 + 1198902212059021 (119905) 11986821(1198861 (119905) + 1198681)2 minus
11988821198682 (1205732 (119905) 11987811986821198862 (119905) + 1198682
minus (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682) + 1198901 + 11989022sdot 12059022 (119905) 11986822(1198862 (119905) + 1198682)2 +
1198881212059021 (119905) 1198782(1198861 (119905) + 1198681)2 +
11988822sdot 12059022 (119905) 1198782(1198862 (119905) + 1198682)2 + 119860 (119905) minus 119889 (119905) (119878 + 1198681 + 1198682) minus 1205721 (119905)sdot 1198681 minus 1205722 (119905) 1198682 le minus1198901119860 (119905)119878 minus 11988811205731 (119905) 1198781198861 (119905) + 1198681 minus 1205721 (119905)sdot (1198861 (119905) + 1198681) minus 1198902119860 (119905)119878 minus 11988821205732 (119905) 1198781198862 (119905) + 1198682 minus 1205722 (119905)sdot (1198862 (119905) + 1198682) + 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905)
+ 12059021 (119905) 1198602119906211988621 (119905) 1198892119897) + 1198901 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )
+ 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905) + 12059022 (119905) 1198602119906211988622 (119905) 1198892119897)
+ 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 ) minus (1198901 + 1198902) (1199031 (119905)119878+ 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878 + 119889 (119905)minus 1205732 (119905)1198862 (119905) ) 1198682 + 119860 (119905) + 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905)le minus3 (119860 (119905) 1205721 (119905) 1205731 (119905) 11988811198901)13minus 3 (119860 (119905) 1205722 (119905) 1205732 (119905) 11988821198902)13 + 1198901 (119889 (119905) + 12059021 (119905)2+ 12059022 (119905)2 ) + 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905)
+ 12059021 (119905) 1198602119906211988621 (119905) 1198892119897) + 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905)
+ 12059022 (119905) 1198602119906211988622 (119905) 1198892119897) + 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )
+ 119860 (119905) + 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905) minus (1198901 + 1198902)sdot (1199031 (119905)119878 + 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878+ 119889 (119905) minus 1205732 (119905)1198862 (119905) ) 1198682 fl R0 (119905) minus (1198901 + 1198902) (1199031 (119905)119878+ 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878 + 119889 (119905)minus 1205732 (119905)1198862 (119905) ) 1198682
(36)
where
R0 (119905) = 3 (119860 (119905) 1205721 (119905) 1205731 (119905) 11988811198901)13+ 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905) + 12059021 (119905) 1198602119906211988621 (119905) 1198892119897)
+ 1198901 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )minus 3 (119860 (119905) 1205722 (119905) 1205732 (119905) 11988821198902)13+ 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905) + 12059022 (119905) 1198602119906211988622 (119905) 1198892119897)
+ 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 ) + 119860 (119905)+ 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905)
(37)
Note that (1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)))119897 gt 119860119906119889119897 and(1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)))119897 gt 119860119906119889119897 hold then119871 (1198811) le R0 (119905) (38)
Define the 119879-periodic function 120596(119905) satisfying1205961015840 (119905) = ⟨R0⟩119879 minusR0 (119905) (39)
So
119871 (1198811 + 120596 (119905)) le ⟨R0⟩119879= minus ⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1)fl minus120582
(40)
Applying Itorsquos formula we can also have
Complexity 7
1198711198812 = minus119860 (119905)119878 + 119889 (119905) + 1205731 (119905) 11986811198861 (119905) + 1198681 +1205732 (119905) 11986821198862 (119905) + 1198682
minus 1199031 (119905) 1198681119878 minus 1199032 (119905) 1198682119878 + 12059021 (119905) 119868212 (1198861 (119905) + 1198681)2+ 12059022 (119905) 119868222 (1198862 (119905) + 1198682)2 le minus119860119897119878 + 119889119906 + 1205731199061 + 1205731199062 + 120590211990612+ 120590211990622
1198711198813 = minus 1205731 (119905) 11987811986811198861 (119905) + 1198681 + (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681 le (119889119906+ 1205721199061 + 1199031199061 ) 1198681
1198711198814 = minus 1205732 (119905) 11987811986821198862 (119905) + 1198682 + (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682 le (119889119906+ 1205721199062 + 1199031199062 ) 1198682
1198711198815 = (119878 + 1198681 + 1198682)120579 (119860 (119905) minus 119889 (119905) 119878 minus 119889 (119905) 1198681 minus 119889 (119905) 1198682minus 1205721 (119905) 1198681 minus 1205722 (119905) 1198682) + 120579 (119878 + 1198681 + 1198682)120579minus1sdot ( 12059021 (119905) 119878211986821(1198861 (119905) + 1198681)2 +
12059022 (119905) 119878211986822(1198862 (119905) + 1198682)2) le 119860119906 (119878 + 1198681+ 1198682)120579 minus 119889119897 (119878 + 1198681 + 1198682)120579+1 + 120579 (119878 + 1198681 + 1198682)120579minus1 (12059021199061+ 12059021199062 ) 1198782 le 119860119906 (119878 + 1198681 + 1198682)120579 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 le 119874 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12
(41)
where
119874 = sup(1198781198681 1198682)isinR
3+
119860119906 (119878 + 1198681 + 1198682)120579
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 lt infin(42)
Therefore119871119881 (119905 119878 1198681 1198682) le minus119872⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1)
+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682 minus 119860119897119878minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874+ 119889119906 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622
= minus119872120582 minus 119860119897119878 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874+ 119889119906 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622
(43)
Now we are in the position to construct a compact subset119880 such that 1198602 in Lemma 2 holds Define the followingbounded closed set
119880 = (119878 1198681 1198682) isin R3+ 120598 le 119878 le 1120598 120598 le 1198681 le 1120598 120598 le 1198682
le 1120598 (44)
where 120598 gt 0 is a sufficiently small number In the set R3+119880we can choose 120598 sufficiently small such that
minus119872120582 minus 119860119897120598 + 119862 le minus1 (45)
minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 120598 + 119863 le minus1 (46)
minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 120598 + 119864 le minus1 (47)
minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119862 le minus1 (48)
minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119866 le minus1 (49)
minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119867 le minus1 (50)
where 119862 119863 119864 119866 and119867 are positive constants which can befound from the following inequations (52) (54) (56) (59)and (61) respectively For the sake of convenience we divideinto six domains
8 Complexity
1198801 = (119878 1198681 1198682) isin R3+ 0 lt 119878 lt 120598
1198802 = (119878 1198681 1198682) isin R3+ 0 lt 1198681 lt 120598
1198803 = (119878 1198681 1198682) isin R3+ 0 lt 1198682 lt 120598
1198804 = (119878 1198681 1198682) isin R3+ 119878 gt 1120598
1198805 = (119878 1198681 1198682) isin R3+ 1198681 gt 1120598
1198806 = (119878 1198681 1198682) isin R3+ 1198682 gt 1120598
(51)
Nextwewill prove that119871119881(119878 1198681 1198682) le minus1 onR3+119880 whichis equivalent to proving it on the above six domains
Case 1 If (119878 1198681 1198682) isin 1198801 one can see that
119871119881 le minus119872120582 minus 119860119897119878 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622 le minus119872120582 minus 119860119897119878 + 119862
le minus119872120582 minus 119860119897120598 + 119862
(52)
where
119862 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(53)
According to (45) we have 119871119881 le minus1 for all (119878 1198681 1198682) isin 1198801Case 2 If (119878 1198681 1198682) isin 1198802 one can get that
119871119881 le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681 + 119863le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 120598 + 119863
(54)
where
119863 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(55)
In view of (46) we can obtain that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198802Case 3 If (119878 1198681 1198682) isin 1198803 we have
119871119881 le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682 + 119864le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 120598 + 119864
(56)
where
119864 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(57)
By (47) we can conclude that 119871119881 le minus1 for all (119878 1198681 1198682) isin1198803Case 4 If (119878 1198681 1198682) isin 1198804 one can derive that
119871119881 le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1 + 119862le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119862
(58)
Complexity 9
Together with (48) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198804Case 5 If (119878 1198681 1198682) isin 1198805 it follows that
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119866le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119866
(59)
where
119866 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(60)
By virtue of (49) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198805Case 6 If (119878 1198681 1198682) isin 1198806 we obtain
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119867le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119867
(61)
where
119867 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(62)
It follows from (50) that 119871119881 le minus1 for all (119878 1198681 1198682) isin 1198806Clearly one can see from (52) (54) (56) (58) (59) and
(61) that for a sufficiently small 120598119871119881 (119878 1198681 1198682) le minus1
(119878 1198681 1198682) isin R3+119880 (63)
Hence 1198602 in Lemma 2 is satisfied This completes the proofof Theorem 5
5 Extinction
In this section we investigate the conditions for the extinc-tion of the two infectious diseases of system (3)
Let
R1 = 119860119906 ⟨1205731⟩119879119889119897 ⟨1198861⟩119879 ⟨119889 + 1205721 + 1199031⟩119879minus 1198602119906 ⟨12059021⟩11987921198892119897 ⟨11988621⟩119879 ⟨119889 + 1205721 + 1199031⟩119879
R2 = 119860119906 ⟨1205732⟩119879119889119897 ⟨1198862⟩119879 ⟨119889 + 1205722 + 1199032⟩119879minus 1198602119906 ⟨12059022⟩11987921198892119897 ⟨11988622⟩119879 ⟨119889 + 1205722 + 1199032⟩119879
(64)
Theorem 6 Let (119878(119905) 1198681(119905) 1198682(119905)) be a solution of system (3)with initial value (119878(0) 1198681(0) 1198682(0)) isin 1198773+
Then if
(i) ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792 ⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 119894 = 1 2 (65)
or(ii) R119894 lt 1⟨1205902119894 ⟩119879 le 119889119897 ⟨1198862119894 ⟩119879 ⟨120573119894⟩119879119860119906 ⟨119886119894⟩119879
119894 = 1 2(66)
hold the two infectious diseases of system (3) go to extinctionas that is
lim119905rarr+infin
119868119894 (119905) = 0 119894 = 1 2 (67)
10 Complexity
Proof Applying Itorsquos formula to system (3) we have
d ln 119868119894 (119905) = ( 120573119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905) minus (119889 (119905) + 120572119894 (119905) + 119903119894 (119905))minus 1205902119894 (119905) 1198782 (119905)2 (119886119894 (119905) + 119868119894 (119905))2) d119905 + 120590119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905)d119861119894 (119905)
119894 = 1 2
(68)
Case (i) Integrating (68) from 0 to 119905 and dividing 119905 on bothsides we obtain
ln 119868119894 (119905)119905 = minus1205902119894 (119905)2119905 int1199050( 119878 (120591)119886119894 (120591) + 119868119894 (120591) minus
120573119894 (120591)1205902119894 (120591))2
d120591minus 1119905 int
119905
0(119889 (120591) + 120572119894 (120591) + 119903119894 (120591)) d120591
+ 1119905 int119905
0
1205732119894 (120591)21205902119894 (120591)d120591 +119872119894 (119905)119905 + ln 119868119894 (0)119905
le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905
(69)
Case (ii) Integrating (68) from 0 to 119905 first and then dividingby 119905 on both sides yield
ln 119868119894 (119905)119905 = 1119905 int119905
0( 120573119894 (120591) 119878 (120591)119886119894 (120591) + 119868119894 (120591)
minus (119889 (120591) + 120572119894 (120591) + 119903119894 (120591))minus 1205902119894 (120591) 1198782 (120591)2 (119886119894 (120591) + 119868119894 (120591))2) d120591 + 119872119894 (119905)119905 + ln 119868119894 (0)119905le ⟨120573119894119860119906119886119894119889119897 minus (119889 + 120572119894 + 119903119894) minus
1205902119894 119860211990621198862119894 1198892119897⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) + 119872119894 (119905)119905+ ln 119868119894 (0)119905
(70)
where 119872119894(119905) = int1199050120590119894(120591)119878(120591)(119886119894(120591) + 119868119894(120591))d119861119894(120591) 119894 = 1 2
which is a local continuous martingale with 119872119894(0) = 0 ByLemma 3 we have
lim119905rarr+infin
119872119894 (119905)119905 = 0 119894 = 1 2 (71)
Taking the limit superior of both sides of (69) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119879 lt 0 (72)
which implies lim119905rarr+infin119868119894(119905) = 0
Taking the superior limit of both sides of (70) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) lt 0 (73)
which implies lim119905rarr+infin119868119894(119905) = 0 119894 = 1 2This completes theproof
Remark 7 Theorem 6 shows that the two diseases will dieout if the white noise disturbance is large or the white noisedisturbance is not large and R119894 lt 1 When ⟨1205902119894 ⟩119879 gt⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectious diseases of system(3) die out almost surely that is to say large white noisestochastic disturbance can lead to the two epidemics beingextinct
6 Numerical Simulations
Now we introduce some numerical simulations exampleswhich illustrate our theoretical results
Example 8 In model (3) let
119860 (119905) = 05 + 01 sin1205871199051198861 (119905) = 03 + 01 sin1205871199051198862 (119905) = 031 + 01 sin1205871199051205731 (119905) = 062 + 01 sin1205871199051205732 (119905) = 062 + 01 sin1205871199051199031 (119905) = 02 + 01 sin1205871199051199032 (119905) = 035 + 01 sin120587119905119889 (119905) = 02 + 01 sin1205871199051205721 (119905) = 02 + 01 sin1205871199051205722 (119905) = 025 + 01 sin1205871199051205901 (119905) = 01 + 005 sin1205871199051205902 (119905) = 01 + 005 sin120587119905
(74)
Note that R gt 1 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897and 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 hold that isthe conditions of Theorem 5 hold Hence system (3) has apositive periodic solution with 119879 = 1 Figure 1(a) shows theperiodicity of the nonautonomous stochastic model (3) with1205901 = 0 and 1205902 = 0 Figure 1(b) shows that solution of thenonautonomous stochastic model (3) with the initial value(119878(119905) 1198681(119905) 1198682(119905)) = (03 015 015) tends to a periodic orbitin the sense of joint distribution
Example 9 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin1205871199051198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin120587119905
Complexity 11
S(t)
I1(t)
I2(t)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
(a)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 1 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (1 02 02) to the nonautonomous stochastic model (3)
0 25 50 75 100t
59 60 61
078
08
082
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
59 60 61
078
08
082
0
1
2
14
14
S(t)I 1(t)I 2(t)
25 50 75 1000t
S(t)
I1(t)
I2(t)
(b)
Figure 2 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 04 + 02 sin1205871199051205902 (119905) = 05 + 02 sin120587119905
(75)
Note that ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897+120572119897119894+119903119897119894⟩119879Therefore conditions(i) of Theorem 6 hold Then the two infectious diseases willgo to extinction Figure 2(a) shows that one of two diseases inthe deterministic SIS epidemic model is extinct and the otheris persistent Figure 2(b) shows that the two diseases will dieout under the large white noise disturbance of model (3)
Example 10 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin120587119905
12 Complexity
25 50 75 1000t
0
05
1
15
2S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
0 25 50 75 100t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 3 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin1205871199051205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 02 + 02 sin1205871199051205902 (119905) = 02 + 02 sin120587119905
(76)
Note that R1 lt 1 R2 lt 1 and ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 That is conditions (ii) of Theorem 6 hold Thenthe two infectious diseases will go to extinction Figure 3(a)
shows that one of two diseases in the deterministic SISepidemic model is extinct and the other is persistent withoutthe white noises Figure 3(b) shows that the two diseases willdie out under a small white noise disturbance of model (3)
7 Discussion and Conclusions
This paper explores the existence of nontrivial positiveT-periodic solution of a nonautonomous stochastic SISepidemic model with nonlinear growth rate and doubleepidemic hypothesis By constructing a suitable stochasticLyapunov function we establish sufficient conditions for theexistence of nontrivial positive 119879-periodic solution of system(3) Furthermore the sufficient conditions for the extinctionof the two diseases are obtained Our results are given asfollows
(1) If 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 and R gt 1 hold the SISmodel has at least one nontrivial positive 119879-periodicsolution where
R = 2sum119894=1
⟨119860120572119894120573119894⟩119879⟨119889 + 120572119894 + 119903119894 + 1205902119894 119860211990621198862119894 1198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩119879 ⟨119860 + 11988611205721 + 11988621205722⟩119879 (77)
(2) If ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectiousdiseases go extinct
(3) If ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 and R119894 lt 1 the two
infectious diseases also go extinct where
R119894 = 119860119906 ⟨120573119894⟩119879119889119897 ⟨119886119894⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879minus 1198602119906 ⟨1205902119894 ⟩11987921198892119897 ⟨1198862119894 ⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879
(78)
Complexity 13
Some interesting questions deserve further investigationOn the one hand we may explore some realistic but complexmodels such as considering the effects of impulsive ordelay perturbations on system (3) On the other hand wecan concern the dynamics of a nonautonomous stochasticSIS epidemic model with two infectious diseases driven byLevy jumps What is more we can also investigate thenonautonomous stochastic SIS epidemic model with twoinfectious diseases by a continuous time Markov chain Wewill investigate these cases in our future work
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (11371230) Joint Innovative Center forSafe andEffectiveMiningTechnology andEquipment of CoalResources Shandong Province the SDUST Research Fund(2014TDJH102) and Shandong Provincial Natural ScienceFoundation China (ZR2015AQ001)
References
[1] W O Kermack and A G McKendrick ldquoA contributions to themathematical theory of epidemics (Part I)rdquo Proceedings of theRoyal Society of London A vol 115 pp 700ndash721 1927
[2] A Gray D Greenhalgh L Hu X Mao and J Pan ldquoA stochasticdifferential equation SIS epidemic modelrdquo SIAM Journal onApplied Mathematics vol 71 no 3 pp 876ndash902 2011
[3] A Korobeinikov and G C Wake ldquoLyapunov functions andglobal stability for SIR SIRS and SIS epidemiological modelsrdquoApplied Mathematics Letters vol 15 no 8 pp 955ndash960 2002
[4] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995
[5] T Zhang X Meng Y Song and T Zhang ldquoA stage-structuredpredator-prey SI model with disease in the prey and impulsiveeffectsrdquoMathematical Modelling and Analysis vol 18 no 4 pp505ndash528 2013
[6] J Li andZMa ldquoQualitative analyses of SIS epidemicmodelwithvaccination and varying total population sizerdquo Mathematicaland Computer Modelling vol 35 no 11-12 pp 1235ndash1243 2002
[7] A drsquoOnofrio ldquoA note on the global behaviour of the network-based SIS epidemic modelrdquo Nonlinear Analysis Real WorldApplications vol 9 no 4 pp 1567ndash1572 2008
[8] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[9] T Zhang X Meng T Zhang and Y Song ldquoGlobal dynamicsfor a new high-dimensional SIR model with distributed delayrdquoApplied Mathematics and Computation vol 218 no 24 pp11806ndash11819 2012
[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006
[11] X Meng S Zhao T Feng and T Zhang ldquoDynamics of a novelnonlinear stochastic SIS epidemic model with double epidemichypothesisrdquo Journal of Mathematical Analysis and Applicationsvol 433 no 1 pp 227ndash242 2016
[12] A Miao X Wang T Zhang W Wang and B Sampath ArunaPradeep ldquoDynamical analysis of a stochastic SIS epidemicmodel with nonlinear incidence rate and double epidemichypothesisrdquo Advances in Difference Equations 2017226 pages2017
[13] X-Z Meng ldquoStability of a novel stochastic epidemic modelwith double epidemic hypothesisrdquo Applied Mathematics andComputation vol 217 no 2 pp 506ndash515 2010
[14] X P Li X Y Lin and Y Q Lin ldquoLyapunov-type conditions andstochastic differential equations driven byG-BrownianmotionrdquoJournal of Mathematical Analysis and Applications vol 439 no1 pp 235ndash255 2016
[15] M Liu and M Fan ldquoPermanence of stochastic Lotka-Volterrasystemsrdquo Journal of Nonlinear Science vol 27 no 2 pp 425ndash452 2017
[16] L D Liu and X Z Meng ldquoOptimal harvesting controland dynamics of two-species stochastic model with delaysrdquoAdvances in Difference Equations vol 2017 18 pages 2017
[17] H Ma and Y Jia ldquoStability analysis for stochastic differentialequations with infiniteMarkovian switchingsrdquo Journal of Math-ematical Analysis and Applications vol 435 no 1 pp 593ndash6052016
[18] G D Liu X H Wang X Z Meng and S J Gao ldquoExtinctionand persistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexity vol 3pp 1ndash15 2017
[19] C Tan and W H Zhang ldquoOn observability and detectabilityof continuous-time stochastic Markov jump systemsrdquo Journalof Systems Science and Complexity vol 28 no 4 pp 830ndash8472015
[20] X Leng T Feng and X Meng ldquoStochastic inequalities andapplications to dynamics analysis of a novel SIVS epidemicmodel with jumpsrdquo Journal of Inequalities and ApplicationsPaper No 138 25 pages 2017
[21] G Li and M Chen ldquoInfinite horizon linear quadratic optimalcontrol for stochastic difference time-delay systemsrdquo Advancesin Difference Equations vol 2015 14 pages 2015
[22] Y Zhao and W Zhang ldquoObserver-based controller design forsingular stochastic Markov jump systems with state dependentnoiserdquo Journal of Systems Science and Complexity vol 29 no 4pp 946ndash958 2016
[23] S Zhang X Meng T Feng and T Zhang ldquoDynamics analysisand numerical simulations of a stochastic non-autonomouspredatorndashprey system with impulsive effectsrdquo Nonlinear Anal-ysis Hybrid Systems vol 26 pp 19ndash37 2017
[24] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013
[25] X Meng L Wang and T Zhang ldquoGlobal dynamics analysis ofa nonlinear impulsive stochastic chemostat system in a pollutedenvironmentrdquo Journal of AppliedAnalysis andComputation vol6 no 3 pp 865ndash875 2016
[26] X Liu Y Li andW Zhang ldquoStochastic linear quadratic optimalcontrol with constraint for discrete-time systemsrdquo AppliedMathematics and Computation vol 228 pp 264ndash270 2014
14 Complexity
[27] X Z Meng and X H Wang ldquoStochastic predator-prey systemsubject to levy jumprdquo Discrete Dynamics in Nature and Societyvol 2016 Article ID 5749892 13 pages 2016
[28] X Lv L Wang and X Meng ldquoGlobal analysis of a new non-linear stochastic differential competition systemwith impulsiveeffectrdquo Advances in Difference Equations vol 2017 296 pages2017
[29] X Zhang D Jiang A Alsaedi and T Hayat ldquoStationarydistribution of stochastic SIS epidemic model with vaccinationunder regime switchingrdquo Applied Mathematics Letters vol 59pp 87ndash93 2016
[30] F Li X Meng and Y Cui ldquoNonlinear stochastic analysis fora stochastic SIS epidemic modelrdquo Journal of Nonlinear Sciencesand Applications vol 10 no 09 pp 5116ndash5124 2017
[31] A Q Miao J Zhang T Q Zhang and B G Pradeep ldquoSampathAruna Pradeep Threshold dynamics of a stochastic SIR modelwith vertical transmission and vaccinationrdquo Computationaland Mathematical Methods in Medicine vol 2017 Article ID4820183 10 pages 2017
[32] Q Liu D Jiang N Shi T Hayat and A Alsaedi ldquoNontrivialperiodic solution of a stochastic non-autonomous SISV epi-demic modelrdquo Physica A Statistical Mechanics and its Applica-tions vol 462 pp 837ndash845 2016
[33] Y L Zhou S L Yuan and D L Zhao ldquoThreshold behavior ofa stochastic SIS model with levy jumpsrdquo Applied Mathematicsand Computation vol 275 pp 255ndash267 2016
[34] T Feng X Meng L Liu and S Gao ldquoApplication of inequal-ities technique to dynamics analysis of a stochastic eco-epidemiology modelrdquo Journal of Inequalities and ApplicationsPaper No 327 29 pages 2016
[35] M Liu and K Wang ldquoPersistence and extinction in stochasticnon-autonomous logistic systemsrdquo Journal of MathematicalAnalysis and Applications vol 375 no 2 pp 443ndash457 2011
[36] Z Bai and Y Zhou ldquoExistence of two periodic solutions for anon-autonomous SIR epidemic modelrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 1 pp 382ndash391 2011
[37] S Gao F Zhang and Y He ldquoThe effects of migratory birdpopulation in a nonautonomous eco-epidemiological modelrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 37 no 6 pp3903ndash3916 2013
[38] T Kuniya ldquoExistence of a nontrivial periodic solution in an age-structured SIR epidemicmodel with time periodic coefficientsrdquoApplied Mathematics Letters vol 27 pp 15ndash20 2014
[39] L Zu D Jiang D OrsquoRegan and B Ge ldquoPeriodic solution fora non-autonomous Lotka-Volterra predator-prey model withrandom perturbationrdquo Journal of Mathematical Analysis andApplications vol 430 no 1 pp 428ndash437 2015
[40] Y Lin D Jiang and T Liu ldquoNontrivial periodic solution ofa stochastic epidemic model with seasonal variationrdquo AppliedMathematics Letters vol 45 pp 103ndash107 2015
[41] X R Mao Stochastic Differential Equations and Their Applica-tions Horwood Chichester UK 1997
[42] R Khasminskii Stochastic Stability of Differential EquationsSpringer Berlin Germany 2011
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Complexity
obviously we have
lim sup119905rarr+infin
119878 (119905) le 119860119906119889119897 lim sup119905rarr+infin
1198681 (119905) le 119860119906119889119897 lim sup119905rarr+infin
1198682 (119905) le 119860119906119889119897 (16)
Since the coefficients of system (3) satisfy the localLipschitz conditions then for any given initial value(119878(0) 1198681(0) 1198682(0)) isin 1198773+ there is a unique local solution(119878(119905) 1198681(119905) 1198682(119905)) on 119905 isin [0 120591119890) where 120591119890 is the explosion timeTo demonstrate that this solution is global we only need toprove that 120591119890 = infin as
Let 1198960 gt 0 be sufficiently large for any initial value119878(0) 1198681(0) and 1198682(0) lying within the interval [11198960 119896] Foreach integer 119896 ge 1198960 define the following stopping time
120591119896 = inf 119905 isin [0 120591119890) min 119878 (119905) 1198681 (119905) 1198682 (119905)le 1119896 ormax 119878 (119905) 1198681 (119905) 1198682 (119905) ge 119896
(17)
where we set inf 0 = infin (as usual 0 denotes the empty set)Clearly 120591119896 is increasing as 119896 rarr infin Let 120591infin = lim119896rarrinfin120591119896 hence120591infin le 120591119896 as Next we only need to verify 120591infin = infin as If thisstatement is false then there exist two constants 119879 gt 0 and120598 isin (0 1) such that
P 120591infin le 119879 gt 120598 (18)
Thus there is an integer 1198961 ge 1198960 such that
P 120591119896 le 119879 ge 120598 119896 ge 1198961 (19)
Define a 1198622-function 119881 R3+ rarr R+ as follows
119881 (119878 1198681 1198682) = 119878 minus 1 minus ln 119878 + 1198681 minus 1 minus ln 1198681 + 1198682 minus 1minus ln 1198682 (20)
the nonnegativity of this function can be obtained from
119906 minus 1 minus ln 119906 ge 0 119906 gt 0 (21)
Applying Itorsquos formula yields
d119881 (119878 1198681 1198682) = 119871119881 (119878 1198681 1198682) d119905+ (21198781198681 minus 119878 minus 1198681) 1205901 (119905)1198861 (119905) + 1198681 1198891198611 (119905)+ (21198781198682 minus 119878 minus 1198682) 1205902 (119905)1198862 (119905) + 1198682 1198891198612 (119905)
(22)
where
119871119881 = (1 minus 1119878)(119860 (119905) minus 119889 (119905) 119878 minus 1205731 (119905) 11987811986811198861 (119905) + 1198681minus 1205732 (119905) 11987811986821198862 (119905) + 1198682 + 1199031 (119905) 1198681 + 1199032 (119905) 1198682) + 12
sdot 12059021 (119905) 11986821(1198861 (119905) + 1198681)2 +12
12059022 (119905) 11986822(1198862 (119905) + 1198682)2 + (1 minus11198681)
sdot ( 1205731 (119905) 11987811986811198861 (119905) + 1198681 minus (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681) + 12sdot 12059021 (119905) 1198782(1198861 (119905) + 1198681)2 + (1 minus
11198682)(1205732 (119905) 11987811986821198862 (119905) + 1198682
minus (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682) + 1212059022 (119905) 1198782(1198862 (119905) + 1198682)2
= 119860 (119905) minus 119889 (119905) 119878 minus 119860 (119905)119878 + 119889 (119905) + 1205731 (119905) 11986811198861 (119905) + 1198681+ 1205732 (119905) 11986821198862 (119905) + 1198682 minus
1199031 (119905) 1198681119878 minus 1199032 (119905) 1198682119878+ 12059021 (119905) 119868212 (1198861 (119905) + 1198681)2 +
12059022 (119905) 119868222 (1198862 (119905) + 1198682)2 minus (119889 (119905)+ 1205721 (119905)) 1198681 minus 1205731 (119905) 1198781198861 (119905) + 1198681 + 119889 (119905) + 1205721 (119905) + 1199031 (119905)+ 12059021 (119905) 11987822 (1198861 (119905) + 1198681)2 minus (119889 (119905) + 1205722 (119905)) 1198682 minus
1205732 (119905) 1198781198862 (119905) + 1198682+ 119889 (119905) + 1205722 (119905) + 1199032 (119905) + 12059022 (119905) 11987822 (1198862 (119905) + 1198682)2 le 119860119906
+ 3119889119906 + 1205721199061 + 1205721199062 + 1199031199061 + 1199031199062 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622+ 12059021199061 1198602119906211988621198971 1198892119897 +
12059021199062 1198602119906211988621198972 1198892119897 fl 119870(23)
where 119870 is a positive constantSo we have
d119881 (119878 1198681 1198682) le 119870d119905+ (21198781198681 minus 119878 minus 1198681) 1205901 (119905)1198861 (119905) + 1198681 1198891198611 (119905)+ (21198781198682 minus 119878 minus 1198682) 1205902 (119905)1198862 (119905) + 1198682 1198891198612 (119905)
(24)
Integrating (24) from 0 to 120591119896 and 119879 and taking expectations onboth sides yield
E119881 (119878 (120591119896 and 119879) 1198681 (120591119896 and 119879) 1198682 (120591119896 and 119879))le 119881 (119878 (0) 1198681 (0) 1198682 (0)) + 119870119879 (25)
Let Ω119896 = 120591119896 le 119879 from inequality (25) we can see that119875(Ω119896) ge 120598We have
119881 (119878 (120591119896 and 119879) 1198681 (120591119896 and 119879) 1198682 (120591119896 and 119879))ge (119896 minus 1 minus ln 119896) and (1119896 minus 1 minus ln 1119896)
(26)
Complexity 5
By (25) and (26) one has
119881 (119878 (0) 1198681 (0) 1198682 (0)) + 119870119879 ge E [1Ω119896 (120596)sdot 119881 (119878 (120591119896 and 119879) 1198681 (120591119896 and 119879) 1198682 (120591119896 and 119879))] ge 120598 (119896minus 1 minus ln 119896) and (1119896 minus 1 minus ln 1119896)
(27)
where 1Ω119896 is the indicator function ofΩ119896
Let 119896 rarr infin we have
infin gt 119881(119878 (0) 1198681 (0) 1198682 (0)) + 119870119879 = infin (28)
So we obtain 120591infin = infinThe proof is completed
4 Existence of Nontrivial 119879-Periodic SolutionIn this section we verify that system (3) admits at least onenontrivial positive 119879-periodic solution Define
R = 2sum119894=1
⟨119860120572119894120573119894⟩119879⟨119889 + 120572119894 + 119903119894 + 1205902119894 119860211990621198862119894 1198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩119879 ⟨119860 + 11988611205721 + 11988621205722⟩119879 (29)
Theorem 5 When (1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)))119897 gt 119860119906119889119897and (1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)))119897 gt 119860119906119889119897 hold if R gt 1then there exists a nontrivial positive T-periodic solution ofsystem (3)
Proof Define a 1198622-function 119881 [0infin) timesR3+ rarr R+
119881 (119905 119878 1198681 1198682) = 119872(minus (1198901 + 1198902) ln 119878 minus 1198881 ln 1198681 minus 1198882 ln 1198682+ 119878 + 1198681 + 1198682 + 120596 (119905)) minus ln 119878 minus 1198681 minus 1198682 + 1120579 + 1 (119878+ 1198681 + 1198682)120579+1 = 119872(1198811 + 120596 (119905)) + 1198812 + 1198813 + 1198814+ 1198815
(30)
where1198881= ⟨11986012057211205731⟩119879⟨119889 + 1205721 + 1199031 + 1205902111986021199062119886211198892119897⟩2119879 ⟨119889 + 120590212 + 120590222⟩119879 1198901= ⟨11986012057211205731⟩119879⟨119889 + 1205721 + 1199031 + 1205902111986021199062119886211198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩2119879 1198882= ⟨11986012057221205732⟩119879⟨119889 + 1205722 + 1199032 + 1205902211986021199062119886221198892119897⟩2119879 ⟨119889 + 120590212 + 120590222⟩119879 1198902= ⟨11986012057221205732⟩119879⟨119889 + 1205722 + 1199032 + 1205902211986021199062119886221198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩2119879
(31)
and 0 lt 120579 lt min1 119889119897(12059021199061 + 12059021199062 )119872 is a sufficiently largepositive constant and satisfies the following conditions
119889119897 minus 120579 (12059021199061 + 12059021199062 ) gt 0minus119872120582 +max 119863 119864 le minus2 (32)
where
120582 = ⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1) 119863 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
119864 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(33)
where
119874 = sup(1198781198681 1198682)isinR
3+
119860119906 (119878 + 1198681 + 1198682)120579
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 lt infin(34)
Next we prove that condition (1198601) in Lemma 2 holds It iseasy to check that 119881(119905 119878 1198681 1198682) is a 119879-periodic function in 119905and satisfies
lim inf119870rarrinfin(1198781198681 1198682)isinR
3+U119896
119881 (119905 119878 1198681 1198682) = infin (35)
where U119896 = (1119896 119896) times (1119896 119896) times (1119896 119896) and 119896 gt 1 is asufficiently large number
6 Complexity
By Itorsquos formula we obtain
1198711198811 = minus1198901 + 1198902119878 (119860 (119905) minus 119889 (119905) 119878 minus 1205731 (119905) 11987811986811198861 (119905) + 1198681minus 1205732 (119905) 11987811986821198862 (119905) + 1198682 + 1199031 (119905) 1198681 + 1199032 (119905) 1198682)minus 11988811198681 (
1205731 (119905) 11987811986811198861 (119905) + I1minus (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681)
+ 1198901 + 1198902212059021 (119905) 11986821(1198861 (119905) + 1198681)2 minus
11988821198682 (1205732 (119905) 11987811986821198862 (119905) + 1198682
minus (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682) + 1198901 + 11989022sdot 12059022 (119905) 11986822(1198862 (119905) + 1198682)2 +
1198881212059021 (119905) 1198782(1198861 (119905) + 1198681)2 +
11988822sdot 12059022 (119905) 1198782(1198862 (119905) + 1198682)2 + 119860 (119905) minus 119889 (119905) (119878 + 1198681 + 1198682) minus 1205721 (119905)sdot 1198681 minus 1205722 (119905) 1198682 le minus1198901119860 (119905)119878 minus 11988811205731 (119905) 1198781198861 (119905) + 1198681 minus 1205721 (119905)sdot (1198861 (119905) + 1198681) minus 1198902119860 (119905)119878 minus 11988821205732 (119905) 1198781198862 (119905) + 1198682 minus 1205722 (119905)sdot (1198862 (119905) + 1198682) + 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905)
+ 12059021 (119905) 1198602119906211988621 (119905) 1198892119897) + 1198901 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )
+ 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905) + 12059022 (119905) 1198602119906211988622 (119905) 1198892119897)
+ 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 ) minus (1198901 + 1198902) (1199031 (119905)119878+ 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878 + 119889 (119905)minus 1205732 (119905)1198862 (119905) ) 1198682 + 119860 (119905) + 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905)le minus3 (119860 (119905) 1205721 (119905) 1205731 (119905) 11988811198901)13minus 3 (119860 (119905) 1205722 (119905) 1205732 (119905) 11988821198902)13 + 1198901 (119889 (119905) + 12059021 (119905)2+ 12059022 (119905)2 ) + 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905)
+ 12059021 (119905) 1198602119906211988621 (119905) 1198892119897) + 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905)
+ 12059022 (119905) 1198602119906211988622 (119905) 1198892119897) + 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )
+ 119860 (119905) + 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905) minus (1198901 + 1198902)sdot (1199031 (119905)119878 + 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878+ 119889 (119905) minus 1205732 (119905)1198862 (119905) ) 1198682 fl R0 (119905) minus (1198901 + 1198902) (1199031 (119905)119878+ 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878 + 119889 (119905)minus 1205732 (119905)1198862 (119905) ) 1198682
(36)
where
R0 (119905) = 3 (119860 (119905) 1205721 (119905) 1205731 (119905) 11988811198901)13+ 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905) + 12059021 (119905) 1198602119906211988621 (119905) 1198892119897)
+ 1198901 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )minus 3 (119860 (119905) 1205722 (119905) 1205732 (119905) 11988821198902)13+ 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905) + 12059022 (119905) 1198602119906211988622 (119905) 1198892119897)
+ 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 ) + 119860 (119905)+ 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905)
(37)
Note that (1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)))119897 gt 119860119906119889119897 and(1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)))119897 gt 119860119906119889119897 hold then119871 (1198811) le R0 (119905) (38)
Define the 119879-periodic function 120596(119905) satisfying1205961015840 (119905) = ⟨R0⟩119879 minusR0 (119905) (39)
So
119871 (1198811 + 120596 (119905)) le ⟨R0⟩119879= minus ⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1)fl minus120582
(40)
Applying Itorsquos formula we can also have
Complexity 7
1198711198812 = minus119860 (119905)119878 + 119889 (119905) + 1205731 (119905) 11986811198861 (119905) + 1198681 +1205732 (119905) 11986821198862 (119905) + 1198682
minus 1199031 (119905) 1198681119878 minus 1199032 (119905) 1198682119878 + 12059021 (119905) 119868212 (1198861 (119905) + 1198681)2+ 12059022 (119905) 119868222 (1198862 (119905) + 1198682)2 le minus119860119897119878 + 119889119906 + 1205731199061 + 1205731199062 + 120590211990612+ 120590211990622
1198711198813 = minus 1205731 (119905) 11987811986811198861 (119905) + 1198681 + (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681 le (119889119906+ 1205721199061 + 1199031199061 ) 1198681
1198711198814 = minus 1205732 (119905) 11987811986821198862 (119905) + 1198682 + (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682 le (119889119906+ 1205721199062 + 1199031199062 ) 1198682
1198711198815 = (119878 + 1198681 + 1198682)120579 (119860 (119905) minus 119889 (119905) 119878 minus 119889 (119905) 1198681 minus 119889 (119905) 1198682minus 1205721 (119905) 1198681 minus 1205722 (119905) 1198682) + 120579 (119878 + 1198681 + 1198682)120579minus1sdot ( 12059021 (119905) 119878211986821(1198861 (119905) + 1198681)2 +
12059022 (119905) 119878211986822(1198862 (119905) + 1198682)2) le 119860119906 (119878 + 1198681+ 1198682)120579 minus 119889119897 (119878 + 1198681 + 1198682)120579+1 + 120579 (119878 + 1198681 + 1198682)120579minus1 (12059021199061+ 12059021199062 ) 1198782 le 119860119906 (119878 + 1198681 + 1198682)120579 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 le 119874 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12
(41)
where
119874 = sup(1198781198681 1198682)isinR
3+
119860119906 (119878 + 1198681 + 1198682)120579
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 lt infin(42)
Therefore119871119881 (119905 119878 1198681 1198682) le minus119872⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1)
+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682 minus 119860119897119878minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874+ 119889119906 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622
= minus119872120582 minus 119860119897119878 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874+ 119889119906 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622
(43)
Now we are in the position to construct a compact subset119880 such that 1198602 in Lemma 2 holds Define the followingbounded closed set
119880 = (119878 1198681 1198682) isin R3+ 120598 le 119878 le 1120598 120598 le 1198681 le 1120598 120598 le 1198682
le 1120598 (44)
where 120598 gt 0 is a sufficiently small number In the set R3+119880we can choose 120598 sufficiently small such that
minus119872120582 minus 119860119897120598 + 119862 le minus1 (45)
minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 120598 + 119863 le minus1 (46)
minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 120598 + 119864 le minus1 (47)
minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119862 le minus1 (48)
minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119866 le minus1 (49)
minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119867 le minus1 (50)
where 119862 119863 119864 119866 and119867 are positive constants which can befound from the following inequations (52) (54) (56) (59)and (61) respectively For the sake of convenience we divideinto six domains
8 Complexity
1198801 = (119878 1198681 1198682) isin R3+ 0 lt 119878 lt 120598
1198802 = (119878 1198681 1198682) isin R3+ 0 lt 1198681 lt 120598
1198803 = (119878 1198681 1198682) isin R3+ 0 lt 1198682 lt 120598
1198804 = (119878 1198681 1198682) isin R3+ 119878 gt 1120598
1198805 = (119878 1198681 1198682) isin R3+ 1198681 gt 1120598
1198806 = (119878 1198681 1198682) isin R3+ 1198682 gt 1120598
(51)
Nextwewill prove that119871119881(119878 1198681 1198682) le minus1 onR3+119880 whichis equivalent to proving it on the above six domains
Case 1 If (119878 1198681 1198682) isin 1198801 one can see that
119871119881 le minus119872120582 minus 119860119897119878 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622 le minus119872120582 minus 119860119897119878 + 119862
le minus119872120582 minus 119860119897120598 + 119862
(52)
where
119862 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(53)
According to (45) we have 119871119881 le minus1 for all (119878 1198681 1198682) isin 1198801Case 2 If (119878 1198681 1198682) isin 1198802 one can get that
119871119881 le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681 + 119863le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 120598 + 119863
(54)
where
119863 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(55)
In view of (46) we can obtain that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198802Case 3 If (119878 1198681 1198682) isin 1198803 we have
119871119881 le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682 + 119864le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 120598 + 119864
(56)
where
119864 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(57)
By (47) we can conclude that 119871119881 le minus1 for all (119878 1198681 1198682) isin1198803Case 4 If (119878 1198681 1198682) isin 1198804 one can derive that
119871119881 le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1 + 119862le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119862
(58)
Complexity 9
Together with (48) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198804Case 5 If (119878 1198681 1198682) isin 1198805 it follows that
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119866le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119866
(59)
where
119866 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(60)
By virtue of (49) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198805Case 6 If (119878 1198681 1198682) isin 1198806 we obtain
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119867le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119867
(61)
where
119867 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(62)
It follows from (50) that 119871119881 le minus1 for all (119878 1198681 1198682) isin 1198806Clearly one can see from (52) (54) (56) (58) (59) and
(61) that for a sufficiently small 120598119871119881 (119878 1198681 1198682) le minus1
(119878 1198681 1198682) isin R3+119880 (63)
Hence 1198602 in Lemma 2 is satisfied This completes the proofof Theorem 5
5 Extinction
In this section we investigate the conditions for the extinc-tion of the two infectious diseases of system (3)
Let
R1 = 119860119906 ⟨1205731⟩119879119889119897 ⟨1198861⟩119879 ⟨119889 + 1205721 + 1199031⟩119879minus 1198602119906 ⟨12059021⟩11987921198892119897 ⟨11988621⟩119879 ⟨119889 + 1205721 + 1199031⟩119879
R2 = 119860119906 ⟨1205732⟩119879119889119897 ⟨1198862⟩119879 ⟨119889 + 1205722 + 1199032⟩119879minus 1198602119906 ⟨12059022⟩11987921198892119897 ⟨11988622⟩119879 ⟨119889 + 1205722 + 1199032⟩119879
(64)
Theorem 6 Let (119878(119905) 1198681(119905) 1198682(119905)) be a solution of system (3)with initial value (119878(0) 1198681(0) 1198682(0)) isin 1198773+
Then if
(i) ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792 ⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 119894 = 1 2 (65)
or(ii) R119894 lt 1⟨1205902119894 ⟩119879 le 119889119897 ⟨1198862119894 ⟩119879 ⟨120573119894⟩119879119860119906 ⟨119886119894⟩119879
119894 = 1 2(66)
hold the two infectious diseases of system (3) go to extinctionas that is
lim119905rarr+infin
119868119894 (119905) = 0 119894 = 1 2 (67)
10 Complexity
Proof Applying Itorsquos formula to system (3) we have
d ln 119868119894 (119905) = ( 120573119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905) minus (119889 (119905) + 120572119894 (119905) + 119903119894 (119905))minus 1205902119894 (119905) 1198782 (119905)2 (119886119894 (119905) + 119868119894 (119905))2) d119905 + 120590119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905)d119861119894 (119905)
119894 = 1 2
(68)
Case (i) Integrating (68) from 0 to 119905 and dividing 119905 on bothsides we obtain
ln 119868119894 (119905)119905 = minus1205902119894 (119905)2119905 int1199050( 119878 (120591)119886119894 (120591) + 119868119894 (120591) minus
120573119894 (120591)1205902119894 (120591))2
d120591minus 1119905 int
119905
0(119889 (120591) + 120572119894 (120591) + 119903119894 (120591)) d120591
+ 1119905 int119905
0
1205732119894 (120591)21205902119894 (120591)d120591 +119872119894 (119905)119905 + ln 119868119894 (0)119905
le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905
(69)
Case (ii) Integrating (68) from 0 to 119905 first and then dividingby 119905 on both sides yield
ln 119868119894 (119905)119905 = 1119905 int119905
0( 120573119894 (120591) 119878 (120591)119886119894 (120591) + 119868119894 (120591)
minus (119889 (120591) + 120572119894 (120591) + 119903119894 (120591))minus 1205902119894 (120591) 1198782 (120591)2 (119886119894 (120591) + 119868119894 (120591))2) d120591 + 119872119894 (119905)119905 + ln 119868119894 (0)119905le ⟨120573119894119860119906119886119894119889119897 minus (119889 + 120572119894 + 119903119894) minus
1205902119894 119860211990621198862119894 1198892119897⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) + 119872119894 (119905)119905+ ln 119868119894 (0)119905
(70)
where 119872119894(119905) = int1199050120590119894(120591)119878(120591)(119886119894(120591) + 119868119894(120591))d119861119894(120591) 119894 = 1 2
which is a local continuous martingale with 119872119894(0) = 0 ByLemma 3 we have
lim119905rarr+infin
119872119894 (119905)119905 = 0 119894 = 1 2 (71)
Taking the limit superior of both sides of (69) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119879 lt 0 (72)
which implies lim119905rarr+infin119868119894(119905) = 0
Taking the superior limit of both sides of (70) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) lt 0 (73)
which implies lim119905rarr+infin119868119894(119905) = 0 119894 = 1 2This completes theproof
Remark 7 Theorem 6 shows that the two diseases will dieout if the white noise disturbance is large or the white noisedisturbance is not large and R119894 lt 1 When ⟨1205902119894 ⟩119879 gt⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectious diseases of system(3) die out almost surely that is to say large white noisestochastic disturbance can lead to the two epidemics beingextinct
6 Numerical Simulations
Now we introduce some numerical simulations exampleswhich illustrate our theoretical results
Example 8 In model (3) let
119860 (119905) = 05 + 01 sin1205871199051198861 (119905) = 03 + 01 sin1205871199051198862 (119905) = 031 + 01 sin1205871199051205731 (119905) = 062 + 01 sin1205871199051205732 (119905) = 062 + 01 sin1205871199051199031 (119905) = 02 + 01 sin1205871199051199032 (119905) = 035 + 01 sin120587119905119889 (119905) = 02 + 01 sin1205871199051205721 (119905) = 02 + 01 sin1205871199051205722 (119905) = 025 + 01 sin1205871199051205901 (119905) = 01 + 005 sin1205871199051205902 (119905) = 01 + 005 sin120587119905
(74)
Note that R gt 1 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897and 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 hold that isthe conditions of Theorem 5 hold Hence system (3) has apositive periodic solution with 119879 = 1 Figure 1(a) shows theperiodicity of the nonautonomous stochastic model (3) with1205901 = 0 and 1205902 = 0 Figure 1(b) shows that solution of thenonautonomous stochastic model (3) with the initial value(119878(119905) 1198681(119905) 1198682(119905)) = (03 015 015) tends to a periodic orbitin the sense of joint distribution
Example 9 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin1205871199051198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin120587119905
Complexity 11
S(t)
I1(t)
I2(t)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
(a)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 1 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (1 02 02) to the nonautonomous stochastic model (3)
0 25 50 75 100t
59 60 61
078
08
082
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
59 60 61
078
08
082
0
1
2
14
14
S(t)I 1(t)I 2(t)
25 50 75 1000t
S(t)
I1(t)
I2(t)
(b)
Figure 2 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 04 + 02 sin1205871199051205902 (119905) = 05 + 02 sin120587119905
(75)
Note that ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897+120572119897119894+119903119897119894⟩119879Therefore conditions(i) of Theorem 6 hold Then the two infectious diseases willgo to extinction Figure 2(a) shows that one of two diseases inthe deterministic SIS epidemic model is extinct and the otheris persistent Figure 2(b) shows that the two diseases will dieout under the large white noise disturbance of model (3)
Example 10 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin120587119905
12 Complexity
25 50 75 1000t
0
05
1
15
2S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
0 25 50 75 100t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 3 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin1205871199051205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 02 + 02 sin1205871199051205902 (119905) = 02 + 02 sin120587119905
(76)
Note that R1 lt 1 R2 lt 1 and ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 That is conditions (ii) of Theorem 6 hold Thenthe two infectious diseases will go to extinction Figure 3(a)
shows that one of two diseases in the deterministic SISepidemic model is extinct and the other is persistent withoutthe white noises Figure 3(b) shows that the two diseases willdie out under a small white noise disturbance of model (3)
7 Discussion and Conclusions
This paper explores the existence of nontrivial positiveT-periodic solution of a nonautonomous stochastic SISepidemic model with nonlinear growth rate and doubleepidemic hypothesis By constructing a suitable stochasticLyapunov function we establish sufficient conditions for theexistence of nontrivial positive 119879-periodic solution of system(3) Furthermore the sufficient conditions for the extinctionof the two diseases are obtained Our results are given asfollows
(1) If 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 and R gt 1 hold the SISmodel has at least one nontrivial positive 119879-periodicsolution where
R = 2sum119894=1
⟨119860120572119894120573119894⟩119879⟨119889 + 120572119894 + 119903119894 + 1205902119894 119860211990621198862119894 1198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩119879 ⟨119860 + 11988611205721 + 11988621205722⟩119879 (77)
(2) If ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectiousdiseases go extinct
(3) If ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 and R119894 lt 1 the two
infectious diseases also go extinct where
R119894 = 119860119906 ⟨120573119894⟩119879119889119897 ⟨119886119894⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879minus 1198602119906 ⟨1205902119894 ⟩11987921198892119897 ⟨1198862119894 ⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879
(78)
Complexity 13
Some interesting questions deserve further investigationOn the one hand we may explore some realistic but complexmodels such as considering the effects of impulsive ordelay perturbations on system (3) On the other hand wecan concern the dynamics of a nonautonomous stochasticSIS epidemic model with two infectious diseases driven byLevy jumps What is more we can also investigate thenonautonomous stochastic SIS epidemic model with twoinfectious diseases by a continuous time Markov chain Wewill investigate these cases in our future work
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (11371230) Joint Innovative Center forSafe andEffectiveMiningTechnology andEquipment of CoalResources Shandong Province the SDUST Research Fund(2014TDJH102) and Shandong Provincial Natural ScienceFoundation China (ZR2015AQ001)
References
[1] W O Kermack and A G McKendrick ldquoA contributions to themathematical theory of epidemics (Part I)rdquo Proceedings of theRoyal Society of London A vol 115 pp 700ndash721 1927
[2] A Gray D Greenhalgh L Hu X Mao and J Pan ldquoA stochasticdifferential equation SIS epidemic modelrdquo SIAM Journal onApplied Mathematics vol 71 no 3 pp 876ndash902 2011
[3] A Korobeinikov and G C Wake ldquoLyapunov functions andglobal stability for SIR SIRS and SIS epidemiological modelsrdquoApplied Mathematics Letters vol 15 no 8 pp 955ndash960 2002
[4] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995
[5] T Zhang X Meng Y Song and T Zhang ldquoA stage-structuredpredator-prey SI model with disease in the prey and impulsiveeffectsrdquoMathematical Modelling and Analysis vol 18 no 4 pp505ndash528 2013
[6] J Li andZMa ldquoQualitative analyses of SIS epidemicmodelwithvaccination and varying total population sizerdquo Mathematicaland Computer Modelling vol 35 no 11-12 pp 1235ndash1243 2002
[7] A drsquoOnofrio ldquoA note on the global behaviour of the network-based SIS epidemic modelrdquo Nonlinear Analysis Real WorldApplications vol 9 no 4 pp 1567ndash1572 2008
[8] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[9] T Zhang X Meng T Zhang and Y Song ldquoGlobal dynamicsfor a new high-dimensional SIR model with distributed delayrdquoApplied Mathematics and Computation vol 218 no 24 pp11806ndash11819 2012
[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006
[11] X Meng S Zhao T Feng and T Zhang ldquoDynamics of a novelnonlinear stochastic SIS epidemic model with double epidemichypothesisrdquo Journal of Mathematical Analysis and Applicationsvol 433 no 1 pp 227ndash242 2016
[12] A Miao X Wang T Zhang W Wang and B Sampath ArunaPradeep ldquoDynamical analysis of a stochastic SIS epidemicmodel with nonlinear incidence rate and double epidemichypothesisrdquo Advances in Difference Equations 2017226 pages2017
[13] X-Z Meng ldquoStability of a novel stochastic epidemic modelwith double epidemic hypothesisrdquo Applied Mathematics andComputation vol 217 no 2 pp 506ndash515 2010
[14] X P Li X Y Lin and Y Q Lin ldquoLyapunov-type conditions andstochastic differential equations driven byG-BrownianmotionrdquoJournal of Mathematical Analysis and Applications vol 439 no1 pp 235ndash255 2016
[15] M Liu and M Fan ldquoPermanence of stochastic Lotka-Volterrasystemsrdquo Journal of Nonlinear Science vol 27 no 2 pp 425ndash452 2017
[16] L D Liu and X Z Meng ldquoOptimal harvesting controland dynamics of two-species stochastic model with delaysrdquoAdvances in Difference Equations vol 2017 18 pages 2017
[17] H Ma and Y Jia ldquoStability analysis for stochastic differentialequations with infiniteMarkovian switchingsrdquo Journal of Math-ematical Analysis and Applications vol 435 no 1 pp 593ndash6052016
[18] G D Liu X H Wang X Z Meng and S J Gao ldquoExtinctionand persistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexity vol 3pp 1ndash15 2017
[19] C Tan and W H Zhang ldquoOn observability and detectabilityof continuous-time stochastic Markov jump systemsrdquo Journalof Systems Science and Complexity vol 28 no 4 pp 830ndash8472015
[20] X Leng T Feng and X Meng ldquoStochastic inequalities andapplications to dynamics analysis of a novel SIVS epidemicmodel with jumpsrdquo Journal of Inequalities and ApplicationsPaper No 138 25 pages 2017
[21] G Li and M Chen ldquoInfinite horizon linear quadratic optimalcontrol for stochastic difference time-delay systemsrdquo Advancesin Difference Equations vol 2015 14 pages 2015
[22] Y Zhao and W Zhang ldquoObserver-based controller design forsingular stochastic Markov jump systems with state dependentnoiserdquo Journal of Systems Science and Complexity vol 29 no 4pp 946ndash958 2016
[23] S Zhang X Meng T Feng and T Zhang ldquoDynamics analysisand numerical simulations of a stochastic non-autonomouspredatorndashprey system with impulsive effectsrdquo Nonlinear Anal-ysis Hybrid Systems vol 26 pp 19ndash37 2017
[24] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013
[25] X Meng L Wang and T Zhang ldquoGlobal dynamics analysis ofa nonlinear impulsive stochastic chemostat system in a pollutedenvironmentrdquo Journal of AppliedAnalysis andComputation vol6 no 3 pp 865ndash875 2016
[26] X Liu Y Li andW Zhang ldquoStochastic linear quadratic optimalcontrol with constraint for discrete-time systemsrdquo AppliedMathematics and Computation vol 228 pp 264ndash270 2014
14 Complexity
[27] X Z Meng and X H Wang ldquoStochastic predator-prey systemsubject to levy jumprdquo Discrete Dynamics in Nature and Societyvol 2016 Article ID 5749892 13 pages 2016
[28] X Lv L Wang and X Meng ldquoGlobal analysis of a new non-linear stochastic differential competition systemwith impulsiveeffectrdquo Advances in Difference Equations vol 2017 296 pages2017
[29] X Zhang D Jiang A Alsaedi and T Hayat ldquoStationarydistribution of stochastic SIS epidemic model with vaccinationunder regime switchingrdquo Applied Mathematics Letters vol 59pp 87ndash93 2016
[30] F Li X Meng and Y Cui ldquoNonlinear stochastic analysis fora stochastic SIS epidemic modelrdquo Journal of Nonlinear Sciencesand Applications vol 10 no 09 pp 5116ndash5124 2017
[31] A Q Miao J Zhang T Q Zhang and B G Pradeep ldquoSampathAruna Pradeep Threshold dynamics of a stochastic SIR modelwith vertical transmission and vaccinationrdquo Computationaland Mathematical Methods in Medicine vol 2017 Article ID4820183 10 pages 2017
[32] Q Liu D Jiang N Shi T Hayat and A Alsaedi ldquoNontrivialperiodic solution of a stochastic non-autonomous SISV epi-demic modelrdquo Physica A Statistical Mechanics and its Applica-tions vol 462 pp 837ndash845 2016
[33] Y L Zhou S L Yuan and D L Zhao ldquoThreshold behavior ofa stochastic SIS model with levy jumpsrdquo Applied Mathematicsand Computation vol 275 pp 255ndash267 2016
[34] T Feng X Meng L Liu and S Gao ldquoApplication of inequal-ities technique to dynamics analysis of a stochastic eco-epidemiology modelrdquo Journal of Inequalities and ApplicationsPaper No 327 29 pages 2016
[35] M Liu and K Wang ldquoPersistence and extinction in stochasticnon-autonomous logistic systemsrdquo Journal of MathematicalAnalysis and Applications vol 375 no 2 pp 443ndash457 2011
[36] Z Bai and Y Zhou ldquoExistence of two periodic solutions for anon-autonomous SIR epidemic modelrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 1 pp 382ndash391 2011
[37] S Gao F Zhang and Y He ldquoThe effects of migratory birdpopulation in a nonautonomous eco-epidemiological modelrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 37 no 6 pp3903ndash3916 2013
[38] T Kuniya ldquoExistence of a nontrivial periodic solution in an age-structured SIR epidemicmodel with time periodic coefficientsrdquoApplied Mathematics Letters vol 27 pp 15ndash20 2014
[39] L Zu D Jiang D OrsquoRegan and B Ge ldquoPeriodic solution fora non-autonomous Lotka-Volterra predator-prey model withrandom perturbationrdquo Journal of Mathematical Analysis andApplications vol 430 no 1 pp 428ndash437 2015
[40] Y Lin D Jiang and T Liu ldquoNontrivial periodic solution ofa stochastic epidemic model with seasonal variationrdquo AppliedMathematics Letters vol 45 pp 103ndash107 2015
[41] X R Mao Stochastic Differential Equations and Their Applica-tions Horwood Chichester UK 1997
[42] R Khasminskii Stochastic Stability of Differential EquationsSpringer Berlin Germany 2011
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Stochastic AnalysisInternational Journal of
Complexity 5
By (25) and (26) one has
119881 (119878 (0) 1198681 (0) 1198682 (0)) + 119870119879 ge E [1Ω119896 (120596)sdot 119881 (119878 (120591119896 and 119879) 1198681 (120591119896 and 119879) 1198682 (120591119896 and 119879))] ge 120598 (119896minus 1 minus ln 119896) and (1119896 minus 1 minus ln 1119896)
(27)
where 1Ω119896 is the indicator function ofΩ119896
Let 119896 rarr infin we have
infin gt 119881(119878 (0) 1198681 (0) 1198682 (0)) + 119870119879 = infin (28)
So we obtain 120591infin = infinThe proof is completed
4 Existence of Nontrivial 119879-Periodic SolutionIn this section we verify that system (3) admits at least onenontrivial positive 119879-periodic solution Define
R = 2sum119894=1
⟨119860120572119894120573119894⟩119879⟨119889 + 120572119894 + 119903119894 + 1205902119894 119860211990621198862119894 1198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩119879 ⟨119860 + 11988611205721 + 11988621205722⟩119879 (29)
Theorem 5 When (1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)))119897 gt 119860119906119889119897and (1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)))119897 gt 119860119906119889119897 hold if R gt 1then there exists a nontrivial positive T-periodic solution ofsystem (3)
Proof Define a 1198622-function 119881 [0infin) timesR3+ rarr R+
119881 (119905 119878 1198681 1198682) = 119872(minus (1198901 + 1198902) ln 119878 minus 1198881 ln 1198681 minus 1198882 ln 1198682+ 119878 + 1198681 + 1198682 + 120596 (119905)) minus ln 119878 minus 1198681 minus 1198682 + 1120579 + 1 (119878+ 1198681 + 1198682)120579+1 = 119872(1198811 + 120596 (119905)) + 1198812 + 1198813 + 1198814+ 1198815
(30)
where1198881= ⟨11986012057211205731⟩119879⟨119889 + 1205721 + 1199031 + 1205902111986021199062119886211198892119897⟩2119879 ⟨119889 + 120590212 + 120590222⟩119879 1198901= ⟨11986012057211205731⟩119879⟨119889 + 1205721 + 1199031 + 1205902111986021199062119886211198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩2119879 1198882= ⟨11986012057221205732⟩119879⟨119889 + 1205722 + 1199032 + 1205902211986021199062119886221198892119897⟩2119879 ⟨119889 + 120590212 + 120590222⟩119879 1198902= ⟨11986012057221205732⟩119879⟨119889 + 1205722 + 1199032 + 1205902211986021199062119886221198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩2119879
(31)
and 0 lt 120579 lt min1 119889119897(12059021199061 + 12059021199062 )119872 is a sufficiently largepositive constant and satisfies the following conditions
119889119897 minus 120579 (12059021199061 + 12059021199062 ) gt 0minus119872120582 +max 119863 119864 le minus2 (32)
where
120582 = ⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1) 119863 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
119864 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(33)
where
119874 = sup(1198781198681 1198682)isinR
3+
119860119906 (119878 + 1198681 + 1198682)120579
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 lt infin(34)
Next we prove that condition (1198601) in Lemma 2 holds It iseasy to check that 119881(119905 119878 1198681 1198682) is a 119879-periodic function in 119905and satisfies
lim inf119870rarrinfin(1198781198681 1198682)isinR
3+U119896
119881 (119905 119878 1198681 1198682) = infin (35)
where U119896 = (1119896 119896) times (1119896 119896) times (1119896 119896) and 119896 gt 1 is asufficiently large number
6 Complexity
By Itorsquos formula we obtain
1198711198811 = minus1198901 + 1198902119878 (119860 (119905) minus 119889 (119905) 119878 minus 1205731 (119905) 11987811986811198861 (119905) + 1198681minus 1205732 (119905) 11987811986821198862 (119905) + 1198682 + 1199031 (119905) 1198681 + 1199032 (119905) 1198682)minus 11988811198681 (
1205731 (119905) 11987811986811198861 (119905) + I1minus (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681)
+ 1198901 + 1198902212059021 (119905) 11986821(1198861 (119905) + 1198681)2 minus
11988821198682 (1205732 (119905) 11987811986821198862 (119905) + 1198682
minus (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682) + 1198901 + 11989022sdot 12059022 (119905) 11986822(1198862 (119905) + 1198682)2 +
1198881212059021 (119905) 1198782(1198861 (119905) + 1198681)2 +
11988822sdot 12059022 (119905) 1198782(1198862 (119905) + 1198682)2 + 119860 (119905) minus 119889 (119905) (119878 + 1198681 + 1198682) minus 1205721 (119905)sdot 1198681 minus 1205722 (119905) 1198682 le minus1198901119860 (119905)119878 minus 11988811205731 (119905) 1198781198861 (119905) + 1198681 minus 1205721 (119905)sdot (1198861 (119905) + 1198681) minus 1198902119860 (119905)119878 minus 11988821205732 (119905) 1198781198862 (119905) + 1198682 minus 1205722 (119905)sdot (1198862 (119905) + 1198682) + 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905)
+ 12059021 (119905) 1198602119906211988621 (119905) 1198892119897) + 1198901 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )
+ 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905) + 12059022 (119905) 1198602119906211988622 (119905) 1198892119897)
+ 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 ) minus (1198901 + 1198902) (1199031 (119905)119878+ 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878 + 119889 (119905)minus 1205732 (119905)1198862 (119905) ) 1198682 + 119860 (119905) + 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905)le minus3 (119860 (119905) 1205721 (119905) 1205731 (119905) 11988811198901)13minus 3 (119860 (119905) 1205722 (119905) 1205732 (119905) 11988821198902)13 + 1198901 (119889 (119905) + 12059021 (119905)2+ 12059022 (119905)2 ) + 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905)
+ 12059021 (119905) 1198602119906211988621 (119905) 1198892119897) + 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905)
+ 12059022 (119905) 1198602119906211988622 (119905) 1198892119897) + 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )
+ 119860 (119905) + 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905) minus (1198901 + 1198902)sdot (1199031 (119905)119878 + 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878+ 119889 (119905) minus 1205732 (119905)1198862 (119905) ) 1198682 fl R0 (119905) minus (1198901 + 1198902) (1199031 (119905)119878+ 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878 + 119889 (119905)minus 1205732 (119905)1198862 (119905) ) 1198682
(36)
where
R0 (119905) = 3 (119860 (119905) 1205721 (119905) 1205731 (119905) 11988811198901)13+ 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905) + 12059021 (119905) 1198602119906211988621 (119905) 1198892119897)
+ 1198901 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )minus 3 (119860 (119905) 1205722 (119905) 1205732 (119905) 11988821198902)13+ 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905) + 12059022 (119905) 1198602119906211988622 (119905) 1198892119897)
+ 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 ) + 119860 (119905)+ 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905)
(37)
Note that (1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)))119897 gt 119860119906119889119897 and(1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)))119897 gt 119860119906119889119897 hold then119871 (1198811) le R0 (119905) (38)
Define the 119879-periodic function 120596(119905) satisfying1205961015840 (119905) = ⟨R0⟩119879 minusR0 (119905) (39)
So
119871 (1198811 + 120596 (119905)) le ⟨R0⟩119879= minus ⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1)fl minus120582
(40)
Applying Itorsquos formula we can also have
Complexity 7
1198711198812 = minus119860 (119905)119878 + 119889 (119905) + 1205731 (119905) 11986811198861 (119905) + 1198681 +1205732 (119905) 11986821198862 (119905) + 1198682
minus 1199031 (119905) 1198681119878 minus 1199032 (119905) 1198682119878 + 12059021 (119905) 119868212 (1198861 (119905) + 1198681)2+ 12059022 (119905) 119868222 (1198862 (119905) + 1198682)2 le minus119860119897119878 + 119889119906 + 1205731199061 + 1205731199062 + 120590211990612+ 120590211990622
1198711198813 = minus 1205731 (119905) 11987811986811198861 (119905) + 1198681 + (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681 le (119889119906+ 1205721199061 + 1199031199061 ) 1198681
1198711198814 = minus 1205732 (119905) 11987811986821198862 (119905) + 1198682 + (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682 le (119889119906+ 1205721199062 + 1199031199062 ) 1198682
1198711198815 = (119878 + 1198681 + 1198682)120579 (119860 (119905) minus 119889 (119905) 119878 minus 119889 (119905) 1198681 minus 119889 (119905) 1198682minus 1205721 (119905) 1198681 minus 1205722 (119905) 1198682) + 120579 (119878 + 1198681 + 1198682)120579minus1sdot ( 12059021 (119905) 119878211986821(1198861 (119905) + 1198681)2 +
12059022 (119905) 119878211986822(1198862 (119905) + 1198682)2) le 119860119906 (119878 + 1198681+ 1198682)120579 minus 119889119897 (119878 + 1198681 + 1198682)120579+1 + 120579 (119878 + 1198681 + 1198682)120579minus1 (12059021199061+ 12059021199062 ) 1198782 le 119860119906 (119878 + 1198681 + 1198682)120579 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 le 119874 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12
(41)
where
119874 = sup(1198781198681 1198682)isinR
3+
119860119906 (119878 + 1198681 + 1198682)120579
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 lt infin(42)
Therefore119871119881 (119905 119878 1198681 1198682) le minus119872⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1)
+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682 minus 119860119897119878minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874+ 119889119906 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622
= minus119872120582 minus 119860119897119878 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874+ 119889119906 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622
(43)
Now we are in the position to construct a compact subset119880 such that 1198602 in Lemma 2 holds Define the followingbounded closed set
119880 = (119878 1198681 1198682) isin R3+ 120598 le 119878 le 1120598 120598 le 1198681 le 1120598 120598 le 1198682
le 1120598 (44)
where 120598 gt 0 is a sufficiently small number In the set R3+119880we can choose 120598 sufficiently small such that
minus119872120582 minus 119860119897120598 + 119862 le minus1 (45)
minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 120598 + 119863 le minus1 (46)
minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 120598 + 119864 le minus1 (47)
minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119862 le minus1 (48)
minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119866 le minus1 (49)
minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119867 le minus1 (50)
where 119862 119863 119864 119866 and119867 are positive constants which can befound from the following inequations (52) (54) (56) (59)and (61) respectively For the sake of convenience we divideinto six domains
8 Complexity
1198801 = (119878 1198681 1198682) isin R3+ 0 lt 119878 lt 120598
1198802 = (119878 1198681 1198682) isin R3+ 0 lt 1198681 lt 120598
1198803 = (119878 1198681 1198682) isin R3+ 0 lt 1198682 lt 120598
1198804 = (119878 1198681 1198682) isin R3+ 119878 gt 1120598
1198805 = (119878 1198681 1198682) isin R3+ 1198681 gt 1120598
1198806 = (119878 1198681 1198682) isin R3+ 1198682 gt 1120598
(51)
Nextwewill prove that119871119881(119878 1198681 1198682) le minus1 onR3+119880 whichis equivalent to proving it on the above six domains
Case 1 If (119878 1198681 1198682) isin 1198801 one can see that
119871119881 le minus119872120582 minus 119860119897119878 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622 le minus119872120582 minus 119860119897119878 + 119862
le minus119872120582 minus 119860119897120598 + 119862
(52)
where
119862 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(53)
According to (45) we have 119871119881 le minus1 for all (119878 1198681 1198682) isin 1198801Case 2 If (119878 1198681 1198682) isin 1198802 one can get that
119871119881 le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681 + 119863le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 120598 + 119863
(54)
where
119863 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(55)
In view of (46) we can obtain that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198802Case 3 If (119878 1198681 1198682) isin 1198803 we have
119871119881 le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682 + 119864le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 120598 + 119864
(56)
where
119864 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(57)
By (47) we can conclude that 119871119881 le minus1 for all (119878 1198681 1198682) isin1198803Case 4 If (119878 1198681 1198682) isin 1198804 one can derive that
119871119881 le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1 + 119862le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119862
(58)
Complexity 9
Together with (48) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198804Case 5 If (119878 1198681 1198682) isin 1198805 it follows that
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119866le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119866
(59)
where
119866 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(60)
By virtue of (49) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198805Case 6 If (119878 1198681 1198682) isin 1198806 we obtain
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119867le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119867
(61)
where
119867 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(62)
It follows from (50) that 119871119881 le minus1 for all (119878 1198681 1198682) isin 1198806Clearly one can see from (52) (54) (56) (58) (59) and
(61) that for a sufficiently small 120598119871119881 (119878 1198681 1198682) le minus1
(119878 1198681 1198682) isin R3+119880 (63)
Hence 1198602 in Lemma 2 is satisfied This completes the proofof Theorem 5
5 Extinction
In this section we investigate the conditions for the extinc-tion of the two infectious diseases of system (3)
Let
R1 = 119860119906 ⟨1205731⟩119879119889119897 ⟨1198861⟩119879 ⟨119889 + 1205721 + 1199031⟩119879minus 1198602119906 ⟨12059021⟩11987921198892119897 ⟨11988621⟩119879 ⟨119889 + 1205721 + 1199031⟩119879
R2 = 119860119906 ⟨1205732⟩119879119889119897 ⟨1198862⟩119879 ⟨119889 + 1205722 + 1199032⟩119879minus 1198602119906 ⟨12059022⟩11987921198892119897 ⟨11988622⟩119879 ⟨119889 + 1205722 + 1199032⟩119879
(64)
Theorem 6 Let (119878(119905) 1198681(119905) 1198682(119905)) be a solution of system (3)with initial value (119878(0) 1198681(0) 1198682(0)) isin 1198773+
Then if
(i) ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792 ⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 119894 = 1 2 (65)
or(ii) R119894 lt 1⟨1205902119894 ⟩119879 le 119889119897 ⟨1198862119894 ⟩119879 ⟨120573119894⟩119879119860119906 ⟨119886119894⟩119879
119894 = 1 2(66)
hold the two infectious diseases of system (3) go to extinctionas that is
lim119905rarr+infin
119868119894 (119905) = 0 119894 = 1 2 (67)
10 Complexity
Proof Applying Itorsquos formula to system (3) we have
d ln 119868119894 (119905) = ( 120573119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905) minus (119889 (119905) + 120572119894 (119905) + 119903119894 (119905))minus 1205902119894 (119905) 1198782 (119905)2 (119886119894 (119905) + 119868119894 (119905))2) d119905 + 120590119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905)d119861119894 (119905)
119894 = 1 2
(68)
Case (i) Integrating (68) from 0 to 119905 and dividing 119905 on bothsides we obtain
ln 119868119894 (119905)119905 = minus1205902119894 (119905)2119905 int1199050( 119878 (120591)119886119894 (120591) + 119868119894 (120591) minus
120573119894 (120591)1205902119894 (120591))2
d120591minus 1119905 int
119905
0(119889 (120591) + 120572119894 (120591) + 119903119894 (120591)) d120591
+ 1119905 int119905
0
1205732119894 (120591)21205902119894 (120591)d120591 +119872119894 (119905)119905 + ln 119868119894 (0)119905
le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905
(69)
Case (ii) Integrating (68) from 0 to 119905 first and then dividingby 119905 on both sides yield
ln 119868119894 (119905)119905 = 1119905 int119905
0( 120573119894 (120591) 119878 (120591)119886119894 (120591) + 119868119894 (120591)
minus (119889 (120591) + 120572119894 (120591) + 119903119894 (120591))minus 1205902119894 (120591) 1198782 (120591)2 (119886119894 (120591) + 119868119894 (120591))2) d120591 + 119872119894 (119905)119905 + ln 119868119894 (0)119905le ⟨120573119894119860119906119886119894119889119897 minus (119889 + 120572119894 + 119903119894) minus
1205902119894 119860211990621198862119894 1198892119897⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) + 119872119894 (119905)119905+ ln 119868119894 (0)119905
(70)
where 119872119894(119905) = int1199050120590119894(120591)119878(120591)(119886119894(120591) + 119868119894(120591))d119861119894(120591) 119894 = 1 2
which is a local continuous martingale with 119872119894(0) = 0 ByLemma 3 we have
lim119905rarr+infin
119872119894 (119905)119905 = 0 119894 = 1 2 (71)
Taking the limit superior of both sides of (69) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119879 lt 0 (72)
which implies lim119905rarr+infin119868119894(119905) = 0
Taking the superior limit of both sides of (70) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) lt 0 (73)
which implies lim119905rarr+infin119868119894(119905) = 0 119894 = 1 2This completes theproof
Remark 7 Theorem 6 shows that the two diseases will dieout if the white noise disturbance is large or the white noisedisturbance is not large and R119894 lt 1 When ⟨1205902119894 ⟩119879 gt⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectious diseases of system(3) die out almost surely that is to say large white noisestochastic disturbance can lead to the two epidemics beingextinct
6 Numerical Simulations
Now we introduce some numerical simulations exampleswhich illustrate our theoretical results
Example 8 In model (3) let
119860 (119905) = 05 + 01 sin1205871199051198861 (119905) = 03 + 01 sin1205871199051198862 (119905) = 031 + 01 sin1205871199051205731 (119905) = 062 + 01 sin1205871199051205732 (119905) = 062 + 01 sin1205871199051199031 (119905) = 02 + 01 sin1205871199051199032 (119905) = 035 + 01 sin120587119905119889 (119905) = 02 + 01 sin1205871199051205721 (119905) = 02 + 01 sin1205871199051205722 (119905) = 025 + 01 sin1205871199051205901 (119905) = 01 + 005 sin1205871199051205902 (119905) = 01 + 005 sin120587119905
(74)
Note that R gt 1 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897and 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 hold that isthe conditions of Theorem 5 hold Hence system (3) has apositive periodic solution with 119879 = 1 Figure 1(a) shows theperiodicity of the nonautonomous stochastic model (3) with1205901 = 0 and 1205902 = 0 Figure 1(b) shows that solution of thenonautonomous stochastic model (3) with the initial value(119878(119905) 1198681(119905) 1198682(119905)) = (03 015 015) tends to a periodic orbitin the sense of joint distribution
Example 9 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin1205871199051198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin120587119905
Complexity 11
S(t)
I1(t)
I2(t)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
(a)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 1 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (1 02 02) to the nonautonomous stochastic model (3)
0 25 50 75 100t
59 60 61
078
08
082
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
59 60 61
078
08
082
0
1
2
14
14
S(t)I 1(t)I 2(t)
25 50 75 1000t
S(t)
I1(t)
I2(t)
(b)
Figure 2 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 04 + 02 sin1205871199051205902 (119905) = 05 + 02 sin120587119905
(75)
Note that ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897+120572119897119894+119903119897119894⟩119879Therefore conditions(i) of Theorem 6 hold Then the two infectious diseases willgo to extinction Figure 2(a) shows that one of two diseases inthe deterministic SIS epidemic model is extinct and the otheris persistent Figure 2(b) shows that the two diseases will dieout under the large white noise disturbance of model (3)
Example 10 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin120587119905
12 Complexity
25 50 75 1000t
0
05
1
15
2S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
0 25 50 75 100t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 3 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin1205871199051205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 02 + 02 sin1205871199051205902 (119905) = 02 + 02 sin120587119905
(76)
Note that R1 lt 1 R2 lt 1 and ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 That is conditions (ii) of Theorem 6 hold Thenthe two infectious diseases will go to extinction Figure 3(a)
shows that one of two diseases in the deterministic SISepidemic model is extinct and the other is persistent withoutthe white noises Figure 3(b) shows that the two diseases willdie out under a small white noise disturbance of model (3)
7 Discussion and Conclusions
This paper explores the existence of nontrivial positiveT-periodic solution of a nonautonomous stochastic SISepidemic model with nonlinear growth rate and doubleepidemic hypothesis By constructing a suitable stochasticLyapunov function we establish sufficient conditions for theexistence of nontrivial positive 119879-periodic solution of system(3) Furthermore the sufficient conditions for the extinctionof the two diseases are obtained Our results are given asfollows
(1) If 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 and R gt 1 hold the SISmodel has at least one nontrivial positive 119879-periodicsolution where
R = 2sum119894=1
⟨119860120572119894120573119894⟩119879⟨119889 + 120572119894 + 119903119894 + 1205902119894 119860211990621198862119894 1198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩119879 ⟨119860 + 11988611205721 + 11988621205722⟩119879 (77)
(2) If ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectiousdiseases go extinct
(3) If ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 and R119894 lt 1 the two
infectious diseases also go extinct where
R119894 = 119860119906 ⟨120573119894⟩119879119889119897 ⟨119886119894⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879minus 1198602119906 ⟨1205902119894 ⟩11987921198892119897 ⟨1198862119894 ⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879
(78)
Complexity 13
Some interesting questions deserve further investigationOn the one hand we may explore some realistic but complexmodels such as considering the effects of impulsive ordelay perturbations on system (3) On the other hand wecan concern the dynamics of a nonautonomous stochasticSIS epidemic model with two infectious diseases driven byLevy jumps What is more we can also investigate thenonautonomous stochastic SIS epidemic model with twoinfectious diseases by a continuous time Markov chain Wewill investigate these cases in our future work
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (11371230) Joint Innovative Center forSafe andEffectiveMiningTechnology andEquipment of CoalResources Shandong Province the SDUST Research Fund(2014TDJH102) and Shandong Provincial Natural ScienceFoundation China (ZR2015AQ001)
References
[1] W O Kermack and A G McKendrick ldquoA contributions to themathematical theory of epidemics (Part I)rdquo Proceedings of theRoyal Society of London A vol 115 pp 700ndash721 1927
[2] A Gray D Greenhalgh L Hu X Mao and J Pan ldquoA stochasticdifferential equation SIS epidemic modelrdquo SIAM Journal onApplied Mathematics vol 71 no 3 pp 876ndash902 2011
[3] A Korobeinikov and G C Wake ldquoLyapunov functions andglobal stability for SIR SIRS and SIS epidemiological modelsrdquoApplied Mathematics Letters vol 15 no 8 pp 955ndash960 2002
[4] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995
[5] T Zhang X Meng Y Song and T Zhang ldquoA stage-structuredpredator-prey SI model with disease in the prey and impulsiveeffectsrdquoMathematical Modelling and Analysis vol 18 no 4 pp505ndash528 2013
[6] J Li andZMa ldquoQualitative analyses of SIS epidemicmodelwithvaccination and varying total population sizerdquo Mathematicaland Computer Modelling vol 35 no 11-12 pp 1235ndash1243 2002
[7] A drsquoOnofrio ldquoA note on the global behaviour of the network-based SIS epidemic modelrdquo Nonlinear Analysis Real WorldApplications vol 9 no 4 pp 1567ndash1572 2008
[8] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[9] T Zhang X Meng T Zhang and Y Song ldquoGlobal dynamicsfor a new high-dimensional SIR model with distributed delayrdquoApplied Mathematics and Computation vol 218 no 24 pp11806ndash11819 2012
[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006
[11] X Meng S Zhao T Feng and T Zhang ldquoDynamics of a novelnonlinear stochastic SIS epidemic model with double epidemichypothesisrdquo Journal of Mathematical Analysis and Applicationsvol 433 no 1 pp 227ndash242 2016
[12] A Miao X Wang T Zhang W Wang and B Sampath ArunaPradeep ldquoDynamical analysis of a stochastic SIS epidemicmodel with nonlinear incidence rate and double epidemichypothesisrdquo Advances in Difference Equations 2017226 pages2017
[13] X-Z Meng ldquoStability of a novel stochastic epidemic modelwith double epidemic hypothesisrdquo Applied Mathematics andComputation vol 217 no 2 pp 506ndash515 2010
[14] X P Li X Y Lin and Y Q Lin ldquoLyapunov-type conditions andstochastic differential equations driven byG-BrownianmotionrdquoJournal of Mathematical Analysis and Applications vol 439 no1 pp 235ndash255 2016
[15] M Liu and M Fan ldquoPermanence of stochastic Lotka-Volterrasystemsrdquo Journal of Nonlinear Science vol 27 no 2 pp 425ndash452 2017
[16] L D Liu and X Z Meng ldquoOptimal harvesting controland dynamics of two-species stochastic model with delaysrdquoAdvances in Difference Equations vol 2017 18 pages 2017
[17] H Ma and Y Jia ldquoStability analysis for stochastic differentialequations with infiniteMarkovian switchingsrdquo Journal of Math-ematical Analysis and Applications vol 435 no 1 pp 593ndash6052016
[18] G D Liu X H Wang X Z Meng and S J Gao ldquoExtinctionand persistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexity vol 3pp 1ndash15 2017
[19] C Tan and W H Zhang ldquoOn observability and detectabilityof continuous-time stochastic Markov jump systemsrdquo Journalof Systems Science and Complexity vol 28 no 4 pp 830ndash8472015
[20] X Leng T Feng and X Meng ldquoStochastic inequalities andapplications to dynamics analysis of a novel SIVS epidemicmodel with jumpsrdquo Journal of Inequalities and ApplicationsPaper No 138 25 pages 2017
[21] G Li and M Chen ldquoInfinite horizon linear quadratic optimalcontrol for stochastic difference time-delay systemsrdquo Advancesin Difference Equations vol 2015 14 pages 2015
[22] Y Zhao and W Zhang ldquoObserver-based controller design forsingular stochastic Markov jump systems with state dependentnoiserdquo Journal of Systems Science and Complexity vol 29 no 4pp 946ndash958 2016
[23] S Zhang X Meng T Feng and T Zhang ldquoDynamics analysisand numerical simulations of a stochastic non-autonomouspredatorndashprey system with impulsive effectsrdquo Nonlinear Anal-ysis Hybrid Systems vol 26 pp 19ndash37 2017
[24] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013
[25] X Meng L Wang and T Zhang ldquoGlobal dynamics analysis ofa nonlinear impulsive stochastic chemostat system in a pollutedenvironmentrdquo Journal of AppliedAnalysis andComputation vol6 no 3 pp 865ndash875 2016
[26] X Liu Y Li andW Zhang ldquoStochastic linear quadratic optimalcontrol with constraint for discrete-time systemsrdquo AppliedMathematics and Computation vol 228 pp 264ndash270 2014
14 Complexity
[27] X Z Meng and X H Wang ldquoStochastic predator-prey systemsubject to levy jumprdquo Discrete Dynamics in Nature and Societyvol 2016 Article ID 5749892 13 pages 2016
[28] X Lv L Wang and X Meng ldquoGlobal analysis of a new non-linear stochastic differential competition systemwith impulsiveeffectrdquo Advances in Difference Equations vol 2017 296 pages2017
[29] X Zhang D Jiang A Alsaedi and T Hayat ldquoStationarydistribution of stochastic SIS epidemic model with vaccinationunder regime switchingrdquo Applied Mathematics Letters vol 59pp 87ndash93 2016
[30] F Li X Meng and Y Cui ldquoNonlinear stochastic analysis fora stochastic SIS epidemic modelrdquo Journal of Nonlinear Sciencesand Applications vol 10 no 09 pp 5116ndash5124 2017
[31] A Q Miao J Zhang T Q Zhang and B G Pradeep ldquoSampathAruna Pradeep Threshold dynamics of a stochastic SIR modelwith vertical transmission and vaccinationrdquo Computationaland Mathematical Methods in Medicine vol 2017 Article ID4820183 10 pages 2017
[32] Q Liu D Jiang N Shi T Hayat and A Alsaedi ldquoNontrivialperiodic solution of a stochastic non-autonomous SISV epi-demic modelrdquo Physica A Statistical Mechanics and its Applica-tions vol 462 pp 837ndash845 2016
[33] Y L Zhou S L Yuan and D L Zhao ldquoThreshold behavior ofa stochastic SIS model with levy jumpsrdquo Applied Mathematicsand Computation vol 275 pp 255ndash267 2016
[34] T Feng X Meng L Liu and S Gao ldquoApplication of inequal-ities technique to dynamics analysis of a stochastic eco-epidemiology modelrdquo Journal of Inequalities and ApplicationsPaper No 327 29 pages 2016
[35] M Liu and K Wang ldquoPersistence and extinction in stochasticnon-autonomous logistic systemsrdquo Journal of MathematicalAnalysis and Applications vol 375 no 2 pp 443ndash457 2011
[36] Z Bai and Y Zhou ldquoExistence of two periodic solutions for anon-autonomous SIR epidemic modelrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 1 pp 382ndash391 2011
[37] S Gao F Zhang and Y He ldquoThe effects of migratory birdpopulation in a nonautonomous eco-epidemiological modelrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 37 no 6 pp3903ndash3916 2013
[38] T Kuniya ldquoExistence of a nontrivial periodic solution in an age-structured SIR epidemicmodel with time periodic coefficientsrdquoApplied Mathematics Letters vol 27 pp 15ndash20 2014
[39] L Zu D Jiang D OrsquoRegan and B Ge ldquoPeriodic solution fora non-autonomous Lotka-Volterra predator-prey model withrandom perturbationrdquo Journal of Mathematical Analysis andApplications vol 430 no 1 pp 428ndash437 2015
[40] Y Lin D Jiang and T Liu ldquoNontrivial periodic solution ofa stochastic epidemic model with seasonal variationrdquo AppliedMathematics Letters vol 45 pp 103ndash107 2015
[41] X R Mao Stochastic Differential Equations and Their Applica-tions Horwood Chichester UK 1997
[42] R Khasminskii Stochastic Stability of Differential EquationsSpringer Berlin Germany 2011
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Algebra
Discrete Dynamics in Nature and Society
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Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Complexity
By Itorsquos formula we obtain
1198711198811 = minus1198901 + 1198902119878 (119860 (119905) minus 119889 (119905) 119878 minus 1205731 (119905) 11987811986811198861 (119905) + 1198681minus 1205732 (119905) 11987811986821198862 (119905) + 1198682 + 1199031 (119905) 1198681 + 1199032 (119905) 1198682)minus 11988811198681 (
1205731 (119905) 11987811986811198861 (119905) + I1minus (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681)
+ 1198901 + 1198902212059021 (119905) 11986821(1198861 (119905) + 1198681)2 minus
11988821198682 (1205732 (119905) 11987811986821198862 (119905) + 1198682
minus (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682) + 1198901 + 11989022sdot 12059022 (119905) 11986822(1198862 (119905) + 1198682)2 +
1198881212059021 (119905) 1198782(1198861 (119905) + 1198681)2 +
11988822sdot 12059022 (119905) 1198782(1198862 (119905) + 1198682)2 + 119860 (119905) minus 119889 (119905) (119878 + 1198681 + 1198682) minus 1205721 (119905)sdot 1198681 minus 1205722 (119905) 1198682 le minus1198901119860 (119905)119878 minus 11988811205731 (119905) 1198781198861 (119905) + 1198681 minus 1205721 (119905)sdot (1198861 (119905) + 1198681) minus 1198902119860 (119905)119878 minus 11988821205732 (119905) 1198781198862 (119905) + 1198682 minus 1205722 (119905)sdot (1198862 (119905) + 1198682) + 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905)
+ 12059021 (119905) 1198602119906211988621 (119905) 1198892119897) + 1198901 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )
+ 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905) + 12059022 (119905) 1198602119906211988622 (119905) 1198892119897)
+ 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 ) minus (1198901 + 1198902) (1199031 (119905)119878+ 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878 + 119889 (119905)minus 1205732 (119905)1198862 (119905) ) 1198682 + 119860 (119905) + 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905)le minus3 (119860 (119905) 1205721 (119905) 1205731 (119905) 11988811198901)13minus 3 (119860 (119905) 1205722 (119905) 1205732 (119905) 11988821198902)13 + 1198901 (119889 (119905) + 12059021 (119905)2+ 12059022 (119905)2 ) + 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905)
+ 12059021 (119905) 1198602119906211988621 (119905) 1198892119897) + 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905)
+ 12059022 (119905) 1198602119906211988622 (119905) 1198892119897) + 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )
+ 119860 (119905) + 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905) minus (1198901 + 1198902)sdot (1199031 (119905)119878 + 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878+ 119889 (119905) minus 1205732 (119905)1198862 (119905) ) 1198682 fl R0 (119905) minus (1198901 + 1198902) (1199031 (119905)119878+ 119889 (119905) minus 1205731 (119905)1198861 (119905) ) 1198681 minus (1198901 + 1198902) (
1199032 (119905)119878 + 119889 (119905)minus 1205732 (119905)1198862 (119905) ) 1198682
(36)
where
R0 (119905) = 3 (119860 (119905) 1205721 (119905) 1205731 (119905) 11988811198901)13+ 1198881 (119889 (119905) + 1205721 (119905) + 1199031 (119905) + 12059021 (119905) 1198602119906211988621 (119905) 1198892119897)
+ 1198901 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 )minus 3 (119860 (119905) 1205722 (119905) 1205732 (119905) 11988821198902)13+ 1198882 (119889 (119905) + 1205722 (119905) + 1199032 (119905) + 12059022 (119905) 1198602119906211988622 (119905) 1198892119897)
+ 1198902 (119889 (119905) + 12059021 (119905)2 + 12059022 (119905)2 ) + 119860 (119905)+ 1198861 (119905) 1205721 (119905) + 1198862 (119905) 1205722 (119905)
(37)
Note that (1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)))119897 gt 119860119906119889119897 and(1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)))119897 gt 119860119906119889119897 hold then119871 (1198811) le R0 (119905) (38)
Define the 119879-periodic function 120596(119905) satisfying1205961015840 (119905) = ⟨R0⟩119879 minusR0 (119905) (39)
So
119871 (1198811 + 120596 (119905)) le ⟨R0⟩119879= minus ⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1)fl minus120582
(40)
Applying Itorsquos formula we can also have
Complexity 7
1198711198812 = minus119860 (119905)119878 + 119889 (119905) + 1205731 (119905) 11986811198861 (119905) + 1198681 +1205732 (119905) 11986821198862 (119905) + 1198682
minus 1199031 (119905) 1198681119878 minus 1199032 (119905) 1198682119878 + 12059021 (119905) 119868212 (1198861 (119905) + 1198681)2+ 12059022 (119905) 119868222 (1198862 (119905) + 1198682)2 le minus119860119897119878 + 119889119906 + 1205731199061 + 1205731199062 + 120590211990612+ 120590211990622
1198711198813 = minus 1205731 (119905) 11987811986811198861 (119905) + 1198681 + (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681 le (119889119906+ 1205721199061 + 1199031199061 ) 1198681
1198711198814 = minus 1205732 (119905) 11987811986821198862 (119905) + 1198682 + (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682 le (119889119906+ 1205721199062 + 1199031199062 ) 1198682
1198711198815 = (119878 + 1198681 + 1198682)120579 (119860 (119905) minus 119889 (119905) 119878 minus 119889 (119905) 1198681 minus 119889 (119905) 1198682minus 1205721 (119905) 1198681 minus 1205722 (119905) 1198682) + 120579 (119878 + 1198681 + 1198682)120579minus1sdot ( 12059021 (119905) 119878211986821(1198861 (119905) + 1198681)2 +
12059022 (119905) 119878211986822(1198862 (119905) + 1198682)2) le 119860119906 (119878 + 1198681+ 1198682)120579 minus 119889119897 (119878 + 1198681 + 1198682)120579+1 + 120579 (119878 + 1198681 + 1198682)120579minus1 (12059021199061+ 12059021199062 ) 1198782 le 119860119906 (119878 + 1198681 + 1198682)120579 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 le 119874 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12
(41)
where
119874 = sup(1198781198681 1198682)isinR
3+
119860119906 (119878 + 1198681 + 1198682)120579
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 lt infin(42)
Therefore119871119881 (119905 119878 1198681 1198682) le minus119872⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1)
+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682 minus 119860119897119878minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874+ 119889119906 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622
= minus119872120582 minus 119860119897119878 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874+ 119889119906 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622
(43)
Now we are in the position to construct a compact subset119880 such that 1198602 in Lemma 2 holds Define the followingbounded closed set
119880 = (119878 1198681 1198682) isin R3+ 120598 le 119878 le 1120598 120598 le 1198681 le 1120598 120598 le 1198682
le 1120598 (44)
where 120598 gt 0 is a sufficiently small number In the set R3+119880we can choose 120598 sufficiently small such that
minus119872120582 minus 119860119897120598 + 119862 le minus1 (45)
minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 120598 + 119863 le minus1 (46)
minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 120598 + 119864 le minus1 (47)
minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119862 le minus1 (48)
minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119866 le minus1 (49)
minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119867 le minus1 (50)
where 119862 119863 119864 119866 and119867 are positive constants which can befound from the following inequations (52) (54) (56) (59)and (61) respectively For the sake of convenience we divideinto six domains
8 Complexity
1198801 = (119878 1198681 1198682) isin R3+ 0 lt 119878 lt 120598
1198802 = (119878 1198681 1198682) isin R3+ 0 lt 1198681 lt 120598
1198803 = (119878 1198681 1198682) isin R3+ 0 lt 1198682 lt 120598
1198804 = (119878 1198681 1198682) isin R3+ 119878 gt 1120598
1198805 = (119878 1198681 1198682) isin R3+ 1198681 gt 1120598
1198806 = (119878 1198681 1198682) isin R3+ 1198682 gt 1120598
(51)
Nextwewill prove that119871119881(119878 1198681 1198682) le minus1 onR3+119880 whichis equivalent to proving it on the above six domains
Case 1 If (119878 1198681 1198682) isin 1198801 one can see that
119871119881 le minus119872120582 minus 119860119897119878 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622 le minus119872120582 minus 119860119897119878 + 119862
le minus119872120582 minus 119860119897120598 + 119862
(52)
where
119862 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(53)
According to (45) we have 119871119881 le minus1 for all (119878 1198681 1198682) isin 1198801Case 2 If (119878 1198681 1198682) isin 1198802 one can get that
119871119881 le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681 + 119863le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 120598 + 119863
(54)
where
119863 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(55)
In view of (46) we can obtain that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198802Case 3 If (119878 1198681 1198682) isin 1198803 we have
119871119881 le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682 + 119864le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 120598 + 119864
(56)
where
119864 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(57)
By (47) we can conclude that 119871119881 le minus1 for all (119878 1198681 1198682) isin1198803Case 4 If (119878 1198681 1198682) isin 1198804 one can derive that
119871119881 le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1 + 119862le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119862
(58)
Complexity 9
Together with (48) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198804Case 5 If (119878 1198681 1198682) isin 1198805 it follows that
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119866le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119866
(59)
where
119866 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(60)
By virtue of (49) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198805Case 6 If (119878 1198681 1198682) isin 1198806 we obtain
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119867le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119867
(61)
where
119867 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(62)
It follows from (50) that 119871119881 le minus1 for all (119878 1198681 1198682) isin 1198806Clearly one can see from (52) (54) (56) (58) (59) and
(61) that for a sufficiently small 120598119871119881 (119878 1198681 1198682) le minus1
(119878 1198681 1198682) isin R3+119880 (63)
Hence 1198602 in Lemma 2 is satisfied This completes the proofof Theorem 5
5 Extinction
In this section we investigate the conditions for the extinc-tion of the two infectious diseases of system (3)
Let
R1 = 119860119906 ⟨1205731⟩119879119889119897 ⟨1198861⟩119879 ⟨119889 + 1205721 + 1199031⟩119879minus 1198602119906 ⟨12059021⟩11987921198892119897 ⟨11988621⟩119879 ⟨119889 + 1205721 + 1199031⟩119879
R2 = 119860119906 ⟨1205732⟩119879119889119897 ⟨1198862⟩119879 ⟨119889 + 1205722 + 1199032⟩119879minus 1198602119906 ⟨12059022⟩11987921198892119897 ⟨11988622⟩119879 ⟨119889 + 1205722 + 1199032⟩119879
(64)
Theorem 6 Let (119878(119905) 1198681(119905) 1198682(119905)) be a solution of system (3)with initial value (119878(0) 1198681(0) 1198682(0)) isin 1198773+
Then if
(i) ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792 ⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 119894 = 1 2 (65)
or(ii) R119894 lt 1⟨1205902119894 ⟩119879 le 119889119897 ⟨1198862119894 ⟩119879 ⟨120573119894⟩119879119860119906 ⟨119886119894⟩119879
119894 = 1 2(66)
hold the two infectious diseases of system (3) go to extinctionas that is
lim119905rarr+infin
119868119894 (119905) = 0 119894 = 1 2 (67)
10 Complexity
Proof Applying Itorsquos formula to system (3) we have
d ln 119868119894 (119905) = ( 120573119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905) minus (119889 (119905) + 120572119894 (119905) + 119903119894 (119905))minus 1205902119894 (119905) 1198782 (119905)2 (119886119894 (119905) + 119868119894 (119905))2) d119905 + 120590119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905)d119861119894 (119905)
119894 = 1 2
(68)
Case (i) Integrating (68) from 0 to 119905 and dividing 119905 on bothsides we obtain
ln 119868119894 (119905)119905 = minus1205902119894 (119905)2119905 int1199050( 119878 (120591)119886119894 (120591) + 119868119894 (120591) minus
120573119894 (120591)1205902119894 (120591))2
d120591minus 1119905 int
119905
0(119889 (120591) + 120572119894 (120591) + 119903119894 (120591)) d120591
+ 1119905 int119905
0
1205732119894 (120591)21205902119894 (120591)d120591 +119872119894 (119905)119905 + ln 119868119894 (0)119905
le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905
(69)
Case (ii) Integrating (68) from 0 to 119905 first and then dividingby 119905 on both sides yield
ln 119868119894 (119905)119905 = 1119905 int119905
0( 120573119894 (120591) 119878 (120591)119886119894 (120591) + 119868119894 (120591)
minus (119889 (120591) + 120572119894 (120591) + 119903119894 (120591))minus 1205902119894 (120591) 1198782 (120591)2 (119886119894 (120591) + 119868119894 (120591))2) d120591 + 119872119894 (119905)119905 + ln 119868119894 (0)119905le ⟨120573119894119860119906119886119894119889119897 minus (119889 + 120572119894 + 119903119894) minus
1205902119894 119860211990621198862119894 1198892119897⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) + 119872119894 (119905)119905+ ln 119868119894 (0)119905
(70)
where 119872119894(119905) = int1199050120590119894(120591)119878(120591)(119886119894(120591) + 119868119894(120591))d119861119894(120591) 119894 = 1 2
which is a local continuous martingale with 119872119894(0) = 0 ByLemma 3 we have
lim119905rarr+infin
119872119894 (119905)119905 = 0 119894 = 1 2 (71)
Taking the limit superior of both sides of (69) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119879 lt 0 (72)
which implies lim119905rarr+infin119868119894(119905) = 0
Taking the superior limit of both sides of (70) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) lt 0 (73)
which implies lim119905rarr+infin119868119894(119905) = 0 119894 = 1 2This completes theproof
Remark 7 Theorem 6 shows that the two diseases will dieout if the white noise disturbance is large or the white noisedisturbance is not large and R119894 lt 1 When ⟨1205902119894 ⟩119879 gt⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectious diseases of system(3) die out almost surely that is to say large white noisestochastic disturbance can lead to the two epidemics beingextinct
6 Numerical Simulations
Now we introduce some numerical simulations exampleswhich illustrate our theoretical results
Example 8 In model (3) let
119860 (119905) = 05 + 01 sin1205871199051198861 (119905) = 03 + 01 sin1205871199051198862 (119905) = 031 + 01 sin1205871199051205731 (119905) = 062 + 01 sin1205871199051205732 (119905) = 062 + 01 sin1205871199051199031 (119905) = 02 + 01 sin1205871199051199032 (119905) = 035 + 01 sin120587119905119889 (119905) = 02 + 01 sin1205871199051205721 (119905) = 02 + 01 sin1205871199051205722 (119905) = 025 + 01 sin1205871199051205901 (119905) = 01 + 005 sin1205871199051205902 (119905) = 01 + 005 sin120587119905
(74)
Note that R gt 1 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897and 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 hold that isthe conditions of Theorem 5 hold Hence system (3) has apositive periodic solution with 119879 = 1 Figure 1(a) shows theperiodicity of the nonautonomous stochastic model (3) with1205901 = 0 and 1205902 = 0 Figure 1(b) shows that solution of thenonautonomous stochastic model (3) with the initial value(119878(119905) 1198681(119905) 1198682(119905)) = (03 015 015) tends to a periodic orbitin the sense of joint distribution
Example 9 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin1205871199051198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin120587119905
Complexity 11
S(t)
I1(t)
I2(t)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
(a)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 1 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (1 02 02) to the nonautonomous stochastic model (3)
0 25 50 75 100t
59 60 61
078
08
082
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
59 60 61
078
08
082
0
1
2
14
14
S(t)I 1(t)I 2(t)
25 50 75 1000t
S(t)
I1(t)
I2(t)
(b)
Figure 2 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 04 + 02 sin1205871199051205902 (119905) = 05 + 02 sin120587119905
(75)
Note that ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897+120572119897119894+119903119897119894⟩119879Therefore conditions(i) of Theorem 6 hold Then the two infectious diseases willgo to extinction Figure 2(a) shows that one of two diseases inthe deterministic SIS epidemic model is extinct and the otheris persistent Figure 2(b) shows that the two diseases will dieout under the large white noise disturbance of model (3)
Example 10 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin120587119905
12 Complexity
25 50 75 1000t
0
05
1
15
2S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
0 25 50 75 100t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 3 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin1205871199051205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 02 + 02 sin1205871199051205902 (119905) = 02 + 02 sin120587119905
(76)
Note that R1 lt 1 R2 lt 1 and ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 That is conditions (ii) of Theorem 6 hold Thenthe two infectious diseases will go to extinction Figure 3(a)
shows that one of two diseases in the deterministic SISepidemic model is extinct and the other is persistent withoutthe white noises Figure 3(b) shows that the two diseases willdie out under a small white noise disturbance of model (3)
7 Discussion and Conclusions
This paper explores the existence of nontrivial positiveT-periodic solution of a nonautonomous stochastic SISepidemic model with nonlinear growth rate and doubleepidemic hypothesis By constructing a suitable stochasticLyapunov function we establish sufficient conditions for theexistence of nontrivial positive 119879-periodic solution of system(3) Furthermore the sufficient conditions for the extinctionof the two diseases are obtained Our results are given asfollows
(1) If 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 and R gt 1 hold the SISmodel has at least one nontrivial positive 119879-periodicsolution where
R = 2sum119894=1
⟨119860120572119894120573119894⟩119879⟨119889 + 120572119894 + 119903119894 + 1205902119894 119860211990621198862119894 1198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩119879 ⟨119860 + 11988611205721 + 11988621205722⟩119879 (77)
(2) If ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectiousdiseases go extinct
(3) If ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 and R119894 lt 1 the two
infectious diseases also go extinct where
R119894 = 119860119906 ⟨120573119894⟩119879119889119897 ⟨119886119894⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879minus 1198602119906 ⟨1205902119894 ⟩11987921198892119897 ⟨1198862119894 ⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879
(78)
Complexity 13
Some interesting questions deserve further investigationOn the one hand we may explore some realistic but complexmodels such as considering the effects of impulsive ordelay perturbations on system (3) On the other hand wecan concern the dynamics of a nonautonomous stochasticSIS epidemic model with two infectious diseases driven byLevy jumps What is more we can also investigate thenonautonomous stochastic SIS epidemic model with twoinfectious diseases by a continuous time Markov chain Wewill investigate these cases in our future work
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (11371230) Joint Innovative Center forSafe andEffectiveMiningTechnology andEquipment of CoalResources Shandong Province the SDUST Research Fund(2014TDJH102) and Shandong Provincial Natural ScienceFoundation China (ZR2015AQ001)
References
[1] W O Kermack and A G McKendrick ldquoA contributions to themathematical theory of epidemics (Part I)rdquo Proceedings of theRoyal Society of London A vol 115 pp 700ndash721 1927
[2] A Gray D Greenhalgh L Hu X Mao and J Pan ldquoA stochasticdifferential equation SIS epidemic modelrdquo SIAM Journal onApplied Mathematics vol 71 no 3 pp 876ndash902 2011
[3] A Korobeinikov and G C Wake ldquoLyapunov functions andglobal stability for SIR SIRS and SIS epidemiological modelsrdquoApplied Mathematics Letters vol 15 no 8 pp 955ndash960 2002
[4] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995
[5] T Zhang X Meng Y Song and T Zhang ldquoA stage-structuredpredator-prey SI model with disease in the prey and impulsiveeffectsrdquoMathematical Modelling and Analysis vol 18 no 4 pp505ndash528 2013
[6] J Li andZMa ldquoQualitative analyses of SIS epidemicmodelwithvaccination and varying total population sizerdquo Mathematicaland Computer Modelling vol 35 no 11-12 pp 1235ndash1243 2002
[7] A drsquoOnofrio ldquoA note on the global behaviour of the network-based SIS epidemic modelrdquo Nonlinear Analysis Real WorldApplications vol 9 no 4 pp 1567ndash1572 2008
[8] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[9] T Zhang X Meng T Zhang and Y Song ldquoGlobal dynamicsfor a new high-dimensional SIR model with distributed delayrdquoApplied Mathematics and Computation vol 218 no 24 pp11806ndash11819 2012
[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006
[11] X Meng S Zhao T Feng and T Zhang ldquoDynamics of a novelnonlinear stochastic SIS epidemic model with double epidemichypothesisrdquo Journal of Mathematical Analysis and Applicationsvol 433 no 1 pp 227ndash242 2016
[12] A Miao X Wang T Zhang W Wang and B Sampath ArunaPradeep ldquoDynamical analysis of a stochastic SIS epidemicmodel with nonlinear incidence rate and double epidemichypothesisrdquo Advances in Difference Equations 2017226 pages2017
[13] X-Z Meng ldquoStability of a novel stochastic epidemic modelwith double epidemic hypothesisrdquo Applied Mathematics andComputation vol 217 no 2 pp 506ndash515 2010
[14] X P Li X Y Lin and Y Q Lin ldquoLyapunov-type conditions andstochastic differential equations driven byG-BrownianmotionrdquoJournal of Mathematical Analysis and Applications vol 439 no1 pp 235ndash255 2016
[15] M Liu and M Fan ldquoPermanence of stochastic Lotka-Volterrasystemsrdquo Journal of Nonlinear Science vol 27 no 2 pp 425ndash452 2017
[16] L D Liu and X Z Meng ldquoOptimal harvesting controland dynamics of two-species stochastic model with delaysrdquoAdvances in Difference Equations vol 2017 18 pages 2017
[17] H Ma and Y Jia ldquoStability analysis for stochastic differentialequations with infiniteMarkovian switchingsrdquo Journal of Math-ematical Analysis and Applications vol 435 no 1 pp 593ndash6052016
[18] G D Liu X H Wang X Z Meng and S J Gao ldquoExtinctionand persistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexity vol 3pp 1ndash15 2017
[19] C Tan and W H Zhang ldquoOn observability and detectabilityof continuous-time stochastic Markov jump systemsrdquo Journalof Systems Science and Complexity vol 28 no 4 pp 830ndash8472015
[20] X Leng T Feng and X Meng ldquoStochastic inequalities andapplications to dynamics analysis of a novel SIVS epidemicmodel with jumpsrdquo Journal of Inequalities and ApplicationsPaper No 138 25 pages 2017
[21] G Li and M Chen ldquoInfinite horizon linear quadratic optimalcontrol for stochastic difference time-delay systemsrdquo Advancesin Difference Equations vol 2015 14 pages 2015
[22] Y Zhao and W Zhang ldquoObserver-based controller design forsingular stochastic Markov jump systems with state dependentnoiserdquo Journal of Systems Science and Complexity vol 29 no 4pp 946ndash958 2016
[23] S Zhang X Meng T Feng and T Zhang ldquoDynamics analysisand numerical simulations of a stochastic non-autonomouspredatorndashprey system with impulsive effectsrdquo Nonlinear Anal-ysis Hybrid Systems vol 26 pp 19ndash37 2017
[24] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013
[25] X Meng L Wang and T Zhang ldquoGlobal dynamics analysis ofa nonlinear impulsive stochastic chemostat system in a pollutedenvironmentrdquo Journal of AppliedAnalysis andComputation vol6 no 3 pp 865ndash875 2016
[26] X Liu Y Li andW Zhang ldquoStochastic linear quadratic optimalcontrol with constraint for discrete-time systemsrdquo AppliedMathematics and Computation vol 228 pp 264ndash270 2014
14 Complexity
[27] X Z Meng and X H Wang ldquoStochastic predator-prey systemsubject to levy jumprdquo Discrete Dynamics in Nature and Societyvol 2016 Article ID 5749892 13 pages 2016
[28] X Lv L Wang and X Meng ldquoGlobal analysis of a new non-linear stochastic differential competition systemwith impulsiveeffectrdquo Advances in Difference Equations vol 2017 296 pages2017
[29] X Zhang D Jiang A Alsaedi and T Hayat ldquoStationarydistribution of stochastic SIS epidemic model with vaccinationunder regime switchingrdquo Applied Mathematics Letters vol 59pp 87ndash93 2016
[30] F Li X Meng and Y Cui ldquoNonlinear stochastic analysis fora stochastic SIS epidemic modelrdquo Journal of Nonlinear Sciencesand Applications vol 10 no 09 pp 5116ndash5124 2017
[31] A Q Miao J Zhang T Q Zhang and B G Pradeep ldquoSampathAruna Pradeep Threshold dynamics of a stochastic SIR modelwith vertical transmission and vaccinationrdquo Computationaland Mathematical Methods in Medicine vol 2017 Article ID4820183 10 pages 2017
[32] Q Liu D Jiang N Shi T Hayat and A Alsaedi ldquoNontrivialperiodic solution of a stochastic non-autonomous SISV epi-demic modelrdquo Physica A Statistical Mechanics and its Applica-tions vol 462 pp 837ndash845 2016
[33] Y L Zhou S L Yuan and D L Zhao ldquoThreshold behavior ofa stochastic SIS model with levy jumpsrdquo Applied Mathematicsand Computation vol 275 pp 255ndash267 2016
[34] T Feng X Meng L Liu and S Gao ldquoApplication of inequal-ities technique to dynamics analysis of a stochastic eco-epidemiology modelrdquo Journal of Inequalities and ApplicationsPaper No 327 29 pages 2016
[35] M Liu and K Wang ldquoPersistence and extinction in stochasticnon-autonomous logistic systemsrdquo Journal of MathematicalAnalysis and Applications vol 375 no 2 pp 443ndash457 2011
[36] Z Bai and Y Zhou ldquoExistence of two periodic solutions for anon-autonomous SIR epidemic modelrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 1 pp 382ndash391 2011
[37] S Gao F Zhang and Y He ldquoThe effects of migratory birdpopulation in a nonautonomous eco-epidemiological modelrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 37 no 6 pp3903ndash3916 2013
[38] T Kuniya ldquoExistence of a nontrivial periodic solution in an age-structured SIR epidemicmodel with time periodic coefficientsrdquoApplied Mathematics Letters vol 27 pp 15ndash20 2014
[39] L Zu D Jiang D OrsquoRegan and B Ge ldquoPeriodic solution fora non-autonomous Lotka-Volterra predator-prey model withrandom perturbationrdquo Journal of Mathematical Analysis andApplications vol 430 no 1 pp 428ndash437 2015
[40] Y Lin D Jiang and T Liu ldquoNontrivial periodic solution ofa stochastic epidemic model with seasonal variationrdquo AppliedMathematics Letters vol 45 pp 103ndash107 2015
[41] X R Mao Stochastic Differential Equations and Their Applica-tions Horwood Chichester UK 1997
[42] R Khasminskii Stochastic Stability of Differential EquationsSpringer Berlin Germany 2011
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Complexity 7
1198711198812 = minus119860 (119905)119878 + 119889 (119905) + 1205731 (119905) 11986811198861 (119905) + 1198681 +1205732 (119905) 11986821198862 (119905) + 1198682
minus 1199031 (119905) 1198681119878 minus 1199032 (119905) 1198682119878 + 12059021 (119905) 119868212 (1198861 (119905) + 1198681)2+ 12059022 (119905) 119868222 (1198862 (119905) + 1198682)2 le minus119860119897119878 + 119889119906 + 1205731199061 + 1205731199062 + 120590211990612+ 120590211990622
1198711198813 = minus 1205731 (119905) 11987811986811198861 (119905) + 1198681 + (119889 (119905) + 1205721 (119905) + 1199031 (119905)) 1198681 le (119889119906+ 1205721199061 + 1199031199061 ) 1198681
1198711198814 = minus 1205732 (119905) 11987811986821198862 (119905) + 1198682 + (119889 (119905) + 1205722 (119905) + 1199032 (119905)) 1198682 le (119889119906+ 1205721199062 + 1199031199062 ) 1198682
1198711198815 = (119878 + 1198681 + 1198682)120579 (119860 (119905) minus 119889 (119905) 119878 minus 119889 (119905) 1198681 minus 119889 (119905) 1198682minus 1205721 (119905) 1198681 minus 1205722 (119905) 1198682) + 120579 (119878 + 1198681 + 1198682)120579minus1sdot ( 12059021 (119905) 119878211986821(1198861 (119905) + 1198681)2 +
12059022 (119905) 119878211986822(1198862 (119905) + 1198682)2) le 119860119906 (119878 + 1198681+ 1198682)120579 minus 119889119897 (119878 + 1198681 + 1198682)120579+1 + 120579 (119878 + 1198681 + 1198682)120579minus1 (12059021199061+ 12059021199062 ) 1198782 le 119860119906 (119878 + 1198681 + 1198682)120579 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 le 119874 minus 12 (119889119897minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12
(41)
where
119874 = sup(1198781198681 1198682)isinR
3+
119860119906 (119878 + 1198681 + 1198682)120579
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) (119878 + 1198681 + 1198682)120579+1 lt infin(42)
Therefore119871119881 (119905 119878 1198681 1198682) le minus119872⟨119860 + 11988611205721 + 11988621205722⟩119879 (R minus 1)
+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682 minus 119860119897119878minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1
minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874+ 119889119906 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622
= minus119872120582 minus 119860119897119878 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874+ 119889119906 + 1205731199061 + 1205731199062 + 120590211990612 + 120590211990622
(43)
Now we are in the position to construct a compact subset119880 such that 1198602 in Lemma 2 holds Define the followingbounded closed set
119880 = (119878 1198681 1198682) isin R3+ 120598 le 119878 le 1120598 120598 le 1198681 le 1120598 120598 le 1198682
le 1120598 (44)
where 120598 gt 0 is a sufficiently small number In the set R3+119880we can choose 120598 sufficiently small such that
minus119872120582 minus 119860119897120598 + 119862 le minus1 (45)
minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 120598 + 119863 le minus1 (46)
minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 120598 + 119864 le minus1 (47)
minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119862 le minus1 (48)
minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119866 le minus1 (49)
minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119867 le minus1 (50)
where 119862 119863 119864 119866 and119867 are positive constants which can befound from the following inequations (52) (54) (56) (59)and (61) respectively For the sake of convenience we divideinto six domains
8 Complexity
1198801 = (119878 1198681 1198682) isin R3+ 0 lt 119878 lt 120598
1198802 = (119878 1198681 1198682) isin R3+ 0 lt 1198681 lt 120598
1198803 = (119878 1198681 1198682) isin R3+ 0 lt 1198682 lt 120598
1198804 = (119878 1198681 1198682) isin R3+ 119878 gt 1120598
1198805 = (119878 1198681 1198682) isin R3+ 1198681 gt 1120598
1198806 = (119878 1198681 1198682) isin R3+ 1198682 gt 1120598
(51)
Nextwewill prove that119871119881(119878 1198681 1198682) le minus1 onR3+119880 whichis equivalent to proving it on the above six domains
Case 1 If (119878 1198681 1198682) isin 1198801 one can see that
119871119881 le minus119872120582 minus 119860119897119878 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622 le minus119872120582 minus 119860119897119878 + 119862
le minus119872120582 minus 119860119897120598 + 119862
(52)
where
119862 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(53)
According to (45) we have 119871119881 le minus1 for all (119878 1198681 1198682) isin 1198801Case 2 If (119878 1198681 1198682) isin 1198802 one can get that
119871119881 le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681 + 119863le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 120598 + 119863
(54)
where
119863 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(55)
In view of (46) we can obtain that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198802Case 3 If (119878 1198681 1198682) isin 1198803 we have
119871119881 le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682 + 119864le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 120598 + 119864
(56)
where
119864 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(57)
By (47) we can conclude that 119871119881 le minus1 for all (119878 1198681 1198682) isin1198803Case 4 If (119878 1198681 1198682) isin 1198804 one can derive that
119871119881 le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1 + 119862le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119862
(58)
Complexity 9
Together with (48) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198804Case 5 If (119878 1198681 1198682) isin 1198805 it follows that
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119866le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119866
(59)
where
119866 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(60)
By virtue of (49) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198805Case 6 If (119878 1198681 1198682) isin 1198806 we obtain
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119867le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119867
(61)
where
119867 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(62)
It follows from (50) that 119871119881 le minus1 for all (119878 1198681 1198682) isin 1198806Clearly one can see from (52) (54) (56) (58) (59) and
(61) that for a sufficiently small 120598119871119881 (119878 1198681 1198682) le minus1
(119878 1198681 1198682) isin R3+119880 (63)
Hence 1198602 in Lemma 2 is satisfied This completes the proofof Theorem 5
5 Extinction
In this section we investigate the conditions for the extinc-tion of the two infectious diseases of system (3)
Let
R1 = 119860119906 ⟨1205731⟩119879119889119897 ⟨1198861⟩119879 ⟨119889 + 1205721 + 1199031⟩119879minus 1198602119906 ⟨12059021⟩11987921198892119897 ⟨11988621⟩119879 ⟨119889 + 1205721 + 1199031⟩119879
R2 = 119860119906 ⟨1205732⟩119879119889119897 ⟨1198862⟩119879 ⟨119889 + 1205722 + 1199032⟩119879minus 1198602119906 ⟨12059022⟩11987921198892119897 ⟨11988622⟩119879 ⟨119889 + 1205722 + 1199032⟩119879
(64)
Theorem 6 Let (119878(119905) 1198681(119905) 1198682(119905)) be a solution of system (3)with initial value (119878(0) 1198681(0) 1198682(0)) isin 1198773+
Then if
(i) ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792 ⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 119894 = 1 2 (65)
or(ii) R119894 lt 1⟨1205902119894 ⟩119879 le 119889119897 ⟨1198862119894 ⟩119879 ⟨120573119894⟩119879119860119906 ⟨119886119894⟩119879
119894 = 1 2(66)
hold the two infectious diseases of system (3) go to extinctionas that is
lim119905rarr+infin
119868119894 (119905) = 0 119894 = 1 2 (67)
10 Complexity
Proof Applying Itorsquos formula to system (3) we have
d ln 119868119894 (119905) = ( 120573119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905) minus (119889 (119905) + 120572119894 (119905) + 119903119894 (119905))minus 1205902119894 (119905) 1198782 (119905)2 (119886119894 (119905) + 119868119894 (119905))2) d119905 + 120590119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905)d119861119894 (119905)
119894 = 1 2
(68)
Case (i) Integrating (68) from 0 to 119905 and dividing 119905 on bothsides we obtain
ln 119868119894 (119905)119905 = minus1205902119894 (119905)2119905 int1199050( 119878 (120591)119886119894 (120591) + 119868119894 (120591) minus
120573119894 (120591)1205902119894 (120591))2
d120591minus 1119905 int
119905
0(119889 (120591) + 120572119894 (120591) + 119903119894 (120591)) d120591
+ 1119905 int119905
0
1205732119894 (120591)21205902119894 (120591)d120591 +119872119894 (119905)119905 + ln 119868119894 (0)119905
le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905
(69)
Case (ii) Integrating (68) from 0 to 119905 first and then dividingby 119905 on both sides yield
ln 119868119894 (119905)119905 = 1119905 int119905
0( 120573119894 (120591) 119878 (120591)119886119894 (120591) + 119868119894 (120591)
minus (119889 (120591) + 120572119894 (120591) + 119903119894 (120591))minus 1205902119894 (120591) 1198782 (120591)2 (119886119894 (120591) + 119868119894 (120591))2) d120591 + 119872119894 (119905)119905 + ln 119868119894 (0)119905le ⟨120573119894119860119906119886119894119889119897 minus (119889 + 120572119894 + 119903119894) minus
1205902119894 119860211990621198862119894 1198892119897⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) + 119872119894 (119905)119905+ ln 119868119894 (0)119905
(70)
where 119872119894(119905) = int1199050120590119894(120591)119878(120591)(119886119894(120591) + 119868119894(120591))d119861119894(120591) 119894 = 1 2
which is a local continuous martingale with 119872119894(0) = 0 ByLemma 3 we have
lim119905rarr+infin
119872119894 (119905)119905 = 0 119894 = 1 2 (71)
Taking the limit superior of both sides of (69) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119879 lt 0 (72)
which implies lim119905rarr+infin119868119894(119905) = 0
Taking the superior limit of both sides of (70) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) lt 0 (73)
which implies lim119905rarr+infin119868119894(119905) = 0 119894 = 1 2This completes theproof
Remark 7 Theorem 6 shows that the two diseases will dieout if the white noise disturbance is large or the white noisedisturbance is not large and R119894 lt 1 When ⟨1205902119894 ⟩119879 gt⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectious diseases of system(3) die out almost surely that is to say large white noisestochastic disturbance can lead to the two epidemics beingextinct
6 Numerical Simulations
Now we introduce some numerical simulations exampleswhich illustrate our theoretical results
Example 8 In model (3) let
119860 (119905) = 05 + 01 sin1205871199051198861 (119905) = 03 + 01 sin1205871199051198862 (119905) = 031 + 01 sin1205871199051205731 (119905) = 062 + 01 sin1205871199051205732 (119905) = 062 + 01 sin1205871199051199031 (119905) = 02 + 01 sin1205871199051199032 (119905) = 035 + 01 sin120587119905119889 (119905) = 02 + 01 sin1205871199051205721 (119905) = 02 + 01 sin1205871199051205722 (119905) = 025 + 01 sin1205871199051205901 (119905) = 01 + 005 sin1205871199051205902 (119905) = 01 + 005 sin120587119905
(74)
Note that R gt 1 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897and 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 hold that isthe conditions of Theorem 5 hold Hence system (3) has apositive periodic solution with 119879 = 1 Figure 1(a) shows theperiodicity of the nonautonomous stochastic model (3) with1205901 = 0 and 1205902 = 0 Figure 1(b) shows that solution of thenonautonomous stochastic model (3) with the initial value(119878(119905) 1198681(119905) 1198682(119905)) = (03 015 015) tends to a periodic orbitin the sense of joint distribution
Example 9 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin1205871199051198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin120587119905
Complexity 11
S(t)
I1(t)
I2(t)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
(a)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 1 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (1 02 02) to the nonautonomous stochastic model (3)
0 25 50 75 100t
59 60 61
078
08
082
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
59 60 61
078
08
082
0
1
2
14
14
S(t)I 1(t)I 2(t)
25 50 75 1000t
S(t)
I1(t)
I2(t)
(b)
Figure 2 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 04 + 02 sin1205871199051205902 (119905) = 05 + 02 sin120587119905
(75)
Note that ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897+120572119897119894+119903119897119894⟩119879Therefore conditions(i) of Theorem 6 hold Then the two infectious diseases willgo to extinction Figure 2(a) shows that one of two diseases inthe deterministic SIS epidemic model is extinct and the otheris persistent Figure 2(b) shows that the two diseases will dieout under the large white noise disturbance of model (3)
Example 10 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin120587119905
12 Complexity
25 50 75 1000t
0
05
1
15
2S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
0 25 50 75 100t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 3 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin1205871199051205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 02 + 02 sin1205871199051205902 (119905) = 02 + 02 sin120587119905
(76)
Note that R1 lt 1 R2 lt 1 and ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 That is conditions (ii) of Theorem 6 hold Thenthe two infectious diseases will go to extinction Figure 3(a)
shows that one of two diseases in the deterministic SISepidemic model is extinct and the other is persistent withoutthe white noises Figure 3(b) shows that the two diseases willdie out under a small white noise disturbance of model (3)
7 Discussion and Conclusions
This paper explores the existence of nontrivial positiveT-periodic solution of a nonautonomous stochastic SISepidemic model with nonlinear growth rate and doubleepidemic hypothesis By constructing a suitable stochasticLyapunov function we establish sufficient conditions for theexistence of nontrivial positive 119879-periodic solution of system(3) Furthermore the sufficient conditions for the extinctionof the two diseases are obtained Our results are given asfollows
(1) If 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 and R gt 1 hold the SISmodel has at least one nontrivial positive 119879-periodicsolution where
R = 2sum119894=1
⟨119860120572119894120573119894⟩119879⟨119889 + 120572119894 + 119903119894 + 1205902119894 119860211990621198862119894 1198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩119879 ⟨119860 + 11988611205721 + 11988621205722⟩119879 (77)
(2) If ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectiousdiseases go extinct
(3) If ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 and R119894 lt 1 the two
infectious diseases also go extinct where
R119894 = 119860119906 ⟨120573119894⟩119879119889119897 ⟨119886119894⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879minus 1198602119906 ⟨1205902119894 ⟩11987921198892119897 ⟨1198862119894 ⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879
(78)
Complexity 13
Some interesting questions deserve further investigationOn the one hand we may explore some realistic but complexmodels such as considering the effects of impulsive ordelay perturbations on system (3) On the other hand wecan concern the dynamics of a nonautonomous stochasticSIS epidemic model with two infectious diseases driven byLevy jumps What is more we can also investigate thenonautonomous stochastic SIS epidemic model with twoinfectious diseases by a continuous time Markov chain Wewill investigate these cases in our future work
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (11371230) Joint Innovative Center forSafe andEffectiveMiningTechnology andEquipment of CoalResources Shandong Province the SDUST Research Fund(2014TDJH102) and Shandong Provincial Natural ScienceFoundation China (ZR2015AQ001)
References
[1] W O Kermack and A G McKendrick ldquoA contributions to themathematical theory of epidemics (Part I)rdquo Proceedings of theRoyal Society of London A vol 115 pp 700ndash721 1927
[2] A Gray D Greenhalgh L Hu X Mao and J Pan ldquoA stochasticdifferential equation SIS epidemic modelrdquo SIAM Journal onApplied Mathematics vol 71 no 3 pp 876ndash902 2011
[3] A Korobeinikov and G C Wake ldquoLyapunov functions andglobal stability for SIR SIRS and SIS epidemiological modelsrdquoApplied Mathematics Letters vol 15 no 8 pp 955ndash960 2002
[4] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995
[5] T Zhang X Meng Y Song and T Zhang ldquoA stage-structuredpredator-prey SI model with disease in the prey and impulsiveeffectsrdquoMathematical Modelling and Analysis vol 18 no 4 pp505ndash528 2013
[6] J Li andZMa ldquoQualitative analyses of SIS epidemicmodelwithvaccination and varying total population sizerdquo Mathematicaland Computer Modelling vol 35 no 11-12 pp 1235ndash1243 2002
[7] A drsquoOnofrio ldquoA note on the global behaviour of the network-based SIS epidemic modelrdquo Nonlinear Analysis Real WorldApplications vol 9 no 4 pp 1567ndash1572 2008
[8] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[9] T Zhang X Meng T Zhang and Y Song ldquoGlobal dynamicsfor a new high-dimensional SIR model with distributed delayrdquoApplied Mathematics and Computation vol 218 no 24 pp11806ndash11819 2012
[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006
[11] X Meng S Zhao T Feng and T Zhang ldquoDynamics of a novelnonlinear stochastic SIS epidemic model with double epidemichypothesisrdquo Journal of Mathematical Analysis and Applicationsvol 433 no 1 pp 227ndash242 2016
[12] A Miao X Wang T Zhang W Wang and B Sampath ArunaPradeep ldquoDynamical analysis of a stochastic SIS epidemicmodel with nonlinear incidence rate and double epidemichypothesisrdquo Advances in Difference Equations 2017226 pages2017
[13] X-Z Meng ldquoStability of a novel stochastic epidemic modelwith double epidemic hypothesisrdquo Applied Mathematics andComputation vol 217 no 2 pp 506ndash515 2010
[14] X P Li X Y Lin and Y Q Lin ldquoLyapunov-type conditions andstochastic differential equations driven byG-BrownianmotionrdquoJournal of Mathematical Analysis and Applications vol 439 no1 pp 235ndash255 2016
[15] M Liu and M Fan ldquoPermanence of stochastic Lotka-Volterrasystemsrdquo Journal of Nonlinear Science vol 27 no 2 pp 425ndash452 2017
[16] L D Liu and X Z Meng ldquoOptimal harvesting controland dynamics of two-species stochastic model with delaysrdquoAdvances in Difference Equations vol 2017 18 pages 2017
[17] H Ma and Y Jia ldquoStability analysis for stochastic differentialequations with infiniteMarkovian switchingsrdquo Journal of Math-ematical Analysis and Applications vol 435 no 1 pp 593ndash6052016
[18] G D Liu X H Wang X Z Meng and S J Gao ldquoExtinctionand persistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexity vol 3pp 1ndash15 2017
[19] C Tan and W H Zhang ldquoOn observability and detectabilityof continuous-time stochastic Markov jump systemsrdquo Journalof Systems Science and Complexity vol 28 no 4 pp 830ndash8472015
[20] X Leng T Feng and X Meng ldquoStochastic inequalities andapplications to dynamics analysis of a novel SIVS epidemicmodel with jumpsrdquo Journal of Inequalities and ApplicationsPaper No 138 25 pages 2017
[21] G Li and M Chen ldquoInfinite horizon linear quadratic optimalcontrol for stochastic difference time-delay systemsrdquo Advancesin Difference Equations vol 2015 14 pages 2015
[22] Y Zhao and W Zhang ldquoObserver-based controller design forsingular stochastic Markov jump systems with state dependentnoiserdquo Journal of Systems Science and Complexity vol 29 no 4pp 946ndash958 2016
[23] S Zhang X Meng T Feng and T Zhang ldquoDynamics analysisand numerical simulations of a stochastic non-autonomouspredatorndashprey system with impulsive effectsrdquo Nonlinear Anal-ysis Hybrid Systems vol 26 pp 19ndash37 2017
[24] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013
[25] X Meng L Wang and T Zhang ldquoGlobal dynamics analysis ofa nonlinear impulsive stochastic chemostat system in a pollutedenvironmentrdquo Journal of AppliedAnalysis andComputation vol6 no 3 pp 865ndash875 2016
[26] X Liu Y Li andW Zhang ldquoStochastic linear quadratic optimalcontrol with constraint for discrete-time systemsrdquo AppliedMathematics and Computation vol 228 pp 264ndash270 2014
14 Complexity
[27] X Z Meng and X H Wang ldquoStochastic predator-prey systemsubject to levy jumprdquo Discrete Dynamics in Nature and Societyvol 2016 Article ID 5749892 13 pages 2016
[28] X Lv L Wang and X Meng ldquoGlobal analysis of a new non-linear stochastic differential competition systemwith impulsiveeffectrdquo Advances in Difference Equations vol 2017 296 pages2017
[29] X Zhang D Jiang A Alsaedi and T Hayat ldquoStationarydistribution of stochastic SIS epidemic model with vaccinationunder regime switchingrdquo Applied Mathematics Letters vol 59pp 87ndash93 2016
[30] F Li X Meng and Y Cui ldquoNonlinear stochastic analysis fora stochastic SIS epidemic modelrdquo Journal of Nonlinear Sciencesand Applications vol 10 no 09 pp 5116ndash5124 2017
[31] A Q Miao J Zhang T Q Zhang and B G Pradeep ldquoSampathAruna Pradeep Threshold dynamics of a stochastic SIR modelwith vertical transmission and vaccinationrdquo Computationaland Mathematical Methods in Medicine vol 2017 Article ID4820183 10 pages 2017
[32] Q Liu D Jiang N Shi T Hayat and A Alsaedi ldquoNontrivialperiodic solution of a stochastic non-autonomous SISV epi-demic modelrdquo Physica A Statistical Mechanics and its Applica-tions vol 462 pp 837ndash845 2016
[33] Y L Zhou S L Yuan and D L Zhao ldquoThreshold behavior ofa stochastic SIS model with levy jumpsrdquo Applied Mathematicsand Computation vol 275 pp 255ndash267 2016
[34] T Feng X Meng L Liu and S Gao ldquoApplication of inequal-ities technique to dynamics analysis of a stochastic eco-epidemiology modelrdquo Journal of Inequalities and ApplicationsPaper No 327 29 pages 2016
[35] M Liu and K Wang ldquoPersistence and extinction in stochasticnon-autonomous logistic systemsrdquo Journal of MathematicalAnalysis and Applications vol 375 no 2 pp 443ndash457 2011
[36] Z Bai and Y Zhou ldquoExistence of two periodic solutions for anon-autonomous SIR epidemic modelrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 1 pp 382ndash391 2011
[37] S Gao F Zhang and Y He ldquoThe effects of migratory birdpopulation in a nonautonomous eco-epidemiological modelrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 37 no 6 pp3903ndash3916 2013
[38] T Kuniya ldquoExistence of a nontrivial periodic solution in an age-structured SIR epidemicmodel with time periodic coefficientsrdquoApplied Mathematics Letters vol 27 pp 15ndash20 2014
[39] L Zu D Jiang D OrsquoRegan and B Ge ldquoPeriodic solution fora non-autonomous Lotka-Volterra predator-prey model withrandom perturbationrdquo Journal of Mathematical Analysis andApplications vol 430 no 1 pp 428ndash437 2015
[40] Y Lin D Jiang and T Liu ldquoNontrivial periodic solution ofa stochastic epidemic model with seasonal variationrdquo AppliedMathematics Letters vol 45 pp 103ndash107 2015
[41] X R Mao Stochastic Differential Equations and Their Applica-tions Horwood Chichester UK 1997
[42] R Khasminskii Stochastic Stability of Differential EquationsSpringer Berlin Germany 2011
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Complexity
1198801 = (119878 1198681 1198682) isin R3+ 0 lt 119878 lt 120598
1198802 = (119878 1198681 1198682) isin R3+ 0 lt 1198681 lt 120598
1198803 = (119878 1198681 1198682) isin R3+ 0 lt 1198682 lt 120598
1198804 = (119878 1198681 1198682) isin R3+ 119878 gt 1120598
1198805 = (119878 1198681 1198682) isin R3+ 1198681 gt 1120598
1198806 = (119878 1198681 1198682) isin R3+ 1198682 gt 1120598
(51)
Nextwewill prove that119871119881(119878 1198681 1198682) le minus1 onR3+119880 whichis equivalent to proving it on the above six domains
Case 1 If (119878 1198681 1198682) isin 1198801 one can see that
119871119881 le minus119872120582 minus 119860119897119878 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622 le minus119872120582 minus 119860119897119878 + 119862
le minus119872120582 minus 119860119897120598 + 119862
(52)
where
119862 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(53)
According to (45) we have 119871119881 le minus1 for all (119878 1198681 1198682) isin 1198801Case 2 If (119878 1198681 1198682) isin 1198802 one can get that
119871119881 le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681 + 119863le minus119872120582 + (119889119906 + 1205721199061 + 1199031199061 ) 120598 + 119863
(54)
where
119863 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(55)
In view of (46) we can obtain that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198802Case 3 If (119878 1198681 1198682) isin 1198803 we have
119871119881 le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682 + (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682 + 119864le minus119872120582 + (119889119906 + 1205721199062 + 1199031199062 ) 120598 + 119864
(56)
where
119864 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(57)
By (47) we can conclude that 119871119881 le minus1 for all (119878 1198681 1198682) isin1198803Case 4 If (119878 1198681 1198682) isin 1198804 one can derive that
119871119881 le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119878120579+1 + 119862le minus119872120582 minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119862
(58)
Complexity 9
Together with (48) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198804Case 5 If (119878 1198681 1198682) isin 1198805 it follows that
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119866le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119866
(59)
where
119866 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(60)
By virtue of (49) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198805Case 6 If (119878 1198681 1198682) isin 1198806 we obtain
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119867le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119867
(61)
where
119867 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(62)
It follows from (50) that 119871119881 le minus1 for all (119878 1198681 1198682) isin 1198806Clearly one can see from (52) (54) (56) (58) (59) and
(61) that for a sufficiently small 120598119871119881 (119878 1198681 1198682) le minus1
(119878 1198681 1198682) isin R3+119880 (63)
Hence 1198602 in Lemma 2 is satisfied This completes the proofof Theorem 5
5 Extinction
In this section we investigate the conditions for the extinc-tion of the two infectious diseases of system (3)
Let
R1 = 119860119906 ⟨1205731⟩119879119889119897 ⟨1198861⟩119879 ⟨119889 + 1205721 + 1199031⟩119879minus 1198602119906 ⟨12059021⟩11987921198892119897 ⟨11988621⟩119879 ⟨119889 + 1205721 + 1199031⟩119879
R2 = 119860119906 ⟨1205732⟩119879119889119897 ⟨1198862⟩119879 ⟨119889 + 1205722 + 1199032⟩119879minus 1198602119906 ⟨12059022⟩11987921198892119897 ⟨11988622⟩119879 ⟨119889 + 1205722 + 1199032⟩119879
(64)
Theorem 6 Let (119878(119905) 1198681(119905) 1198682(119905)) be a solution of system (3)with initial value (119878(0) 1198681(0) 1198682(0)) isin 1198773+
Then if
(i) ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792 ⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 119894 = 1 2 (65)
or(ii) R119894 lt 1⟨1205902119894 ⟩119879 le 119889119897 ⟨1198862119894 ⟩119879 ⟨120573119894⟩119879119860119906 ⟨119886119894⟩119879
119894 = 1 2(66)
hold the two infectious diseases of system (3) go to extinctionas that is
lim119905rarr+infin
119868119894 (119905) = 0 119894 = 1 2 (67)
10 Complexity
Proof Applying Itorsquos formula to system (3) we have
d ln 119868119894 (119905) = ( 120573119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905) minus (119889 (119905) + 120572119894 (119905) + 119903119894 (119905))minus 1205902119894 (119905) 1198782 (119905)2 (119886119894 (119905) + 119868119894 (119905))2) d119905 + 120590119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905)d119861119894 (119905)
119894 = 1 2
(68)
Case (i) Integrating (68) from 0 to 119905 and dividing 119905 on bothsides we obtain
ln 119868119894 (119905)119905 = minus1205902119894 (119905)2119905 int1199050( 119878 (120591)119886119894 (120591) + 119868119894 (120591) minus
120573119894 (120591)1205902119894 (120591))2
d120591minus 1119905 int
119905
0(119889 (120591) + 120572119894 (120591) + 119903119894 (120591)) d120591
+ 1119905 int119905
0
1205732119894 (120591)21205902119894 (120591)d120591 +119872119894 (119905)119905 + ln 119868119894 (0)119905
le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905
(69)
Case (ii) Integrating (68) from 0 to 119905 first and then dividingby 119905 on both sides yield
ln 119868119894 (119905)119905 = 1119905 int119905
0( 120573119894 (120591) 119878 (120591)119886119894 (120591) + 119868119894 (120591)
minus (119889 (120591) + 120572119894 (120591) + 119903119894 (120591))minus 1205902119894 (120591) 1198782 (120591)2 (119886119894 (120591) + 119868119894 (120591))2) d120591 + 119872119894 (119905)119905 + ln 119868119894 (0)119905le ⟨120573119894119860119906119886119894119889119897 minus (119889 + 120572119894 + 119903119894) minus
1205902119894 119860211990621198862119894 1198892119897⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) + 119872119894 (119905)119905+ ln 119868119894 (0)119905
(70)
where 119872119894(119905) = int1199050120590119894(120591)119878(120591)(119886119894(120591) + 119868119894(120591))d119861119894(120591) 119894 = 1 2
which is a local continuous martingale with 119872119894(0) = 0 ByLemma 3 we have
lim119905rarr+infin
119872119894 (119905)119905 = 0 119894 = 1 2 (71)
Taking the limit superior of both sides of (69) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119879 lt 0 (72)
which implies lim119905rarr+infin119868119894(119905) = 0
Taking the superior limit of both sides of (70) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) lt 0 (73)
which implies lim119905rarr+infin119868119894(119905) = 0 119894 = 1 2This completes theproof
Remark 7 Theorem 6 shows that the two diseases will dieout if the white noise disturbance is large or the white noisedisturbance is not large and R119894 lt 1 When ⟨1205902119894 ⟩119879 gt⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectious diseases of system(3) die out almost surely that is to say large white noisestochastic disturbance can lead to the two epidemics beingextinct
6 Numerical Simulations
Now we introduce some numerical simulations exampleswhich illustrate our theoretical results
Example 8 In model (3) let
119860 (119905) = 05 + 01 sin1205871199051198861 (119905) = 03 + 01 sin1205871199051198862 (119905) = 031 + 01 sin1205871199051205731 (119905) = 062 + 01 sin1205871199051205732 (119905) = 062 + 01 sin1205871199051199031 (119905) = 02 + 01 sin1205871199051199032 (119905) = 035 + 01 sin120587119905119889 (119905) = 02 + 01 sin1205871199051205721 (119905) = 02 + 01 sin1205871199051205722 (119905) = 025 + 01 sin1205871199051205901 (119905) = 01 + 005 sin1205871199051205902 (119905) = 01 + 005 sin120587119905
(74)
Note that R gt 1 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897and 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 hold that isthe conditions of Theorem 5 hold Hence system (3) has apositive periodic solution with 119879 = 1 Figure 1(a) shows theperiodicity of the nonautonomous stochastic model (3) with1205901 = 0 and 1205902 = 0 Figure 1(b) shows that solution of thenonautonomous stochastic model (3) with the initial value(119878(119905) 1198681(119905) 1198682(119905)) = (03 015 015) tends to a periodic orbitin the sense of joint distribution
Example 9 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin1205871199051198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin120587119905
Complexity 11
S(t)
I1(t)
I2(t)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
(a)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 1 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (1 02 02) to the nonautonomous stochastic model (3)
0 25 50 75 100t
59 60 61
078
08
082
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
59 60 61
078
08
082
0
1
2
14
14
S(t)I 1(t)I 2(t)
25 50 75 1000t
S(t)
I1(t)
I2(t)
(b)
Figure 2 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 04 + 02 sin1205871199051205902 (119905) = 05 + 02 sin120587119905
(75)
Note that ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897+120572119897119894+119903119897119894⟩119879Therefore conditions(i) of Theorem 6 hold Then the two infectious diseases willgo to extinction Figure 2(a) shows that one of two diseases inthe deterministic SIS epidemic model is extinct and the otheris persistent Figure 2(b) shows that the two diseases will dieout under the large white noise disturbance of model (3)
Example 10 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin120587119905
12 Complexity
25 50 75 1000t
0
05
1
15
2S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
0 25 50 75 100t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 3 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin1205871199051205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 02 + 02 sin1205871199051205902 (119905) = 02 + 02 sin120587119905
(76)
Note that R1 lt 1 R2 lt 1 and ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 That is conditions (ii) of Theorem 6 hold Thenthe two infectious diseases will go to extinction Figure 3(a)
shows that one of two diseases in the deterministic SISepidemic model is extinct and the other is persistent withoutthe white noises Figure 3(b) shows that the two diseases willdie out under a small white noise disturbance of model (3)
7 Discussion and Conclusions
This paper explores the existence of nontrivial positiveT-periodic solution of a nonautonomous stochastic SISepidemic model with nonlinear growth rate and doubleepidemic hypothesis By constructing a suitable stochasticLyapunov function we establish sufficient conditions for theexistence of nontrivial positive 119879-periodic solution of system(3) Furthermore the sufficient conditions for the extinctionof the two diseases are obtained Our results are given asfollows
(1) If 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 and R gt 1 hold the SISmodel has at least one nontrivial positive 119879-periodicsolution where
R = 2sum119894=1
⟨119860120572119894120573119894⟩119879⟨119889 + 120572119894 + 119903119894 + 1205902119894 119860211990621198862119894 1198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩119879 ⟨119860 + 11988611205721 + 11988621205722⟩119879 (77)
(2) If ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectiousdiseases go extinct
(3) If ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 and R119894 lt 1 the two
infectious diseases also go extinct where
R119894 = 119860119906 ⟨120573119894⟩119879119889119897 ⟨119886119894⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879minus 1198602119906 ⟨1205902119894 ⟩11987921198892119897 ⟨1198862119894 ⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879
(78)
Complexity 13
Some interesting questions deserve further investigationOn the one hand we may explore some realistic but complexmodels such as considering the effects of impulsive ordelay perturbations on system (3) On the other hand wecan concern the dynamics of a nonautonomous stochasticSIS epidemic model with two infectious diseases driven byLevy jumps What is more we can also investigate thenonautonomous stochastic SIS epidemic model with twoinfectious diseases by a continuous time Markov chain Wewill investigate these cases in our future work
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (11371230) Joint Innovative Center forSafe andEffectiveMiningTechnology andEquipment of CoalResources Shandong Province the SDUST Research Fund(2014TDJH102) and Shandong Provincial Natural ScienceFoundation China (ZR2015AQ001)
References
[1] W O Kermack and A G McKendrick ldquoA contributions to themathematical theory of epidemics (Part I)rdquo Proceedings of theRoyal Society of London A vol 115 pp 700ndash721 1927
[2] A Gray D Greenhalgh L Hu X Mao and J Pan ldquoA stochasticdifferential equation SIS epidemic modelrdquo SIAM Journal onApplied Mathematics vol 71 no 3 pp 876ndash902 2011
[3] A Korobeinikov and G C Wake ldquoLyapunov functions andglobal stability for SIR SIRS and SIS epidemiological modelsrdquoApplied Mathematics Letters vol 15 no 8 pp 955ndash960 2002
[4] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995
[5] T Zhang X Meng Y Song and T Zhang ldquoA stage-structuredpredator-prey SI model with disease in the prey and impulsiveeffectsrdquoMathematical Modelling and Analysis vol 18 no 4 pp505ndash528 2013
[6] J Li andZMa ldquoQualitative analyses of SIS epidemicmodelwithvaccination and varying total population sizerdquo Mathematicaland Computer Modelling vol 35 no 11-12 pp 1235ndash1243 2002
[7] A drsquoOnofrio ldquoA note on the global behaviour of the network-based SIS epidemic modelrdquo Nonlinear Analysis Real WorldApplications vol 9 no 4 pp 1567ndash1572 2008
[8] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[9] T Zhang X Meng T Zhang and Y Song ldquoGlobal dynamicsfor a new high-dimensional SIR model with distributed delayrdquoApplied Mathematics and Computation vol 218 no 24 pp11806ndash11819 2012
[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006
[11] X Meng S Zhao T Feng and T Zhang ldquoDynamics of a novelnonlinear stochastic SIS epidemic model with double epidemichypothesisrdquo Journal of Mathematical Analysis and Applicationsvol 433 no 1 pp 227ndash242 2016
[12] A Miao X Wang T Zhang W Wang and B Sampath ArunaPradeep ldquoDynamical analysis of a stochastic SIS epidemicmodel with nonlinear incidence rate and double epidemichypothesisrdquo Advances in Difference Equations 2017226 pages2017
[13] X-Z Meng ldquoStability of a novel stochastic epidemic modelwith double epidemic hypothesisrdquo Applied Mathematics andComputation vol 217 no 2 pp 506ndash515 2010
[14] X P Li X Y Lin and Y Q Lin ldquoLyapunov-type conditions andstochastic differential equations driven byG-BrownianmotionrdquoJournal of Mathematical Analysis and Applications vol 439 no1 pp 235ndash255 2016
[15] M Liu and M Fan ldquoPermanence of stochastic Lotka-Volterrasystemsrdquo Journal of Nonlinear Science vol 27 no 2 pp 425ndash452 2017
[16] L D Liu and X Z Meng ldquoOptimal harvesting controland dynamics of two-species stochastic model with delaysrdquoAdvances in Difference Equations vol 2017 18 pages 2017
[17] H Ma and Y Jia ldquoStability analysis for stochastic differentialequations with infiniteMarkovian switchingsrdquo Journal of Math-ematical Analysis and Applications vol 435 no 1 pp 593ndash6052016
[18] G D Liu X H Wang X Z Meng and S J Gao ldquoExtinctionand persistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexity vol 3pp 1ndash15 2017
[19] C Tan and W H Zhang ldquoOn observability and detectabilityof continuous-time stochastic Markov jump systemsrdquo Journalof Systems Science and Complexity vol 28 no 4 pp 830ndash8472015
[20] X Leng T Feng and X Meng ldquoStochastic inequalities andapplications to dynamics analysis of a novel SIVS epidemicmodel with jumpsrdquo Journal of Inequalities and ApplicationsPaper No 138 25 pages 2017
[21] G Li and M Chen ldquoInfinite horizon linear quadratic optimalcontrol for stochastic difference time-delay systemsrdquo Advancesin Difference Equations vol 2015 14 pages 2015
[22] Y Zhao and W Zhang ldquoObserver-based controller design forsingular stochastic Markov jump systems with state dependentnoiserdquo Journal of Systems Science and Complexity vol 29 no 4pp 946ndash958 2016
[23] S Zhang X Meng T Feng and T Zhang ldquoDynamics analysisand numerical simulations of a stochastic non-autonomouspredatorndashprey system with impulsive effectsrdquo Nonlinear Anal-ysis Hybrid Systems vol 26 pp 19ndash37 2017
[24] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013
[25] X Meng L Wang and T Zhang ldquoGlobal dynamics analysis ofa nonlinear impulsive stochastic chemostat system in a pollutedenvironmentrdquo Journal of AppliedAnalysis andComputation vol6 no 3 pp 865ndash875 2016
[26] X Liu Y Li andW Zhang ldquoStochastic linear quadratic optimalcontrol with constraint for discrete-time systemsrdquo AppliedMathematics and Computation vol 228 pp 264ndash270 2014
14 Complexity
[27] X Z Meng and X H Wang ldquoStochastic predator-prey systemsubject to levy jumprdquo Discrete Dynamics in Nature and Societyvol 2016 Article ID 5749892 13 pages 2016
[28] X Lv L Wang and X Meng ldquoGlobal analysis of a new non-linear stochastic differential competition systemwith impulsiveeffectrdquo Advances in Difference Equations vol 2017 296 pages2017
[29] X Zhang D Jiang A Alsaedi and T Hayat ldquoStationarydistribution of stochastic SIS epidemic model with vaccinationunder regime switchingrdquo Applied Mathematics Letters vol 59pp 87ndash93 2016
[30] F Li X Meng and Y Cui ldquoNonlinear stochastic analysis fora stochastic SIS epidemic modelrdquo Journal of Nonlinear Sciencesand Applications vol 10 no 09 pp 5116ndash5124 2017
[31] A Q Miao J Zhang T Q Zhang and B G Pradeep ldquoSampathAruna Pradeep Threshold dynamics of a stochastic SIR modelwith vertical transmission and vaccinationrdquo Computationaland Mathematical Methods in Medicine vol 2017 Article ID4820183 10 pages 2017
[32] Q Liu D Jiang N Shi T Hayat and A Alsaedi ldquoNontrivialperiodic solution of a stochastic non-autonomous SISV epi-demic modelrdquo Physica A Statistical Mechanics and its Applica-tions vol 462 pp 837ndash845 2016
[33] Y L Zhou S L Yuan and D L Zhao ldquoThreshold behavior ofa stochastic SIS model with levy jumpsrdquo Applied Mathematicsand Computation vol 275 pp 255ndash267 2016
[34] T Feng X Meng L Liu and S Gao ldquoApplication of inequal-ities technique to dynamics analysis of a stochastic eco-epidemiology modelrdquo Journal of Inequalities and ApplicationsPaper No 327 29 pages 2016
[35] M Liu and K Wang ldquoPersistence and extinction in stochasticnon-autonomous logistic systemsrdquo Journal of MathematicalAnalysis and Applications vol 375 no 2 pp 443ndash457 2011
[36] Z Bai and Y Zhou ldquoExistence of two periodic solutions for anon-autonomous SIR epidemic modelrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 1 pp 382ndash391 2011
[37] S Gao F Zhang and Y He ldquoThe effects of migratory birdpopulation in a nonautonomous eco-epidemiological modelrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 37 no 6 pp3903ndash3916 2013
[38] T Kuniya ldquoExistence of a nontrivial periodic solution in an age-structured SIR epidemicmodel with time periodic coefficientsrdquoApplied Mathematics Letters vol 27 pp 15ndash20 2014
[39] L Zu D Jiang D OrsquoRegan and B Ge ldquoPeriodic solution fora non-autonomous Lotka-Volterra predator-prey model withrandom perturbationrdquo Journal of Mathematical Analysis andApplications vol 430 no 1 pp 428ndash437 2015
[40] Y Lin D Jiang and T Liu ldquoNontrivial periodic solution ofa stochastic epidemic model with seasonal variationrdquo AppliedMathematics Letters vol 45 pp 103ndash107 2015
[41] X R Mao Stochastic Differential Equations and Their Applica-tions Horwood Chichester UK 1997
[42] R Khasminskii Stochastic Stability of Differential EquationsSpringer Berlin Germany 2011
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Complexity 9
Together with (48) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198804Case 5 If (119878 1198681 1198682) isin 1198805 it follows that
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + 119866le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119866
(59)
where
119866 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(60)
By virtue of (49) we can deduce that 119871119881 le minus1 for all(119878 1198681 1198682) isin 1198805Case 6 If (119878 1198681 1198682) isin 1198806 we obtain
119871119881 le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12+ (119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11+ (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061+ 1205731199062 + 120590211990612 + 120590211990622
le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119867le minus119872120582 minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 1120598120579+1 + 119867
(61)
where
119867 = sup(1198781198681 1198682)isin119877
3+
(119889119906 + 1205721199061 + 1199031199061 ) 1198681minus 12 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+11 + (119889119906 + 1205721199062 + 1199031199062 ) 1198682minus 14 (119889119897 minus 120579 (12059021199061 + 12059021199062 )) 119868120579+12 + 119874 + 119889119906 + 1205731199061 + 1205731199062+ 120590211990612 + 120590211990622
(62)
It follows from (50) that 119871119881 le minus1 for all (119878 1198681 1198682) isin 1198806Clearly one can see from (52) (54) (56) (58) (59) and
(61) that for a sufficiently small 120598119871119881 (119878 1198681 1198682) le minus1
(119878 1198681 1198682) isin R3+119880 (63)
Hence 1198602 in Lemma 2 is satisfied This completes the proofof Theorem 5
5 Extinction
In this section we investigate the conditions for the extinc-tion of the two infectious diseases of system (3)
Let
R1 = 119860119906 ⟨1205731⟩119879119889119897 ⟨1198861⟩119879 ⟨119889 + 1205721 + 1199031⟩119879minus 1198602119906 ⟨12059021⟩11987921198892119897 ⟨11988621⟩119879 ⟨119889 + 1205721 + 1199031⟩119879
R2 = 119860119906 ⟨1205732⟩119879119889119897 ⟨1198862⟩119879 ⟨119889 + 1205722 + 1199032⟩119879minus 1198602119906 ⟨12059022⟩11987921198892119897 ⟨11988622⟩119879 ⟨119889 + 1205722 + 1199032⟩119879
(64)
Theorem 6 Let (119878(119905) 1198681(119905) 1198682(119905)) be a solution of system (3)with initial value (119878(0) 1198681(0) 1198682(0)) isin 1198773+
Then if
(i) ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792 ⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 119894 = 1 2 (65)
or(ii) R119894 lt 1⟨1205902119894 ⟩119879 le 119889119897 ⟨1198862119894 ⟩119879 ⟨120573119894⟩119879119860119906 ⟨119886119894⟩119879
119894 = 1 2(66)
hold the two infectious diseases of system (3) go to extinctionas that is
lim119905rarr+infin
119868119894 (119905) = 0 119894 = 1 2 (67)
10 Complexity
Proof Applying Itorsquos formula to system (3) we have
d ln 119868119894 (119905) = ( 120573119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905) minus (119889 (119905) + 120572119894 (119905) + 119903119894 (119905))minus 1205902119894 (119905) 1198782 (119905)2 (119886119894 (119905) + 119868119894 (119905))2) d119905 + 120590119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905)d119861119894 (119905)
119894 = 1 2
(68)
Case (i) Integrating (68) from 0 to 119905 and dividing 119905 on bothsides we obtain
ln 119868119894 (119905)119905 = minus1205902119894 (119905)2119905 int1199050( 119878 (120591)119886119894 (120591) + 119868119894 (120591) minus
120573119894 (120591)1205902119894 (120591))2
d120591minus 1119905 int
119905
0(119889 (120591) + 120572119894 (120591) + 119903119894 (120591)) d120591
+ 1119905 int119905
0
1205732119894 (120591)21205902119894 (120591)d120591 +119872119894 (119905)119905 + ln 119868119894 (0)119905
le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905
(69)
Case (ii) Integrating (68) from 0 to 119905 first and then dividingby 119905 on both sides yield
ln 119868119894 (119905)119905 = 1119905 int119905
0( 120573119894 (120591) 119878 (120591)119886119894 (120591) + 119868119894 (120591)
minus (119889 (120591) + 120572119894 (120591) + 119903119894 (120591))minus 1205902119894 (120591) 1198782 (120591)2 (119886119894 (120591) + 119868119894 (120591))2) d120591 + 119872119894 (119905)119905 + ln 119868119894 (0)119905le ⟨120573119894119860119906119886119894119889119897 minus (119889 + 120572119894 + 119903119894) minus
1205902119894 119860211990621198862119894 1198892119897⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) + 119872119894 (119905)119905+ ln 119868119894 (0)119905
(70)
where 119872119894(119905) = int1199050120590119894(120591)119878(120591)(119886119894(120591) + 119868119894(120591))d119861119894(120591) 119894 = 1 2
which is a local continuous martingale with 119872119894(0) = 0 ByLemma 3 we have
lim119905rarr+infin
119872119894 (119905)119905 = 0 119894 = 1 2 (71)
Taking the limit superior of both sides of (69) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119879 lt 0 (72)
which implies lim119905rarr+infin119868119894(119905) = 0
Taking the superior limit of both sides of (70) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) lt 0 (73)
which implies lim119905rarr+infin119868119894(119905) = 0 119894 = 1 2This completes theproof
Remark 7 Theorem 6 shows that the two diseases will dieout if the white noise disturbance is large or the white noisedisturbance is not large and R119894 lt 1 When ⟨1205902119894 ⟩119879 gt⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectious diseases of system(3) die out almost surely that is to say large white noisestochastic disturbance can lead to the two epidemics beingextinct
6 Numerical Simulations
Now we introduce some numerical simulations exampleswhich illustrate our theoretical results
Example 8 In model (3) let
119860 (119905) = 05 + 01 sin1205871199051198861 (119905) = 03 + 01 sin1205871199051198862 (119905) = 031 + 01 sin1205871199051205731 (119905) = 062 + 01 sin1205871199051205732 (119905) = 062 + 01 sin1205871199051199031 (119905) = 02 + 01 sin1205871199051199032 (119905) = 035 + 01 sin120587119905119889 (119905) = 02 + 01 sin1205871199051205721 (119905) = 02 + 01 sin1205871199051205722 (119905) = 025 + 01 sin1205871199051205901 (119905) = 01 + 005 sin1205871199051205902 (119905) = 01 + 005 sin120587119905
(74)
Note that R gt 1 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897and 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 hold that isthe conditions of Theorem 5 hold Hence system (3) has apositive periodic solution with 119879 = 1 Figure 1(a) shows theperiodicity of the nonautonomous stochastic model (3) with1205901 = 0 and 1205902 = 0 Figure 1(b) shows that solution of thenonautonomous stochastic model (3) with the initial value(119878(119905) 1198681(119905) 1198682(119905)) = (03 015 015) tends to a periodic orbitin the sense of joint distribution
Example 9 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin1205871199051198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin120587119905
Complexity 11
S(t)
I1(t)
I2(t)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
(a)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 1 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (1 02 02) to the nonautonomous stochastic model (3)
0 25 50 75 100t
59 60 61
078
08
082
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
59 60 61
078
08
082
0
1
2
14
14
S(t)I 1(t)I 2(t)
25 50 75 1000t
S(t)
I1(t)
I2(t)
(b)
Figure 2 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 04 + 02 sin1205871199051205902 (119905) = 05 + 02 sin120587119905
(75)
Note that ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897+120572119897119894+119903119897119894⟩119879Therefore conditions(i) of Theorem 6 hold Then the two infectious diseases willgo to extinction Figure 2(a) shows that one of two diseases inthe deterministic SIS epidemic model is extinct and the otheris persistent Figure 2(b) shows that the two diseases will dieout under the large white noise disturbance of model (3)
Example 10 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin120587119905
12 Complexity
25 50 75 1000t
0
05
1
15
2S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
0 25 50 75 100t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 3 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin1205871199051205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 02 + 02 sin1205871199051205902 (119905) = 02 + 02 sin120587119905
(76)
Note that R1 lt 1 R2 lt 1 and ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 That is conditions (ii) of Theorem 6 hold Thenthe two infectious diseases will go to extinction Figure 3(a)
shows that one of two diseases in the deterministic SISepidemic model is extinct and the other is persistent withoutthe white noises Figure 3(b) shows that the two diseases willdie out under a small white noise disturbance of model (3)
7 Discussion and Conclusions
This paper explores the existence of nontrivial positiveT-periodic solution of a nonautonomous stochastic SISepidemic model with nonlinear growth rate and doubleepidemic hypothesis By constructing a suitable stochasticLyapunov function we establish sufficient conditions for theexistence of nontrivial positive 119879-periodic solution of system(3) Furthermore the sufficient conditions for the extinctionof the two diseases are obtained Our results are given asfollows
(1) If 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 and R gt 1 hold the SISmodel has at least one nontrivial positive 119879-periodicsolution where
R = 2sum119894=1
⟨119860120572119894120573119894⟩119879⟨119889 + 120572119894 + 119903119894 + 1205902119894 119860211990621198862119894 1198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩119879 ⟨119860 + 11988611205721 + 11988621205722⟩119879 (77)
(2) If ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectiousdiseases go extinct
(3) If ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 and R119894 lt 1 the two
infectious diseases also go extinct where
R119894 = 119860119906 ⟨120573119894⟩119879119889119897 ⟨119886119894⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879minus 1198602119906 ⟨1205902119894 ⟩11987921198892119897 ⟨1198862119894 ⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879
(78)
Complexity 13
Some interesting questions deserve further investigationOn the one hand we may explore some realistic but complexmodels such as considering the effects of impulsive ordelay perturbations on system (3) On the other hand wecan concern the dynamics of a nonautonomous stochasticSIS epidemic model with two infectious diseases driven byLevy jumps What is more we can also investigate thenonautonomous stochastic SIS epidemic model with twoinfectious diseases by a continuous time Markov chain Wewill investigate these cases in our future work
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (11371230) Joint Innovative Center forSafe andEffectiveMiningTechnology andEquipment of CoalResources Shandong Province the SDUST Research Fund(2014TDJH102) and Shandong Provincial Natural ScienceFoundation China (ZR2015AQ001)
References
[1] W O Kermack and A G McKendrick ldquoA contributions to themathematical theory of epidemics (Part I)rdquo Proceedings of theRoyal Society of London A vol 115 pp 700ndash721 1927
[2] A Gray D Greenhalgh L Hu X Mao and J Pan ldquoA stochasticdifferential equation SIS epidemic modelrdquo SIAM Journal onApplied Mathematics vol 71 no 3 pp 876ndash902 2011
[3] A Korobeinikov and G C Wake ldquoLyapunov functions andglobal stability for SIR SIRS and SIS epidemiological modelsrdquoApplied Mathematics Letters vol 15 no 8 pp 955ndash960 2002
[4] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995
[5] T Zhang X Meng Y Song and T Zhang ldquoA stage-structuredpredator-prey SI model with disease in the prey and impulsiveeffectsrdquoMathematical Modelling and Analysis vol 18 no 4 pp505ndash528 2013
[6] J Li andZMa ldquoQualitative analyses of SIS epidemicmodelwithvaccination and varying total population sizerdquo Mathematicaland Computer Modelling vol 35 no 11-12 pp 1235ndash1243 2002
[7] A drsquoOnofrio ldquoA note on the global behaviour of the network-based SIS epidemic modelrdquo Nonlinear Analysis Real WorldApplications vol 9 no 4 pp 1567ndash1572 2008
[8] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[9] T Zhang X Meng T Zhang and Y Song ldquoGlobal dynamicsfor a new high-dimensional SIR model with distributed delayrdquoApplied Mathematics and Computation vol 218 no 24 pp11806ndash11819 2012
[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006
[11] X Meng S Zhao T Feng and T Zhang ldquoDynamics of a novelnonlinear stochastic SIS epidemic model with double epidemichypothesisrdquo Journal of Mathematical Analysis and Applicationsvol 433 no 1 pp 227ndash242 2016
[12] A Miao X Wang T Zhang W Wang and B Sampath ArunaPradeep ldquoDynamical analysis of a stochastic SIS epidemicmodel with nonlinear incidence rate and double epidemichypothesisrdquo Advances in Difference Equations 2017226 pages2017
[13] X-Z Meng ldquoStability of a novel stochastic epidemic modelwith double epidemic hypothesisrdquo Applied Mathematics andComputation vol 217 no 2 pp 506ndash515 2010
[14] X P Li X Y Lin and Y Q Lin ldquoLyapunov-type conditions andstochastic differential equations driven byG-BrownianmotionrdquoJournal of Mathematical Analysis and Applications vol 439 no1 pp 235ndash255 2016
[15] M Liu and M Fan ldquoPermanence of stochastic Lotka-Volterrasystemsrdquo Journal of Nonlinear Science vol 27 no 2 pp 425ndash452 2017
[16] L D Liu and X Z Meng ldquoOptimal harvesting controland dynamics of two-species stochastic model with delaysrdquoAdvances in Difference Equations vol 2017 18 pages 2017
[17] H Ma and Y Jia ldquoStability analysis for stochastic differentialequations with infiniteMarkovian switchingsrdquo Journal of Math-ematical Analysis and Applications vol 435 no 1 pp 593ndash6052016
[18] G D Liu X H Wang X Z Meng and S J Gao ldquoExtinctionand persistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexity vol 3pp 1ndash15 2017
[19] C Tan and W H Zhang ldquoOn observability and detectabilityof continuous-time stochastic Markov jump systemsrdquo Journalof Systems Science and Complexity vol 28 no 4 pp 830ndash8472015
[20] X Leng T Feng and X Meng ldquoStochastic inequalities andapplications to dynamics analysis of a novel SIVS epidemicmodel with jumpsrdquo Journal of Inequalities and ApplicationsPaper No 138 25 pages 2017
[21] G Li and M Chen ldquoInfinite horizon linear quadratic optimalcontrol for stochastic difference time-delay systemsrdquo Advancesin Difference Equations vol 2015 14 pages 2015
[22] Y Zhao and W Zhang ldquoObserver-based controller design forsingular stochastic Markov jump systems with state dependentnoiserdquo Journal of Systems Science and Complexity vol 29 no 4pp 946ndash958 2016
[23] S Zhang X Meng T Feng and T Zhang ldquoDynamics analysisand numerical simulations of a stochastic non-autonomouspredatorndashprey system with impulsive effectsrdquo Nonlinear Anal-ysis Hybrid Systems vol 26 pp 19ndash37 2017
[24] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013
[25] X Meng L Wang and T Zhang ldquoGlobal dynamics analysis ofa nonlinear impulsive stochastic chemostat system in a pollutedenvironmentrdquo Journal of AppliedAnalysis andComputation vol6 no 3 pp 865ndash875 2016
[26] X Liu Y Li andW Zhang ldquoStochastic linear quadratic optimalcontrol with constraint for discrete-time systemsrdquo AppliedMathematics and Computation vol 228 pp 264ndash270 2014
14 Complexity
[27] X Z Meng and X H Wang ldquoStochastic predator-prey systemsubject to levy jumprdquo Discrete Dynamics in Nature and Societyvol 2016 Article ID 5749892 13 pages 2016
[28] X Lv L Wang and X Meng ldquoGlobal analysis of a new non-linear stochastic differential competition systemwith impulsiveeffectrdquo Advances in Difference Equations vol 2017 296 pages2017
[29] X Zhang D Jiang A Alsaedi and T Hayat ldquoStationarydistribution of stochastic SIS epidemic model with vaccinationunder regime switchingrdquo Applied Mathematics Letters vol 59pp 87ndash93 2016
[30] F Li X Meng and Y Cui ldquoNonlinear stochastic analysis fora stochastic SIS epidemic modelrdquo Journal of Nonlinear Sciencesand Applications vol 10 no 09 pp 5116ndash5124 2017
[31] A Q Miao J Zhang T Q Zhang and B G Pradeep ldquoSampathAruna Pradeep Threshold dynamics of a stochastic SIR modelwith vertical transmission and vaccinationrdquo Computationaland Mathematical Methods in Medicine vol 2017 Article ID4820183 10 pages 2017
[32] Q Liu D Jiang N Shi T Hayat and A Alsaedi ldquoNontrivialperiodic solution of a stochastic non-autonomous SISV epi-demic modelrdquo Physica A Statistical Mechanics and its Applica-tions vol 462 pp 837ndash845 2016
[33] Y L Zhou S L Yuan and D L Zhao ldquoThreshold behavior ofa stochastic SIS model with levy jumpsrdquo Applied Mathematicsand Computation vol 275 pp 255ndash267 2016
[34] T Feng X Meng L Liu and S Gao ldquoApplication of inequal-ities technique to dynamics analysis of a stochastic eco-epidemiology modelrdquo Journal of Inequalities and ApplicationsPaper No 327 29 pages 2016
[35] M Liu and K Wang ldquoPersistence and extinction in stochasticnon-autonomous logistic systemsrdquo Journal of MathematicalAnalysis and Applications vol 375 no 2 pp 443ndash457 2011
[36] Z Bai and Y Zhou ldquoExistence of two periodic solutions for anon-autonomous SIR epidemic modelrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 1 pp 382ndash391 2011
[37] S Gao F Zhang and Y He ldquoThe effects of migratory birdpopulation in a nonautonomous eco-epidemiological modelrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 37 no 6 pp3903ndash3916 2013
[38] T Kuniya ldquoExistence of a nontrivial periodic solution in an age-structured SIR epidemicmodel with time periodic coefficientsrdquoApplied Mathematics Letters vol 27 pp 15ndash20 2014
[39] L Zu D Jiang D OrsquoRegan and B Ge ldquoPeriodic solution fora non-autonomous Lotka-Volterra predator-prey model withrandom perturbationrdquo Journal of Mathematical Analysis andApplications vol 430 no 1 pp 428ndash437 2015
[40] Y Lin D Jiang and T Liu ldquoNontrivial periodic solution ofa stochastic epidemic model with seasonal variationrdquo AppliedMathematics Letters vol 45 pp 103ndash107 2015
[41] X R Mao Stochastic Differential Equations and Their Applica-tions Horwood Chichester UK 1997
[42] R Khasminskii Stochastic Stability of Differential EquationsSpringer Berlin Germany 2011
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Complexity
Proof Applying Itorsquos formula to system (3) we have
d ln 119868119894 (119905) = ( 120573119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905) minus (119889 (119905) + 120572119894 (119905) + 119903119894 (119905))minus 1205902119894 (119905) 1198782 (119905)2 (119886119894 (119905) + 119868119894 (119905))2) d119905 + 120590119894 (119905) 119878 (119905)119886119894 (119905) + 119868119894 (119905)d119861119894 (119905)
119894 = 1 2
(68)
Case (i) Integrating (68) from 0 to 119905 and dividing 119905 on bothsides we obtain
ln 119868119894 (119905)119905 = minus1205902119894 (119905)2119905 int1199050( 119878 (120591)119886119894 (120591) + 119868119894 (120591) minus
120573119894 (120591)1205902119894 (120591))2
d120591minus 1119905 int
119905
0(119889 (120591) + 120572119894 (120591) + 119903119894 (120591)) d120591
+ 1119905 int119905
0
1205732119894 (120591)21205902119894 (120591)d120591 +119872119894 (119905)119905 + ln 119868119894 (0)119905
le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905
(69)
Case (ii) Integrating (68) from 0 to 119905 first and then dividingby 119905 on both sides yield
ln 119868119894 (119905)119905 = 1119905 int119905
0( 120573119894 (120591) 119878 (120591)119886119894 (120591) + 119868119894 (120591)
minus (119889 (120591) + 120572119894 (120591) + 119903119894 (120591))minus 1205902119894 (120591) 1198782 (120591)2 (119886119894 (120591) + 119868119894 (120591))2) d120591 + 119872119894 (119905)119905 + ln 119868119894 (0)119905le ⟨120573119894119860119906119886119894119889119897 minus (119889 + 120572119894 + 119903119894) minus
1205902119894 119860211990621198862119894 1198892119897⟩119905 +119872119894 (119905)119905
+ ln 119868119894 (0)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) + 119872119894 (119905)119905+ ln 119868119894 (0)119905
(70)
where 119872119894(119905) = int1199050120590119894(120591)119878(120591)(119886119894(120591) + 119868119894(120591))d119861119894(120591) 119894 = 1 2
which is a local continuous martingale with 119872119894(0) = 0 ByLemma 3 we have
lim119905rarr+infin
119872119894 (119905)119905 = 0 119894 = 1 2 (71)
Taking the limit superior of both sides of (69) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le minus⟨119889 + 120572119894 + 119903119894 minus 120573211989421205902119894 ⟩119879 lt 0 (72)
which implies lim119905rarr+infin119868119894(119905) = 0
Taking the superior limit of both sides of (70) leads to
lim sup119905rarr+infin
ln 119868119894 (119905)119905 le ⟨119889 + 120572119894 + 119903119894⟩119879 (R119894 minus 1) lt 0 (73)
which implies lim119905rarr+infin119868119894(119905) = 0 119894 = 1 2This completes theproof
Remark 7 Theorem 6 shows that the two diseases will dieout if the white noise disturbance is large or the white noisedisturbance is not large and R119894 lt 1 When ⟨1205902119894 ⟩119879 gt⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectious diseases of system(3) die out almost surely that is to say large white noisestochastic disturbance can lead to the two epidemics beingextinct
6 Numerical Simulations
Now we introduce some numerical simulations exampleswhich illustrate our theoretical results
Example 8 In model (3) let
119860 (119905) = 05 + 01 sin1205871199051198861 (119905) = 03 + 01 sin1205871199051198862 (119905) = 031 + 01 sin1205871199051205731 (119905) = 062 + 01 sin1205871199051205732 (119905) = 062 + 01 sin1205871199051199031 (119905) = 02 + 01 sin1205871199051199032 (119905) = 035 + 01 sin120587119905119889 (119905) = 02 + 01 sin1205871199051205721 (119905) = 02 + 01 sin1205871199051205722 (119905) = 025 + 01 sin1205871199051205901 (119905) = 01 + 005 sin1205871199051205902 (119905) = 01 + 005 sin120587119905
(74)
Note that R gt 1 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897and 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 hold that isthe conditions of Theorem 5 hold Hence system (3) has apositive periodic solution with 119879 = 1 Figure 1(a) shows theperiodicity of the nonautonomous stochastic model (3) with1205901 = 0 and 1205902 = 0 Figure 1(b) shows that solution of thenonautonomous stochastic model (3) with the initial value(119878(119905) 1198681(119905) 1198682(119905)) = (03 015 015) tends to a periodic orbitin the sense of joint distribution
Example 9 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin1205871199051198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin120587119905
Complexity 11
S(t)
I1(t)
I2(t)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
(a)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 1 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (1 02 02) to the nonautonomous stochastic model (3)
0 25 50 75 100t
59 60 61
078
08
082
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
59 60 61
078
08
082
0
1
2
14
14
S(t)I 1(t)I 2(t)
25 50 75 1000t
S(t)
I1(t)
I2(t)
(b)
Figure 2 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 04 + 02 sin1205871199051205902 (119905) = 05 + 02 sin120587119905
(75)
Note that ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897+120572119897119894+119903119897119894⟩119879Therefore conditions(i) of Theorem 6 hold Then the two infectious diseases willgo to extinction Figure 2(a) shows that one of two diseases inthe deterministic SIS epidemic model is extinct and the otheris persistent Figure 2(b) shows that the two diseases will dieout under the large white noise disturbance of model (3)
Example 10 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin120587119905
12 Complexity
25 50 75 1000t
0
05
1
15
2S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
0 25 50 75 100t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 3 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin1205871199051205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 02 + 02 sin1205871199051205902 (119905) = 02 + 02 sin120587119905
(76)
Note that R1 lt 1 R2 lt 1 and ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 That is conditions (ii) of Theorem 6 hold Thenthe two infectious diseases will go to extinction Figure 3(a)
shows that one of two diseases in the deterministic SISepidemic model is extinct and the other is persistent withoutthe white noises Figure 3(b) shows that the two diseases willdie out under a small white noise disturbance of model (3)
7 Discussion and Conclusions
This paper explores the existence of nontrivial positiveT-periodic solution of a nonautonomous stochastic SISepidemic model with nonlinear growth rate and doubleepidemic hypothesis By constructing a suitable stochasticLyapunov function we establish sufficient conditions for theexistence of nontrivial positive 119879-periodic solution of system(3) Furthermore the sufficient conditions for the extinctionof the two diseases are obtained Our results are given asfollows
(1) If 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 and R gt 1 hold the SISmodel has at least one nontrivial positive 119879-periodicsolution where
R = 2sum119894=1
⟨119860120572119894120573119894⟩119879⟨119889 + 120572119894 + 119903119894 + 1205902119894 119860211990621198862119894 1198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩119879 ⟨119860 + 11988611205721 + 11988621205722⟩119879 (77)
(2) If ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectiousdiseases go extinct
(3) If ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 and R119894 lt 1 the two
infectious diseases also go extinct where
R119894 = 119860119906 ⟨120573119894⟩119879119889119897 ⟨119886119894⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879minus 1198602119906 ⟨1205902119894 ⟩11987921198892119897 ⟨1198862119894 ⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879
(78)
Complexity 13
Some interesting questions deserve further investigationOn the one hand we may explore some realistic but complexmodels such as considering the effects of impulsive ordelay perturbations on system (3) On the other hand wecan concern the dynamics of a nonautonomous stochasticSIS epidemic model with two infectious diseases driven byLevy jumps What is more we can also investigate thenonautonomous stochastic SIS epidemic model with twoinfectious diseases by a continuous time Markov chain Wewill investigate these cases in our future work
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (11371230) Joint Innovative Center forSafe andEffectiveMiningTechnology andEquipment of CoalResources Shandong Province the SDUST Research Fund(2014TDJH102) and Shandong Provincial Natural ScienceFoundation China (ZR2015AQ001)
References
[1] W O Kermack and A G McKendrick ldquoA contributions to themathematical theory of epidemics (Part I)rdquo Proceedings of theRoyal Society of London A vol 115 pp 700ndash721 1927
[2] A Gray D Greenhalgh L Hu X Mao and J Pan ldquoA stochasticdifferential equation SIS epidemic modelrdquo SIAM Journal onApplied Mathematics vol 71 no 3 pp 876ndash902 2011
[3] A Korobeinikov and G C Wake ldquoLyapunov functions andglobal stability for SIR SIRS and SIS epidemiological modelsrdquoApplied Mathematics Letters vol 15 no 8 pp 955ndash960 2002
[4] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995
[5] T Zhang X Meng Y Song and T Zhang ldquoA stage-structuredpredator-prey SI model with disease in the prey and impulsiveeffectsrdquoMathematical Modelling and Analysis vol 18 no 4 pp505ndash528 2013
[6] J Li andZMa ldquoQualitative analyses of SIS epidemicmodelwithvaccination and varying total population sizerdquo Mathematicaland Computer Modelling vol 35 no 11-12 pp 1235ndash1243 2002
[7] A drsquoOnofrio ldquoA note on the global behaviour of the network-based SIS epidemic modelrdquo Nonlinear Analysis Real WorldApplications vol 9 no 4 pp 1567ndash1572 2008
[8] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[9] T Zhang X Meng T Zhang and Y Song ldquoGlobal dynamicsfor a new high-dimensional SIR model with distributed delayrdquoApplied Mathematics and Computation vol 218 no 24 pp11806ndash11819 2012
[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006
[11] X Meng S Zhao T Feng and T Zhang ldquoDynamics of a novelnonlinear stochastic SIS epidemic model with double epidemichypothesisrdquo Journal of Mathematical Analysis and Applicationsvol 433 no 1 pp 227ndash242 2016
[12] A Miao X Wang T Zhang W Wang and B Sampath ArunaPradeep ldquoDynamical analysis of a stochastic SIS epidemicmodel with nonlinear incidence rate and double epidemichypothesisrdquo Advances in Difference Equations 2017226 pages2017
[13] X-Z Meng ldquoStability of a novel stochastic epidemic modelwith double epidemic hypothesisrdquo Applied Mathematics andComputation vol 217 no 2 pp 506ndash515 2010
[14] X P Li X Y Lin and Y Q Lin ldquoLyapunov-type conditions andstochastic differential equations driven byG-BrownianmotionrdquoJournal of Mathematical Analysis and Applications vol 439 no1 pp 235ndash255 2016
[15] M Liu and M Fan ldquoPermanence of stochastic Lotka-Volterrasystemsrdquo Journal of Nonlinear Science vol 27 no 2 pp 425ndash452 2017
[16] L D Liu and X Z Meng ldquoOptimal harvesting controland dynamics of two-species stochastic model with delaysrdquoAdvances in Difference Equations vol 2017 18 pages 2017
[17] H Ma and Y Jia ldquoStability analysis for stochastic differentialequations with infiniteMarkovian switchingsrdquo Journal of Math-ematical Analysis and Applications vol 435 no 1 pp 593ndash6052016
[18] G D Liu X H Wang X Z Meng and S J Gao ldquoExtinctionand persistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexity vol 3pp 1ndash15 2017
[19] C Tan and W H Zhang ldquoOn observability and detectabilityof continuous-time stochastic Markov jump systemsrdquo Journalof Systems Science and Complexity vol 28 no 4 pp 830ndash8472015
[20] X Leng T Feng and X Meng ldquoStochastic inequalities andapplications to dynamics analysis of a novel SIVS epidemicmodel with jumpsrdquo Journal of Inequalities and ApplicationsPaper No 138 25 pages 2017
[21] G Li and M Chen ldquoInfinite horizon linear quadratic optimalcontrol for stochastic difference time-delay systemsrdquo Advancesin Difference Equations vol 2015 14 pages 2015
[22] Y Zhao and W Zhang ldquoObserver-based controller design forsingular stochastic Markov jump systems with state dependentnoiserdquo Journal of Systems Science and Complexity vol 29 no 4pp 946ndash958 2016
[23] S Zhang X Meng T Feng and T Zhang ldquoDynamics analysisand numerical simulations of a stochastic non-autonomouspredatorndashprey system with impulsive effectsrdquo Nonlinear Anal-ysis Hybrid Systems vol 26 pp 19ndash37 2017
[24] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013
[25] X Meng L Wang and T Zhang ldquoGlobal dynamics analysis ofa nonlinear impulsive stochastic chemostat system in a pollutedenvironmentrdquo Journal of AppliedAnalysis andComputation vol6 no 3 pp 865ndash875 2016
[26] X Liu Y Li andW Zhang ldquoStochastic linear quadratic optimalcontrol with constraint for discrete-time systemsrdquo AppliedMathematics and Computation vol 228 pp 264ndash270 2014
14 Complexity
[27] X Z Meng and X H Wang ldquoStochastic predator-prey systemsubject to levy jumprdquo Discrete Dynamics in Nature and Societyvol 2016 Article ID 5749892 13 pages 2016
[28] X Lv L Wang and X Meng ldquoGlobal analysis of a new non-linear stochastic differential competition systemwith impulsiveeffectrdquo Advances in Difference Equations vol 2017 296 pages2017
[29] X Zhang D Jiang A Alsaedi and T Hayat ldquoStationarydistribution of stochastic SIS epidemic model with vaccinationunder regime switchingrdquo Applied Mathematics Letters vol 59pp 87ndash93 2016
[30] F Li X Meng and Y Cui ldquoNonlinear stochastic analysis fora stochastic SIS epidemic modelrdquo Journal of Nonlinear Sciencesand Applications vol 10 no 09 pp 5116ndash5124 2017
[31] A Q Miao J Zhang T Q Zhang and B G Pradeep ldquoSampathAruna Pradeep Threshold dynamics of a stochastic SIR modelwith vertical transmission and vaccinationrdquo Computationaland Mathematical Methods in Medicine vol 2017 Article ID4820183 10 pages 2017
[32] Q Liu D Jiang N Shi T Hayat and A Alsaedi ldquoNontrivialperiodic solution of a stochastic non-autonomous SISV epi-demic modelrdquo Physica A Statistical Mechanics and its Applica-tions vol 462 pp 837ndash845 2016
[33] Y L Zhou S L Yuan and D L Zhao ldquoThreshold behavior ofa stochastic SIS model with levy jumpsrdquo Applied Mathematicsand Computation vol 275 pp 255ndash267 2016
[34] T Feng X Meng L Liu and S Gao ldquoApplication of inequal-ities technique to dynamics analysis of a stochastic eco-epidemiology modelrdquo Journal of Inequalities and ApplicationsPaper No 327 29 pages 2016
[35] M Liu and K Wang ldquoPersistence and extinction in stochasticnon-autonomous logistic systemsrdquo Journal of MathematicalAnalysis and Applications vol 375 no 2 pp 443ndash457 2011
[36] Z Bai and Y Zhou ldquoExistence of two periodic solutions for anon-autonomous SIR epidemic modelrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 1 pp 382ndash391 2011
[37] S Gao F Zhang and Y He ldquoThe effects of migratory birdpopulation in a nonautonomous eco-epidemiological modelrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 37 no 6 pp3903ndash3916 2013
[38] T Kuniya ldquoExistence of a nontrivial periodic solution in an age-structured SIR epidemicmodel with time periodic coefficientsrdquoApplied Mathematics Letters vol 27 pp 15ndash20 2014
[39] L Zu D Jiang D OrsquoRegan and B Ge ldquoPeriodic solution fora non-autonomous Lotka-Volterra predator-prey model withrandom perturbationrdquo Journal of Mathematical Analysis andApplications vol 430 no 1 pp 428ndash437 2015
[40] Y Lin D Jiang and T Liu ldquoNontrivial periodic solution ofa stochastic epidemic model with seasonal variationrdquo AppliedMathematics Letters vol 45 pp 103ndash107 2015
[41] X R Mao Stochastic Differential Equations and Their Applica-tions Horwood Chichester UK 1997
[42] R Khasminskii Stochastic Stability of Differential EquationsSpringer Berlin Germany 2011
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Complexity 11
S(t)
I1(t)
I2(t)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
(a)
50 100 150 2000t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 1 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (1 02 02) to the nonautonomous stochastic model (3)
0 25 50 75 100t
59 60 61
078
08
082
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
59 60 61
078
08
082
0
1
2
14
14
S(t)I 1(t)I 2(t)
25 50 75 1000t
S(t)
I1(t)
I2(t)
(b)
Figure 2 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 04 + 02 sin1205871199051205902 (119905) = 05 + 02 sin120587119905
(75)
Note that ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897+120572119897119894+119903119897119894⟩119879Therefore conditions(i) of Theorem 6 hold Then the two infectious diseases willgo to extinction Figure 2(a) shows that one of two diseases inthe deterministic SIS epidemic model is extinct and the otheris persistent Figure 2(b) shows that the two diseases will dieout under the large white noise disturbance of model (3)
Example 10 Choose the parameters in model (3) as follows
119860 (119905) = 02 + 02 sin1205871199051198861 (119905) = 023 + 02 sin120587119905
12 Complexity
25 50 75 1000t
0
05
1
15
2S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
0 25 50 75 100t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 3 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin1205871199051205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 02 + 02 sin1205871199051205902 (119905) = 02 + 02 sin120587119905
(76)
Note that R1 lt 1 R2 lt 1 and ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 That is conditions (ii) of Theorem 6 hold Thenthe two infectious diseases will go to extinction Figure 3(a)
shows that one of two diseases in the deterministic SISepidemic model is extinct and the other is persistent withoutthe white noises Figure 3(b) shows that the two diseases willdie out under a small white noise disturbance of model (3)
7 Discussion and Conclusions
This paper explores the existence of nontrivial positiveT-periodic solution of a nonautonomous stochastic SISepidemic model with nonlinear growth rate and doubleepidemic hypothesis By constructing a suitable stochasticLyapunov function we establish sufficient conditions for theexistence of nontrivial positive 119879-periodic solution of system(3) Furthermore the sufficient conditions for the extinctionof the two diseases are obtained Our results are given asfollows
(1) If 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 and R gt 1 hold the SISmodel has at least one nontrivial positive 119879-periodicsolution where
R = 2sum119894=1
⟨119860120572119894120573119894⟩119879⟨119889 + 120572119894 + 119903119894 + 1205902119894 119860211990621198862119894 1198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩119879 ⟨119860 + 11988611205721 + 11988621205722⟩119879 (77)
(2) If ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectiousdiseases go extinct
(3) If ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 and R119894 lt 1 the two
infectious diseases also go extinct where
R119894 = 119860119906 ⟨120573119894⟩119879119889119897 ⟨119886119894⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879minus 1198602119906 ⟨1205902119894 ⟩11987921198892119897 ⟨1198862119894 ⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879
(78)
Complexity 13
Some interesting questions deserve further investigationOn the one hand we may explore some realistic but complexmodels such as considering the effects of impulsive ordelay perturbations on system (3) On the other hand wecan concern the dynamics of a nonautonomous stochasticSIS epidemic model with two infectious diseases driven byLevy jumps What is more we can also investigate thenonautonomous stochastic SIS epidemic model with twoinfectious diseases by a continuous time Markov chain Wewill investigate these cases in our future work
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (11371230) Joint Innovative Center forSafe andEffectiveMiningTechnology andEquipment of CoalResources Shandong Province the SDUST Research Fund(2014TDJH102) and Shandong Provincial Natural ScienceFoundation China (ZR2015AQ001)
References
[1] W O Kermack and A G McKendrick ldquoA contributions to themathematical theory of epidemics (Part I)rdquo Proceedings of theRoyal Society of London A vol 115 pp 700ndash721 1927
[2] A Gray D Greenhalgh L Hu X Mao and J Pan ldquoA stochasticdifferential equation SIS epidemic modelrdquo SIAM Journal onApplied Mathematics vol 71 no 3 pp 876ndash902 2011
[3] A Korobeinikov and G C Wake ldquoLyapunov functions andglobal stability for SIR SIRS and SIS epidemiological modelsrdquoApplied Mathematics Letters vol 15 no 8 pp 955ndash960 2002
[4] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995
[5] T Zhang X Meng Y Song and T Zhang ldquoA stage-structuredpredator-prey SI model with disease in the prey and impulsiveeffectsrdquoMathematical Modelling and Analysis vol 18 no 4 pp505ndash528 2013
[6] J Li andZMa ldquoQualitative analyses of SIS epidemicmodelwithvaccination and varying total population sizerdquo Mathematicaland Computer Modelling vol 35 no 11-12 pp 1235ndash1243 2002
[7] A drsquoOnofrio ldquoA note on the global behaviour of the network-based SIS epidemic modelrdquo Nonlinear Analysis Real WorldApplications vol 9 no 4 pp 1567ndash1572 2008
[8] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[9] T Zhang X Meng T Zhang and Y Song ldquoGlobal dynamicsfor a new high-dimensional SIR model with distributed delayrdquoApplied Mathematics and Computation vol 218 no 24 pp11806ndash11819 2012
[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006
[11] X Meng S Zhao T Feng and T Zhang ldquoDynamics of a novelnonlinear stochastic SIS epidemic model with double epidemichypothesisrdquo Journal of Mathematical Analysis and Applicationsvol 433 no 1 pp 227ndash242 2016
[12] A Miao X Wang T Zhang W Wang and B Sampath ArunaPradeep ldquoDynamical analysis of a stochastic SIS epidemicmodel with nonlinear incidence rate and double epidemichypothesisrdquo Advances in Difference Equations 2017226 pages2017
[13] X-Z Meng ldquoStability of a novel stochastic epidemic modelwith double epidemic hypothesisrdquo Applied Mathematics andComputation vol 217 no 2 pp 506ndash515 2010
[14] X P Li X Y Lin and Y Q Lin ldquoLyapunov-type conditions andstochastic differential equations driven byG-BrownianmotionrdquoJournal of Mathematical Analysis and Applications vol 439 no1 pp 235ndash255 2016
[15] M Liu and M Fan ldquoPermanence of stochastic Lotka-Volterrasystemsrdquo Journal of Nonlinear Science vol 27 no 2 pp 425ndash452 2017
[16] L D Liu and X Z Meng ldquoOptimal harvesting controland dynamics of two-species stochastic model with delaysrdquoAdvances in Difference Equations vol 2017 18 pages 2017
[17] H Ma and Y Jia ldquoStability analysis for stochastic differentialequations with infiniteMarkovian switchingsrdquo Journal of Math-ematical Analysis and Applications vol 435 no 1 pp 593ndash6052016
[18] G D Liu X H Wang X Z Meng and S J Gao ldquoExtinctionand persistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexity vol 3pp 1ndash15 2017
[19] C Tan and W H Zhang ldquoOn observability and detectabilityof continuous-time stochastic Markov jump systemsrdquo Journalof Systems Science and Complexity vol 28 no 4 pp 830ndash8472015
[20] X Leng T Feng and X Meng ldquoStochastic inequalities andapplications to dynamics analysis of a novel SIVS epidemicmodel with jumpsrdquo Journal of Inequalities and ApplicationsPaper No 138 25 pages 2017
[21] G Li and M Chen ldquoInfinite horizon linear quadratic optimalcontrol for stochastic difference time-delay systemsrdquo Advancesin Difference Equations vol 2015 14 pages 2015
[22] Y Zhao and W Zhang ldquoObserver-based controller design forsingular stochastic Markov jump systems with state dependentnoiserdquo Journal of Systems Science and Complexity vol 29 no 4pp 946ndash958 2016
[23] S Zhang X Meng T Feng and T Zhang ldquoDynamics analysisand numerical simulations of a stochastic non-autonomouspredatorndashprey system with impulsive effectsrdquo Nonlinear Anal-ysis Hybrid Systems vol 26 pp 19ndash37 2017
[24] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013
[25] X Meng L Wang and T Zhang ldquoGlobal dynamics analysis ofa nonlinear impulsive stochastic chemostat system in a pollutedenvironmentrdquo Journal of AppliedAnalysis andComputation vol6 no 3 pp 865ndash875 2016
[26] X Liu Y Li andW Zhang ldquoStochastic linear quadratic optimalcontrol with constraint for discrete-time systemsrdquo AppliedMathematics and Computation vol 228 pp 264ndash270 2014
14 Complexity
[27] X Z Meng and X H Wang ldquoStochastic predator-prey systemsubject to levy jumprdquo Discrete Dynamics in Nature and Societyvol 2016 Article ID 5749892 13 pages 2016
[28] X Lv L Wang and X Meng ldquoGlobal analysis of a new non-linear stochastic differential competition systemwith impulsiveeffectrdquo Advances in Difference Equations vol 2017 296 pages2017
[29] X Zhang D Jiang A Alsaedi and T Hayat ldquoStationarydistribution of stochastic SIS epidemic model with vaccinationunder regime switchingrdquo Applied Mathematics Letters vol 59pp 87ndash93 2016
[30] F Li X Meng and Y Cui ldquoNonlinear stochastic analysis fora stochastic SIS epidemic modelrdquo Journal of Nonlinear Sciencesand Applications vol 10 no 09 pp 5116ndash5124 2017
[31] A Q Miao J Zhang T Q Zhang and B G Pradeep ldquoSampathAruna Pradeep Threshold dynamics of a stochastic SIR modelwith vertical transmission and vaccinationrdquo Computationaland Mathematical Methods in Medicine vol 2017 Article ID4820183 10 pages 2017
[32] Q Liu D Jiang N Shi T Hayat and A Alsaedi ldquoNontrivialperiodic solution of a stochastic non-autonomous SISV epi-demic modelrdquo Physica A Statistical Mechanics and its Applica-tions vol 462 pp 837ndash845 2016
[33] Y L Zhou S L Yuan and D L Zhao ldquoThreshold behavior ofa stochastic SIS model with levy jumpsrdquo Applied Mathematicsand Computation vol 275 pp 255ndash267 2016
[34] T Feng X Meng L Liu and S Gao ldquoApplication of inequal-ities technique to dynamics analysis of a stochastic eco-epidemiology modelrdquo Journal of Inequalities and ApplicationsPaper No 327 29 pages 2016
[35] M Liu and K Wang ldquoPersistence and extinction in stochasticnon-autonomous logistic systemsrdquo Journal of MathematicalAnalysis and Applications vol 375 no 2 pp 443ndash457 2011
[36] Z Bai and Y Zhou ldquoExistence of two periodic solutions for anon-autonomous SIR epidemic modelrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 1 pp 382ndash391 2011
[37] S Gao F Zhang and Y He ldquoThe effects of migratory birdpopulation in a nonautonomous eco-epidemiological modelrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 37 no 6 pp3903ndash3916 2013
[38] T Kuniya ldquoExistence of a nontrivial periodic solution in an age-structured SIR epidemicmodel with time periodic coefficientsrdquoApplied Mathematics Letters vol 27 pp 15ndash20 2014
[39] L Zu D Jiang D OrsquoRegan and B Ge ldquoPeriodic solution fora non-autonomous Lotka-Volterra predator-prey model withrandom perturbationrdquo Journal of Mathematical Analysis andApplications vol 430 no 1 pp 428ndash437 2015
[40] Y Lin D Jiang and T Liu ldquoNontrivial periodic solution ofa stochastic epidemic model with seasonal variationrdquo AppliedMathematics Letters vol 45 pp 103ndash107 2015
[41] X R Mao Stochastic Differential Equations and Their Applica-tions Horwood Chichester UK 1997
[42] R Khasminskii Stochastic Stability of Differential EquationsSpringer Berlin Germany 2011
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Complexity
25 50 75 1000t
0
05
1
15
2S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(a)
0 25 50 75 100t
0
05
1
15
2
S(t)I 1(t)I 2(t)
S(t)
I1(t)
I2(t)
(b)
Figure 3 The solution (119878(119905) 1198681(119905) 1198682(119905)) = (03 025 025) to the nonautonomous stochastic model (3)
1198862 (119905) = 025 + 02 sin1205871199051205731 (119905) = 02 + 02 sin1205871199051205732 (119905) = 03 + 02 sin1205871199051199031 (119905) = 02 + 02 sin1205871199051199032 (119905) = 03 + 02 sin120587119905119889 (119905) = 025 + 02 sin1205871199051205721 (119905) = 02 + 02 sin1205871199051205722 (119905) = 03 + 02 sin1205871199051205901 (119905) = 02 + 02 sin1205871199051205902 (119905) = 02 + 02 sin120587119905
(76)
Note that R1 lt 1 R2 lt 1 and ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 That is conditions (ii) of Theorem 6 hold Thenthe two infectious diseases will go to extinction Figure 3(a)
shows that one of two diseases in the deterministic SISepidemic model is extinct and the other is persistent withoutthe white noises Figure 3(b) shows that the two diseases willdie out under a small white noise disturbance of model (3)
7 Discussion and Conclusions
This paper explores the existence of nontrivial positiveT-periodic solution of a nonautonomous stochastic SISepidemic model with nonlinear growth rate and doubleepidemic hypothesis By constructing a suitable stochasticLyapunov function we establish sufficient conditions for theexistence of nontrivial positive 119879-periodic solution of system(3) Furthermore the sufficient conditions for the extinctionof the two diseases are obtained Our results are given asfollows
(1) If 1198861(119905)1199031(119905)(1205731(119905) minus 1198861(119905)119889(119905)) gt 119860119906119889119897 1198862(119905)1199032(119905)(1205732(119905) minus 1198862(119905)119889(119905)) gt 119860119906119889119897 and R gt 1 hold the SISmodel has at least one nontrivial positive 119879-periodicsolution where
R = 2sum119894=1
⟨119860120572119894120573119894⟩119879⟨119889 + 120572119894 + 119903119894 + 1205902119894 119860211990621198862119894 1198892119897⟩119879 ⟨119889 + 120590212 + 120590222⟩119879 ⟨119860 + 11988611205721 + 11988621205722⟩119879 (77)
(2) If ⟨1205902119894 ⟩119879 gt ⟨1205732119894 ⟩1198792⟨119889119897 + 120572119897119894 + 119903119897119894⟩119879 the two infectiousdiseases go extinct
(3) If ⟨1205902119894 ⟩119879 le 119889119897⟨1198862119894 ⟩119879⟨1205732119894 ⟩119879119860119906⟨119886119894⟩119879 and R119894 lt 1 the two
infectious diseases also go extinct where
R119894 = 119860119906 ⟨120573119894⟩119879119889119897 ⟨119886119894⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879minus 1198602119906 ⟨1205902119894 ⟩11987921198892119897 ⟨1198862119894 ⟩119879 ⟨119889 + 120572119894 + 119903119894⟩119879
(78)
Complexity 13
Some interesting questions deserve further investigationOn the one hand we may explore some realistic but complexmodels such as considering the effects of impulsive ordelay perturbations on system (3) On the other hand wecan concern the dynamics of a nonautonomous stochasticSIS epidemic model with two infectious diseases driven byLevy jumps What is more we can also investigate thenonautonomous stochastic SIS epidemic model with twoinfectious diseases by a continuous time Markov chain Wewill investigate these cases in our future work
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (11371230) Joint Innovative Center forSafe andEffectiveMiningTechnology andEquipment of CoalResources Shandong Province the SDUST Research Fund(2014TDJH102) and Shandong Provincial Natural ScienceFoundation China (ZR2015AQ001)
References
[1] W O Kermack and A G McKendrick ldquoA contributions to themathematical theory of epidemics (Part I)rdquo Proceedings of theRoyal Society of London A vol 115 pp 700ndash721 1927
[2] A Gray D Greenhalgh L Hu X Mao and J Pan ldquoA stochasticdifferential equation SIS epidemic modelrdquo SIAM Journal onApplied Mathematics vol 71 no 3 pp 876ndash902 2011
[3] A Korobeinikov and G C Wake ldquoLyapunov functions andglobal stability for SIR SIRS and SIS epidemiological modelsrdquoApplied Mathematics Letters vol 15 no 8 pp 955ndash960 2002
[4] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995
[5] T Zhang X Meng Y Song and T Zhang ldquoA stage-structuredpredator-prey SI model with disease in the prey and impulsiveeffectsrdquoMathematical Modelling and Analysis vol 18 no 4 pp505ndash528 2013
[6] J Li andZMa ldquoQualitative analyses of SIS epidemicmodelwithvaccination and varying total population sizerdquo Mathematicaland Computer Modelling vol 35 no 11-12 pp 1235ndash1243 2002
[7] A drsquoOnofrio ldquoA note on the global behaviour of the network-based SIS epidemic modelrdquo Nonlinear Analysis Real WorldApplications vol 9 no 4 pp 1567ndash1572 2008
[8] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[9] T Zhang X Meng T Zhang and Y Song ldquoGlobal dynamicsfor a new high-dimensional SIR model with distributed delayrdquoApplied Mathematics and Computation vol 218 no 24 pp11806ndash11819 2012
[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006
[11] X Meng S Zhao T Feng and T Zhang ldquoDynamics of a novelnonlinear stochastic SIS epidemic model with double epidemichypothesisrdquo Journal of Mathematical Analysis and Applicationsvol 433 no 1 pp 227ndash242 2016
[12] A Miao X Wang T Zhang W Wang and B Sampath ArunaPradeep ldquoDynamical analysis of a stochastic SIS epidemicmodel with nonlinear incidence rate and double epidemichypothesisrdquo Advances in Difference Equations 2017226 pages2017
[13] X-Z Meng ldquoStability of a novel stochastic epidemic modelwith double epidemic hypothesisrdquo Applied Mathematics andComputation vol 217 no 2 pp 506ndash515 2010
[14] X P Li X Y Lin and Y Q Lin ldquoLyapunov-type conditions andstochastic differential equations driven byG-BrownianmotionrdquoJournal of Mathematical Analysis and Applications vol 439 no1 pp 235ndash255 2016
[15] M Liu and M Fan ldquoPermanence of stochastic Lotka-Volterrasystemsrdquo Journal of Nonlinear Science vol 27 no 2 pp 425ndash452 2017
[16] L D Liu and X Z Meng ldquoOptimal harvesting controland dynamics of two-species stochastic model with delaysrdquoAdvances in Difference Equations vol 2017 18 pages 2017
[17] H Ma and Y Jia ldquoStability analysis for stochastic differentialequations with infiniteMarkovian switchingsrdquo Journal of Math-ematical Analysis and Applications vol 435 no 1 pp 593ndash6052016
[18] G D Liu X H Wang X Z Meng and S J Gao ldquoExtinctionand persistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexity vol 3pp 1ndash15 2017
[19] C Tan and W H Zhang ldquoOn observability and detectabilityof continuous-time stochastic Markov jump systemsrdquo Journalof Systems Science and Complexity vol 28 no 4 pp 830ndash8472015
[20] X Leng T Feng and X Meng ldquoStochastic inequalities andapplications to dynamics analysis of a novel SIVS epidemicmodel with jumpsrdquo Journal of Inequalities and ApplicationsPaper No 138 25 pages 2017
[21] G Li and M Chen ldquoInfinite horizon linear quadratic optimalcontrol for stochastic difference time-delay systemsrdquo Advancesin Difference Equations vol 2015 14 pages 2015
[22] Y Zhao and W Zhang ldquoObserver-based controller design forsingular stochastic Markov jump systems with state dependentnoiserdquo Journal of Systems Science and Complexity vol 29 no 4pp 946ndash958 2016
[23] S Zhang X Meng T Feng and T Zhang ldquoDynamics analysisand numerical simulations of a stochastic non-autonomouspredatorndashprey system with impulsive effectsrdquo Nonlinear Anal-ysis Hybrid Systems vol 26 pp 19ndash37 2017
[24] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013
[25] X Meng L Wang and T Zhang ldquoGlobal dynamics analysis ofa nonlinear impulsive stochastic chemostat system in a pollutedenvironmentrdquo Journal of AppliedAnalysis andComputation vol6 no 3 pp 865ndash875 2016
[26] X Liu Y Li andW Zhang ldquoStochastic linear quadratic optimalcontrol with constraint for discrete-time systemsrdquo AppliedMathematics and Computation vol 228 pp 264ndash270 2014
14 Complexity
[27] X Z Meng and X H Wang ldquoStochastic predator-prey systemsubject to levy jumprdquo Discrete Dynamics in Nature and Societyvol 2016 Article ID 5749892 13 pages 2016
[28] X Lv L Wang and X Meng ldquoGlobal analysis of a new non-linear stochastic differential competition systemwith impulsiveeffectrdquo Advances in Difference Equations vol 2017 296 pages2017
[29] X Zhang D Jiang A Alsaedi and T Hayat ldquoStationarydistribution of stochastic SIS epidemic model with vaccinationunder regime switchingrdquo Applied Mathematics Letters vol 59pp 87ndash93 2016
[30] F Li X Meng and Y Cui ldquoNonlinear stochastic analysis fora stochastic SIS epidemic modelrdquo Journal of Nonlinear Sciencesand Applications vol 10 no 09 pp 5116ndash5124 2017
[31] A Q Miao J Zhang T Q Zhang and B G Pradeep ldquoSampathAruna Pradeep Threshold dynamics of a stochastic SIR modelwith vertical transmission and vaccinationrdquo Computationaland Mathematical Methods in Medicine vol 2017 Article ID4820183 10 pages 2017
[32] Q Liu D Jiang N Shi T Hayat and A Alsaedi ldquoNontrivialperiodic solution of a stochastic non-autonomous SISV epi-demic modelrdquo Physica A Statistical Mechanics and its Applica-tions vol 462 pp 837ndash845 2016
[33] Y L Zhou S L Yuan and D L Zhao ldquoThreshold behavior ofa stochastic SIS model with levy jumpsrdquo Applied Mathematicsand Computation vol 275 pp 255ndash267 2016
[34] T Feng X Meng L Liu and S Gao ldquoApplication of inequal-ities technique to dynamics analysis of a stochastic eco-epidemiology modelrdquo Journal of Inequalities and ApplicationsPaper No 327 29 pages 2016
[35] M Liu and K Wang ldquoPersistence and extinction in stochasticnon-autonomous logistic systemsrdquo Journal of MathematicalAnalysis and Applications vol 375 no 2 pp 443ndash457 2011
[36] Z Bai and Y Zhou ldquoExistence of two periodic solutions for anon-autonomous SIR epidemic modelrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 1 pp 382ndash391 2011
[37] S Gao F Zhang and Y He ldquoThe effects of migratory birdpopulation in a nonautonomous eco-epidemiological modelrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 37 no 6 pp3903ndash3916 2013
[38] T Kuniya ldquoExistence of a nontrivial periodic solution in an age-structured SIR epidemicmodel with time periodic coefficientsrdquoApplied Mathematics Letters vol 27 pp 15ndash20 2014
[39] L Zu D Jiang D OrsquoRegan and B Ge ldquoPeriodic solution fora non-autonomous Lotka-Volterra predator-prey model withrandom perturbationrdquo Journal of Mathematical Analysis andApplications vol 430 no 1 pp 428ndash437 2015
[40] Y Lin D Jiang and T Liu ldquoNontrivial periodic solution ofa stochastic epidemic model with seasonal variationrdquo AppliedMathematics Letters vol 45 pp 103ndash107 2015
[41] X R Mao Stochastic Differential Equations and Their Applica-tions Horwood Chichester UK 1997
[42] R Khasminskii Stochastic Stability of Differential EquationsSpringer Berlin Germany 2011
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Complexity 13
Some interesting questions deserve further investigationOn the one hand we may explore some realistic but complexmodels such as considering the effects of impulsive ordelay perturbations on system (3) On the other hand wecan concern the dynamics of a nonautonomous stochasticSIS epidemic model with two infectious diseases driven byLevy jumps What is more we can also investigate thenonautonomous stochastic SIS epidemic model with twoinfectious diseases by a continuous time Markov chain Wewill investigate these cases in our future work
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (11371230) Joint Innovative Center forSafe andEffectiveMiningTechnology andEquipment of CoalResources Shandong Province the SDUST Research Fund(2014TDJH102) and Shandong Provincial Natural ScienceFoundation China (ZR2015AQ001)
References
[1] W O Kermack and A G McKendrick ldquoA contributions to themathematical theory of epidemics (Part I)rdquo Proceedings of theRoyal Society of London A vol 115 pp 700ndash721 1927
[2] A Gray D Greenhalgh L Hu X Mao and J Pan ldquoA stochasticdifferential equation SIS epidemic modelrdquo SIAM Journal onApplied Mathematics vol 71 no 3 pp 876ndash902 2011
[3] A Korobeinikov and G C Wake ldquoLyapunov functions andglobal stability for SIR SIRS and SIS epidemiological modelsrdquoApplied Mathematics Letters vol 15 no 8 pp 955ndash960 2002
[4] H W Hethcote and P van den Driessche ldquoAn SIS epidemicmodel with variable population size and a delayrdquo Journal ofMathematical Biology vol 34 no 2 pp 177ndash194 1995
[5] T Zhang X Meng Y Song and T Zhang ldquoA stage-structuredpredator-prey SI model with disease in the prey and impulsiveeffectsrdquoMathematical Modelling and Analysis vol 18 no 4 pp505ndash528 2013
[6] J Li andZMa ldquoQualitative analyses of SIS epidemicmodelwithvaccination and varying total population sizerdquo Mathematicaland Computer Modelling vol 35 no 11-12 pp 1235ndash1243 2002
[7] A drsquoOnofrio ldquoA note on the global behaviour of the network-based SIS epidemic modelrdquo Nonlinear Analysis Real WorldApplications vol 9 no 4 pp 1567ndash1572 2008
[8] H W Hethcote and P van den Driessche ldquoTwo SIS epidemio-logic models with delaysrdquo Journal of Mathematical Biology vol40 no 1 pp 3ndash26 2000
[9] T Zhang X Meng T Zhang and Y Song ldquoGlobal dynamicsfor a new high-dimensional SIR model with distributed delayrdquoApplied Mathematics and Computation vol 218 no 24 pp11806ndash11819 2012
[10] S Gao L Chen J J Nieto and A Torres ldquoAnalysis of adelayed epidemic model with pulse vaccination and saturationincidencerdquo Vaccine vol 24 no 35-36 pp 6037ndash6045 2006
[11] X Meng S Zhao T Feng and T Zhang ldquoDynamics of a novelnonlinear stochastic SIS epidemic model with double epidemichypothesisrdquo Journal of Mathematical Analysis and Applicationsvol 433 no 1 pp 227ndash242 2016
[12] A Miao X Wang T Zhang W Wang and B Sampath ArunaPradeep ldquoDynamical analysis of a stochastic SIS epidemicmodel with nonlinear incidence rate and double epidemichypothesisrdquo Advances in Difference Equations 2017226 pages2017
[13] X-Z Meng ldquoStability of a novel stochastic epidemic modelwith double epidemic hypothesisrdquo Applied Mathematics andComputation vol 217 no 2 pp 506ndash515 2010
[14] X P Li X Y Lin and Y Q Lin ldquoLyapunov-type conditions andstochastic differential equations driven byG-BrownianmotionrdquoJournal of Mathematical Analysis and Applications vol 439 no1 pp 235ndash255 2016
[15] M Liu and M Fan ldquoPermanence of stochastic Lotka-Volterrasystemsrdquo Journal of Nonlinear Science vol 27 no 2 pp 425ndash452 2017
[16] L D Liu and X Z Meng ldquoOptimal harvesting controland dynamics of two-species stochastic model with delaysrdquoAdvances in Difference Equations vol 2017 18 pages 2017
[17] H Ma and Y Jia ldquoStability analysis for stochastic differentialequations with infiniteMarkovian switchingsrdquo Journal of Math-ematical Analysis and Applications vol 435 no 1 pp 593ndash6052016
[18] G D Liu X H Wang X Z Meng and S J Gao ldquoExtinctionand persistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexity vol 3pp 1ndash15 2017
[19] C Tan and W H Zhang ldquoOn observability and detectabilityof continuous-time stochastic Markov jump systemsrdquo Journalof Systems Science and Complexity vol 28 no 4 pp 830ndash8472015
[20] X Leng T Feng and X Meng ldquoStochastic inequalities andapplications to dynamics analysis of a novel SIVS epidemicmodel with jumpsrdquo Journal of Inequalities and ApplicationsPaper No 138 25 pages 2017
[21] G Li and M Chen ldquoInfinite horizon linear quadratic optimalcontrol for stochastic difference time-delay systemsrdquo Advancesin Difference Equations vol 2015 14 pages 2015
[22] Y Zhao and W Zhang ldquoObserver-based controller design forsingular stochastic Markov jump systems with state dependentnoiserdquo Journal of Systems Science and Complexity vol 29 no 4pp 946ndash958 2016
[23] S Zhang X Meng T Feng and T Zhang ldquoDynamics analysisand numerical simulations of a stochastic non-autonomouspredatorndashprey system with impulsive effectsrdquo Nonlinear Anal-ysis Hybrid Systems vol 26 pp 19ndash37 2017
[24] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013
[25] X Meng L Wang and T Zhang ldquoGlobal dynamics analysis ofa nonlinear impulsive stochastic chemostat system in a pollutedenvironmentrdquo Journal of AppliedAnalysis andComputation vol6 no 3 pp 865ndash875 2016
[26] X Liu Y Li andW Zhang ldquoStochastic linear quadratic optimalcontrol with constraint for discrete-time systemsrdquo AppliedMathematics and Computation vol 228 pp 264ndash270 2014
14 Complexity
[27] X Z Meng and X H Wang ldquoStochastic predator-prey systemsubject to levy jumprdquo Discrete Dynamics in Nature and Societyvol 2016 Article ID 5749892 13 pages 2016
[28] X Lv L Wang and X Meng ldquoGlobal analysis of a new non-linear stochastic differential competition systemwith impulsiveeffectrdquo Advances in Difference Equations vol 2017 296 pages2017
[29] X Zhang D Jiang A Alsaedi and T Hayat ldquoStationarydistribution of stochastic SIS epidemic model with vaccinationunder regime switchingrdquo Applied Mathematics Letters vol 59pp 87ndash93 2016
[30] F Li X Meng and Y Cui ldquoNonlinear stochastic analysis fora stochastic SIS epidemic modelrdquo Journal of Nonlinear Sciencesand Applications vol 10 no 09 pp 5116ndash5124 2017
[31] A Q Miao J Zhang T Q Zhang and B G Pradeep ldquoSampathAruna Pradeep Threshold dynamics of a stochastic SIR modelwith vertical transmission and vaccinationrdquo Computationaland Mathematical Methods in Medicine vol 2017 Article ID4820183 10 pages 2017
[32] Q Liu D Jiang N Shi T Hayat and A Alsaedi ldquoNontrivialperiodic solution of a stochastic non-autonomous SISV epi-demic modelrdquo Physica A Statistical Mechanics and its Applica-tions vol 462 pp 837ndash845 2016
[33] Y L Zhou S L Yuan and D L Zhao ldquoThreshold behavior ofa stochastic SIS model with levy jumpsrdquo Applied Mathematicsand Computation vol 275 pp 255ndash267 2016
[34] T Feng X Meng L Liu and S Gao ldquoApplication of inequal-ities technique to dynamics analysis of a stochastic eco-epidemiology modelrdquo Journal of Inequalities and ApplicationsPaper No 327 29 pages 2016
[35] M Liu and K Wang ldquoPersistence and extinction in stochasticnon-autonomous logistic systemsrdquo Journal of MathematicalAnalysis and Applications vol 375 no 2 pp 443ndash457 2011
[36] Z Bai and Y Zhou ldquoExistence of two periodic solutions for anon-autonomous SIR epidemic modelrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 1 pp 382ndash391 2011
[37] S Gao F Zhang and Y He ldquoThe effects of migratory birdpopulation in a nonautonomous eco-epidemiological modelrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 37 no 6 pp3903ndash3916 2013
[38] T Kuniya ldquoExistence of a nontrivial periodic solution in an age-structured SIR epidemicmodel with time periodic coefficientsrdquoApplied Mathematics Letters vol 27 pp 15ndash20 2014
[39] L Zu D Jiang D OrsquoRegan and B Ge ldquoPeriodic solution fora non-autonomous Lotka-Volterra predator-prey model withrandom perturbationrdquo Journal of Mathematical Analysis andApplications vol 430 no 1 pp 428ndash437 2015
[40] Y Lin D Jiang and T Liu ldquoNontrivial periodic solution ofa stochastic epidemic model with seasonal variationrdquo AppliedMathematics Letters vol 45 pp 103ndash107 2015
[41] X R Mao Stochastic Differential Equations and Their Applica-tions Horwood Chichester UK 1997
[42] R Khasminskii Stochastic Stability of Differential EquationsSpringer Berlin Germany 2011
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Complexity
[27] X Z Meng and X H Wang ldquoStochastic predator-prey systemsubject to levy jumprdquo Discrete Dynamics in Nature and Societyvol 2016 Article ID 5749892 13 pages 2016
[28] X Lv L Wang and X Meng ldquoGlobal analysis of a new non-linear stochastic differential competition systemwith impulsiveeffectrdquo Advances in Difference Equations vol 2017 296 pages2017
[29] X Zhang D Jiang A Alsaedi and T Hayat ldquoStationarydistribution of stochastic SIS epidemic model with vaccinationunder regime switchingrdquo Applied Mathematics Letters vol 59pp 87ndash93 2016
[30] F Li X Meng and Y Cui ldquoNonlinear stochastic analysis fora stochastic SIS epidemic modelrdquo Journal of Nonlinear Sciencesand Applications vol 10 no 09 pp 5116ndash5124 2017
[31] A Q Miao J Zhang T Q Zhang and B G Pradeep ldquoSampathAruna Pradeep Threshold dynamics of a stochastic SIR modelwith vertical transmission and vaccinationrdquo Computationaland Mathematical Methods in Medicine vol 2017 Article ID4820183 10 pages 2017
[32] Q Liu D Jiang N Shi T Hayat and A Alsaedi ldquoNontrivialperiodic solution of a stochastic non-autonomous SISV epi-demic modelrdquo Physica A Statistical Mechanics and its Applica-tions vol 462 pp 837ndash845 2016
[33] Y L Zhou S L Yuan and D L Zhao ldquoThreshold behavior ofa stochastic SIS model with levy jumpsrdquo Applied Mathematicsand Computation vol 275 pp 255ndash267 2016
[34] T Feng X Meng L Liu and S Gao ldquoApplication of inequal-ities technique to dynamics analysis of a stochastic eco-epidemiology modelrdquo Journal of Inequalities and ApplicationsPaper No 327 29 pages 2016
[35] M Liu and K Wang ldquoPersistence and extinction in stochasticnon-autonomous logistic systemsrdquo Journal of MathematicalAnalysis and Applications vol 375 no 2 pp 443ndash457 2011
[36] Z Bai and Y Zhou ldquoExistence of two periodic solutions for anon-autonomous SIR epidemic modelrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 35 no 1 pp 382ndash391 2011
[37] S Gao F Zhang and Y He ldquoThe effects of migratory birdpopulation in a nonautonomous eco-epidemiological modelrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 37 no 6 pp3903ndash3916 2013
[38] T Kuniya ldquoExistence of a nontrivial periodic solution in an age-structured SIR epidemicmodel with time periodic coefficientsrdquoApplied Mathematics Letters vol 27 pp 15ndash20 2014
[39] L Zu D Jiang D OrsquoRegan and B Ge ldquoPeriodic solution fora non-autonomous Lotka-Volterra predator-prey model withrandom perturbationrdquo Journal of Mathematical Analysis andApplications vol 430 no 1 pp 428ndash437 2015
[40] Y Lin D Jiang and T Liu ldquoNontrivial periodic solution ofa stochastic epidemic model with seasonal variationrdquo AppliedMathematics Letters vol 45 pp 103ndash107 2015
[41] X R Mao Stochastic Differential Equations and Their Applica-tions Horwood Chichester UK 1997
[42] R Khasminskii Stochastic Stability of Differential EquationsSpringer Berlin Germany 2011
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
