# Dynamics of a Nonautonomous Stochastic SIS Epidemic Model with Double Epidemic Hypothesis

Authors: Haokun Qi; Lidan Liu; Xinzhu Meng
Journal: Complexity
 (2017)

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## Abstract

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.


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## Body

## 1. Introduction

The 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]. The SIS model is defined in that individuals start off susceptible, at some stage catch the disease, and after a short infectious period become susceptible again [2]. Therefore, 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)S˙t=A-dSt-β1StI1ta1+I1t-β2StI2ta2+I2t+r1I1t+r2I2t,I˙1t=β1StI1ta1+I1t-d+α1+r1I1t,I˙2t=β2StI2ta2+I2t-d+α2+r2I2t,where S(t), I1(t), and I2(t) represent the number of susceptibles and infected individuals with viruses A and B at time t, respectively. The parameters in model (1) have the following meanings: A is the total input susceptible population size, d represents the natural death rate of S, I1, and I2, βi represents the disease transmission coefficient between compartments S and Ii(i=1,2), r1 and r2 are the recovery rates of the two diseases, and α1 and α2 are mortality rates due to diseases, respectively. Functions β1S(t)I1(t)/(a1+I2(t)) and β2S(t)I2(t)/(a2+I2(t)) represent two different types of saturated incidence rates for the two epidemic diseases I1(t) and I2(t). All parameter values are nonnegative.In the real world, population systems and epidemic systems are inevitably infected by some uncertain environmental disturbances. Hence, many authors have introduced stochastic 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, 30] they obtained thresholds of the stochastic system which determines the extinction and persistence of the epidemic. Zhang et al. [29] proved that there is a unique ergodic stationary distribution of his model. We assume that environment fluctuations will manifest themselves mainly as fluctuations in the saturated response rate, so that βiS(t)Ii(t)/(ai+Ii(t))→βiS(t)Ii(t)/(ai+Ii(t))+(σiS(t)Ii(t)/(ai+Ii(t)))B˙i(t)(i=1,2), where B(t)=(B1(t),B2(t)) is a standard Brownian motion with intensity σi>0(i=1,2). Therefore, a stochastic model is described by [11](2)dSt=A-dSt-β1StI1ta1+I1t-β2StI2ta2+I2t+r1I1t+r2I2tdt-σ1StI1ta1+I1tdB1t-σ2StI2ta2+I2tdB2t,dI1t=β1StI1ta1+I1t-d+α1+r1I1tdt+σ1StI1ta1+I1tdB1t,dI2t=β2StI2ta2+I2t-d+α2+r2I2tdt+σ2StI2ta2+I2tdB2t.However, many infectious diseases of human fluctuate over time and often show the seasonal morbidity. Therefore, the existence of periodic solutions of some nonautonomous epidemic models was explored [35–37]. Recently, many scholars focused on nonautonomous stochastic periodic systems. With the development of stochastic differential equations and application of Has’minskii theory, the existence of stochastic periodic solution has been studied [23, 38, 39]. In [23], Zhang et al. considered a nonautonomous stochastic Lotka-Volterra predator-prey model with impulsive effects; they got thresholds for stochastic persistence and extinction of the system. Authors of [38–40] investigated periodic solution of a stochastic nonautonomous epidemic model.Based on the discussion above, in this paper, we consider a nonautonomous stochastic SIS model with periodic coefficients(3)dSt=At-dtSt-β1tStI1ta1t+I1t-β2tStI2ta2t+I2t+r1tI1t+r2tI2tdt-σ1tStI1ta1t+I1tdB1t-σ2tStI2ta2t+I2tdB2t,dI1t=β1tStI1ta1t+I1t-dt+α1t+r1tI1tdt+σ1tStI1ta1t+I1tdB1t,dI2t=β2tStI2ta2t+I2t-dt+α2t+r2tI2tdt+σ2tStI2ta2t+I2tdB2t,where the parameter functions A(t), d(t), βi(t), ai(t), ri(t), αi(t), σi(t)(i=1,2) are positive, nonconstant, and continuous periodic functions with positive period T.To the best of our knowledge, there are only few works on research of nonautonomous stochastic epidemic models with nonlinear saturated incidence rate and double epidemic hypothesis. Therefore, based on an autonomous stochastic epidemic model, we propose a nonautonomous stochastic model and investigate the existence of stochastic periodic solution and the extinction of the two epidemic diseases.This paper is organized as follows. In Section2, we give some definitions and known results. In Section 3, we prove that system (3) has a unique global positive solution. In Section 4, we present sufficient condition for the existence a nontrivial positive periodic solution of system (3). In Section 5, we obtain the sufficient conditions of system (3) for extinction of the two epidemic diseases. In Section 6, we carry out a series of numerical simulations to illustrate our theoretical findings.


## 2. Preliminaries

Throughout this paper, let(Ω,F,P) be a complete probability space with a filtration {Ft}t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F0 contains all P-null sets). The function Bi(t)(i=1,2,3,4) is defined on this complete probability space.For simplicity, some notations are given first. Iff(t) is an integrable function defined on [0,∞), define ft=(1/t)∫0tf(s)ds,t>0. If f(t) is a bounded function on [0,∞), definefl=inft∈[0,∞)⁡f(t) and fu=supt∈[0,∞)⁡f(t).Here we present some basic theory in stochastic differential equations which are introduced in [41].In general, consider thel-dimensional stochastic differential equation(4)dXt=fXt,tdt+gXt,tdBt,t≥t0,with initial value x(t0)=x0∈Rl. B(t) stands for a l-dimensional standard Brownian motion defined on the complete probability space (Ω,F,{Ft}t≥0,P). Denote by C2,1(Rl×[t0,∞];R+) the family of all nonnegative functions V(X,t) defined on Rl×[t0,∞] such that they are continuously twice differentiable in X and once in t. The differential operator L of (4) is defined by [41] (5)L=∂∂t+∑fiX,t∂∂Xi+12∑i,j=1lgTX,tgX,tij∂2∂Xi∂Xj.If L acts on a function V∈C2,1(Rl×[t0,∞];R+), then (6)LVX,t=VtX,t+VXX,tfX,t+12tracegTX,tVXXX,tgX,t,where Vt=∂V/∂t, VX=(∂V/∂x1,…,∂V/∂xl), and VXX=(∂2V/∂Xi∂Xj)l×l. In view of Itô’s formula, if X(t)∈Rl, then (7)dVXt,t=LVxt,tdt+VXXt,tgXt,tdBt.Definition 1 (see [42]).
A stochastic processξ(t)=ξ(t,ω)(-∞<t<+∞) is said to be periodic with period T if for every finite sequence of numbers t1,t2,…,tn the joint distribution of random variables ξ(t1+h),ξ(t2+h),…,ξ(tn+h) is independent of h, where h=kT,k=±1,±2,….It is shown in [42] that a Markov process x(t) is T-periodic if and only if its transition probability function is T-periodic and the function P0(t,A)=P{x(t)∈A} satisfies the equation (8)P0s,A=∫RlP0s,dxPs,x,s+T,A=P0s+T,A.Consider the following equation:(9)Xt=Xt0+∫t0tbs,Xsds+∑r=1k∫t0tσrs,XsdBrs,X∈Rl.Lemma 2 (see [42]).
Suppose that coefficients of (9) are T-periodic in t and satisfy the condition (10)bs,x-bs,y+∑r=1kσrs,x-σrs,y≤Px-y,bs,x+∑r=1kσrs,x≤P1+x,in every cylinder I×U, where P is a constant. And suppose further that there exists a function V(t,x)∈C2 in Rd which is T-periodic in t and satisfies the following conditions: (A1)
infx≥RV(t,x)→∞ as R→∞.(A2)
LV(t,x)≤-1 outside some compact set, where the operator L is given by (11)L=∂∂t+∑i=1lbit,x∂∂xi+12∑i,j=1laijt,x∂2∂xi∂xj,aij=∑r=1kσrit,xσrjt,x.Then there exists a solution of (9) which is a T-periodic Markov process.Lemma 3 (see [2], strong law of large numbers).
LetM={Mt}t≥0 be a real-valued continuous local martingale vanishing at t=0.
Then(12)limt→∞⁡M,Mt=∞a.s.⟹limt→∞⁡MtM,Mt=0a.s.and also (13)limsupt→∞⁡M,Mtt<∞a.s.⟹limt→∞⁡Mtt=0a.s.


## 3. Existence and Uniqueness of the Global Positive Solution

In this section, we prove that system (3) has a unique global positive solution.Theorem 4.
For any initial value(S(0),I1(0),I2(0))∈R+3, there is a unique positive solution (S(t),I1(t),I2(t)) of (3) on t≥0 and the solution will remain in R+3 with probability one.Proof.
From system (3), we can get (14)dSt+I1t+I2tdt=At-dtSt+I1t+I2t-α1tI1t+α2tI2t≤At-dtSt+I1t+I2t≤Au-dlSt+I1t+I2t.Then(15)limt→+∞⁡St+I1t+I2t≤Audl;obviously, we have (16)limsupt→+∞⁡St≤Audl,limsupt→+∞⁡I1t≤Audl,limsupt→+∞⁡I2t≤Audl.
Since the coefficients of system (3) satisfy the local Lipschitz conditions, then for any given initial value (S(0),I1(0),I2(0))∈R+3, there is a unique local solution (S(t),I1(t),I2(t)) on t∈[0,τe), where τe is the explosion time. To demonstrate that this solution is global, we only need to prove that τe=∞ a.s.
Letk0>0 be sufficiently large for any initial value S(0),I1(0), and I2(0) lying within the interval [1/k0,k]. For each integer k≥k0, define the following stopping time: (17)τk=inf⁡t∈0,τe:min⁡St,I1t,I2t≤1kormax⁡St,I1t,I2t≥k,where we set inf⁡∅=∞ (as usual ∅ denotes the empty set). Clearly, τk is increasing as k→∞. Let τ∞=limk→∞τk; hence τ∞≤τk a.s. Next, we only need to verify τ∞=∞ a.s. If this statement is false, then there exist two constants T>0 and ϵ∈(0,1) such that (18)Pτ∞≤T>ϵ.Thus there is an integer k1≥k0 such that(19)Pτk≤T≥ϵ,k≥k1.Define a C2-function V: R+3→R+ as follows: (20)VS,I1,I2=S-1-ln⁡S+I1-1-ln⁡I1+I2-1-ln⁡I2;the nonnegativity of this function can be obtained from (21)u-1-ln⁡u≥0,u>0.Applying Itô’s formula yields (22)dVS,I1,I2=LVS,I1,I2dt+2SI1-S-I1σ1ta1t+I1dB1t+2SI2-S-I2σ2ta2t+I2dB2t,where(23)LV=1-1SAt-dtS-β1tSI1a1t+I1-β2tSI2a2t+I2+r1tI1+r2tI2+12σ12tI12a1t+I12+12σ22tI22a2t+I22+1-1I1β1tSI1a1t+I1-dt+α1t+r1tI1+12σ12tS2a1t+I12+1-1I2β2tSI2a2t+I2-dt+α2t+r2tI2+12σ22tS2a2t+I22=At-dtS-AtS+dt+β1tI1a1t+I1+β2tI2a2t+I2-r1tI1S-r2tI2S+σ12tI122a1t+I12+σ22tI222a2t+I22-dt+α1tI1-β1tSa1t+I1+dt+α1t+r1t+σ12tS22a1t+I12-dt+α2tI2-β2tSa2t+I2+dt+α2t+r2t+σ22tS22a2t+I22≤Au+3du+α1u+α2u+r1u+r2u+β1u+β2u+σ12u2+σ22u2+σ12uA2u2a12ld2l+σ22uA2u2a22ld2l≔K,where K is a positive constant.
So we have(24)dVS,I1,I2≤Kdt+2SI1-S-I1σ1ta1t+I1dB1t+2SI2-S-I2σ2ta2t+I2dB2t.Integrating (24) from 0 to τk∧T and taking expectations on both sides yield(25)EVSτk∧T,I1τk∧T,I2τk∧T≤VS0,I10,I20+KT.Let Ωk={τk≤T}; from inequality (25) we can see that P(Ωk)≥ϵ. We have(26)VSτk∧T,I1τk∧T,I2τk∧T≥k-1-ln⁡k∧1k-1-ln⁡1k.By (25) and (26), one has (27)VS0,I10,I20+KT≥E1ΩkωVSτk∧T,I1τk∧T,I2τk∧T≥ϵk-1-ln⁡k∧1k-1-ln⁡1k,where 1Ωk is the indicator function of Ωk.
Letk→∞; we have (28)∞>VS0,I10,I20+KT=∞.So we obtain τ∞=∞. The proof is completed.


## 4. Existence of NontrivialT-Periodic Solution

In this section, we verify that system (3) admits at least one nontrivial positive T-periodic solution. Define(29)R=∑i=12AαiβiTd+αi+ri+σi2A2u/2ai2d2lTd+σ12/2+σ22/2TA+a1α1+a2α2T.Theorem 5.
Whena1(t)r1(t)/(β1(t)-a1(t)d(t))l>Au/dl and a2(t)r2(t)/(β2(t)-a2(t)d(t))l>Au/dl hold, if R>1, then there exists a nontrivial positive T-periodic solution of system (3).Proof.
Define aC2-function V¯:[0,∞)×R+3→R+: (30)V¯t,S,I1,I2=M-e1+e2ln⁡S-c1ln⁡I1-c2ln⁡I2+S+I1+I2+ωt-ln⁡S-I1-I2+1θ+1S+I1+I2θ+1=MV1+ωt+V2+V3+V4+V5,where(31)c1=Aα1β1Td+α1+r1+σ12A2u/2a12d2lT2d+σ12/2+σ22/2T,e1=Aα1β1Td+α1+r1+σ12A2u/2a12d2lTd+σ12/2+σ22/2T2,c2=Aα2β2Td+α2+r2+σ22A2u/2a22d2lT2d+σ12/2+σ22/2T,e2=Aα2β2Td+α2+r2+σ22A2u/2a22d2lTd+σ12/2+σ22/2T2,and 0<θ<min⁡1,dl/(σ12u+σ22u); M is a sufficiently large positive constant and satisfies the following conditions: (32)dl-θσ12u+σ22u>0,-Mλ+max⁡D,E≤-2,where(33)λ=A+a1α1+a2α2TR-1,D=supS,I1,I2∈R+3⁡du+α2u+r2uI2-12dl-θσ12u+σ22uI2θ+1+O+du+β1u+β2u+σ12u2+σ22u2,E=supS,I1,I2∈R+3⁡du+α1u+r1uI1-12dl-θσ12u+σ22uI1θ+1+O+du+β1u+β2u+σ12u2+σ22u2,where(34)O=supS,I1,I2∈R+3⁡AuS+I1+I2θ-12dl-θσ12u+σ22uS+I1+I2θ+1<∞.Next we prove that condition (A1) in Lemma 2 holds. It is easy to check that V¯(t,S,I1,I2) is a T-periodic function in t and satisfies (35)liminfK→∞,S,I1,I2∈R+3\Uk⁡V¯t,S,I1,I2=∞,where Uk=1/k,k×1/k,k×1/k,k and k>1 is a sufficiently large number.
By Itô’s formula, we obtain(36)LV1=-e1+e2SAt-dtS-β1tSI1a1t+I1-β2tSI2a2t+I2+r1tI1+r2tI2-c1I1β1tSI1a1t+I1-dt+α1t+r1tI1+e1+e22σ12tI12a1t+I12-c2I2β2tSI2a2t+I2-dt+α2t+r2tI2+e1+e22σ22tI22a2t+I22+c12σ12tS2a1t+I12+c22σ22tS2a2t+I22+At-dtS+I1+I2-α1tI1-α2tI2≤-e1AtS-c1β1tSa1t+I1-α1ta1t+I1-e2AtS-c2β2tSa2t+I2-α2ta2t+I2+c1dt+α1t+r1t+σ12tA2u2a12td2l+e1dt+σ12t2+σ22t2+c2dt+α2t+r2t+σ22tA2u2a22td2l+e2dt+σ12t2+σ22t2-e1+e2r1tS+dt-β1ta1tI1-e1+e2r2tS+dt-β2ta2tI2+At+a1tα1t+a2tα2t≤-3Atα1tβ1tc1e11/3-3Atα2tβ2tc2e21/3+e1dt+σ12t2+σ22t2+c1dt+α1t+r1t+σ12tA2u2a12td2l+c2dt+α2t+r2t+σ22tA2u2a22td2l+e2dt+σ12t2+σ22t2+At+a1tα1t+a2tα2t-e1+e2r1tS+dt-β1ta1tI1-e1+e2r2tS+dt-β2ta2tI2≔R0t-e1+e2r1tS+dt-β1ta1tI1-e1+e2r2tS+dt-β2ta2tI2,where(37)R0t=3Atα1tβ1tc1e11/3+c1dt+α1t+r1t+σ12tA2u2a12td2l+e1dt+σ12t2+σ22t2-3Atα2tβ2tc2e21/3+c2dt+α2t+r2t+σ22tA2u2a22td2l+e2dt+σ12t2+σ22t2+At+a1tα1t+a2tα2t.Note that a1(t)r1(t)/(β1(t)-a1(t)d(t))l>Au/dl and a2(t)r2(t)/(β2(t)-a2(t)d(t))l>Au/dl hold; then (38)LV1≤R0t.Define the T-periodic function ω(t) satisfying (39)ω′t=R0T-R0t.So(40)LV1+ωt≤R0T=-A+a1α1+a2α2TR-1≔-λ.Applying Itô’s formula, we can also have (41)LV2=-AtS+dt+β1tI1a1t+I1+β2tI2a2t+I2-r1tI1S-r2tI2S+σ12tI122a1t+I12+σ22tI222a2t+I22≤-AlS+du+β1u+β2u+σ12u2+σ22u2,LV3=-β1tSI1a1t+I1+dt+α1t+r1tI1≤du+α1u+r1uI1,LV4=-β2tSI2a2t+I2+dt+α2t+r2tI2≤du+α2u+r2uI2,LV5=S+I1+I2θAt-dtS-dtI1-dtI2-α1tI1-α2tI2+θS+I1+I2θ-1σ12tS2I12a1t+I12+σ22tS2I22a2t+I22≤AuS+I1+I2θ-dlS+I1+I2θ+1+θS+I1+I2θ-1σ12u+σ22uS2≤AuS+I1+I2θ-12dl-θσ12u+σ22uS+I1+I2θ+1-12dl-θσ12u+σ22uS+I1+I2θ+1≤O-12dl-θσ12u+σ22uSθ+1-12dl-θσ12u+σ22uI1θ+1-12dl-θσ12u+σ22uI2θ+1,where (42)O=supS,I1,I2∈R+3⁡AuS+I1+I2θ-12dl-θσ12u+σ22uS+I1+I2θ+1<∞.Therefore(43)LV¯t,S,I1,I2≤-MA+a1α1+a2α2TR-1+du+α1u+r1uI1+du+α2u+r2uI2-AlS-12dl-θσ12u+σ22uSθ+1-12dl-θσ12u+σ22uI1θ+1-12dl-θσ12u+σ22uI2θ+1+O+du+β1u+β2u+σ12u2+σ22u2=-Mλ-AlS+du+α1u+r1uI1+du+α2u+r2uI2-12dl-θσ12u+σ22uSθ+1-12dl-θσ12u+σ22uI1θ+1-12dl-θσ12u+σ22uI2θ+1+O+du+β1u+β2u+σ12u2+σ22u2.Now, we are in the position to construct a compact subset U such that A2 in Lemma 2 holds. Define the following bounded closed set: (44)U=S,I1,I2∈R+3:ϵ≤S≤1ϵ,ϵ≤I1≤1ϵ,ϵ≤I2≤1ϵ,where ϵ>0 is a sufficiently small number. In the set R+3\U, we can choose ϵ sufficiently small such that(45)-Mλ-Alϵ+C≤-1,(46)-Mλ+du+α1u+r1uϵ+D≤-1,(47)-Mλ+du+α2u+r2uϵ+E≤-1,(48)-Mλ-12dl-θσ12u+σ22u1ϵθ+1+C≤-1,(49)-Mλ-14dl-θσ12u+σ22u1ϵθ+1+G≤-1,(50)-Mλ-14dl-θσ12u+σ22u1ϵθ+1+H≤-1,where C, D, E, G, and H are positive constants which can be found from the following inequations (52), (54), (56), (59), and (61), respectively. For the sake of convenience, we divide into six domains, (51)U1=S,I1,I2∈R+3,0<S<ϵ,U2=S,I1,I2∈R+3,0<I1<ϵ,U3=S,I1,I2∈R+3,0<I2<ϵ,U4=S,I1,I2∈R+3,S>1ϵ,U5=S,I1,I2∈R+3,I1>1ϵ,U6=S,I1,I2∈R+3,I2>1ϵ.
Next we will prove thatLV¯(S,I1,I2)≤-1 on R+3\U, which is equivalent to proving it on the above six domains.
Case  1. If (S,I1,I2)∈U1, one can see that(52)LV¯≤-Mλ-AlS+du+α1u+r1uI1+du+α2u+r2uI2-12dl-θσ12u+σ22uI1θ+1-12dl-θσ12u+σ22uI2θ+1+O+du+β1u+β2u+σ12u2+σ22u2≤-Mλ-AlS+C≤-Mλ-Alϵ+C,where (53)C=supS,I1,I2∈R+3⁡du+α1u+r1uI1+du+α2u+r2uI2-12dl-θσ12u+σ22uI1θ+1-12dl-θσ12u+σ22uI2θ+1+O+du+β1u+β2u+σ12u2+σ22u2.
According to (45), we have LV¯≤-1 for all (S,I1,I2)∈U1.
Case  2. If (S,I1,I2)∈U2, one can get that(54)LV¯≤-Mλ+du+α1u+r1uI1+du+α2u+r2uI2-12dl-θσ12u+σ22uI2θ+1+O+du+β1u+β2u+σ12u2+σ22u2≤-Mλ+du+α1u+r1uI1+D≤-Mλ+du+α1u+r1uϵ+D,where (55)D=supS,I1,I2∈R+3⁡du+α2u+r2uI2-12dl-θσ12u+σ22uI2θ+1+O+du+β1u+β2u+σ12u2+σ22u2.
In view of (46), we can obtain that LV¯≤-1 for all (S,I1,I2)∈U2.
Case  3. If (S,I1,I2)∈U3, we have(56)LV¯≤-Mλ+du+α2u+r2uI2+du+α1u+r1uI1-12dl-θσ12u+σ22uI1θ+1+O+du+β1u+β2u+σ12u2+σ22u2≤-Mλ+du+α2u+r2uI2+E≤-Mλ+du+α2u+r2uϵ+E,where (57)E=supS,I1,I2∈R+3⁡du+α1u+r1uI1-12dl-θσ12u+σ22uI1θ+1+O+du+β1u+β2u+σ12u2+σ22u2.
By (47), we can conclude that LV¯≤-1 for all (S,I1,I2)∈U3.
Case  4. If (S,I1,I2)∈U4, one can derive that(58)LV¯≤-Mλ-12dl-θσ12u+σ22uSθ+1+du+α1u+r1uI1-12dl-θσ12u+σ22uI1θ+1+du+α2u+r2uI2-12dl-θσ12u+σ22uI2θ+1+O+du+β1u+β2u+σ12u2+σ22u2≤-Mλ-12dl-θσ12u+σ22uSθ+1+C≤-Mλ-12dl-θσ12u+σ22u1ϵθ+1+C.
Together with (48), we can deduce that LV¯≤-1 for all (S,I1,I2)∈U4.
Case  5. If (S,I1,I2)∈U5, it follows that(59)LV¯≤-Mλ-14dl-θσ12u+σ22uI1θ+1+du+α1u+r1uI1-14dl-θσ12u+σ22uI1θ+1+du+α2u+r2uI2-12dl-θσ12u+σ22uI2θ+1+O+du+β1u+β2u+σ12u2+σ22u2≤-Mλ-14dl-θσ12u+σ22uI1θ+1+G≤-Mλ-14dl-θσ12u+σ22u1ϵθ+1+G,where (60)G=supS,I1,I2∈R+3⁡du+α1u+r1uI1-14dl-θσ12u+σ22uI1θ+1+du+α2u+r2uI2-12dl-θσ12u+σ22uI2θ+1+O+du+β1u+β2u+σ12u2+σ22u2.
By virtue of (49), we can deduce that LV¯≤-1 for all (S,I1,I2)∈U5.
Case  6. If (S,I1,I2)∈U6, we obtain(61)LV¯≤-Mλ-14dl-θσ12u+σ22uI2θ+1+du+α1u+r1uI1-12dl-θσ12u+σ22uI1θ+1+du+α2u+r2uI2-14dl-θσ12u+σ22uI2θ+1+O+du+β1u+β2u+σ12u2+σ22u2≤-Mλ-14dl-θσ12u+σ22uI2θ+1+H≤-Mλ-14dl-θσ12u+σ22u1ϵθ+1+H,where (62)H=supS,I1,I2∈R+3⁡du+α1u+r1uI1-12dl-θσ12u+σ22uI1θ+1+du+α2u+r2uI2-14dl-θσ12u+σ22uI2θ+1+O+du+β1u+β2u+σ12u2+σ22u2.It follows from (50) that LV¯≤-1 for all (S,I1,I2)∈U6.
Clearly, one can see from (52), (54), (56), (58), (59), and (61) that, for a sufficiently small ϵ, (63)LV¯S,I1,I2≤-1,S,I1,I2∈R+3\U.Hence A2 in Lemma 2 is satisfied. This completes the proof of Theorem 5.


## 5. Extinction

In this section, we investigate the conditions for the extinction of the two infectious diseases of system (3).Let(64)R1=Auβ1Tdla1Td+α1+r1T-A2uσ12T2d2la12Td+α1+r1T,R2=Auβ2Tdla2Td+α2+r2T-A2uσ22T2d2la22Td+α2+r2T.Theorem 6.
Let(S(t),I1(t),I2(t)) be a solution of system (3) with initial value (S(0),I1(0),I2(0))∈R+3.
Then if(65)i:σi2T>βi2T2dl+αil+rilT,i=1,2,or(66)ii:Ri<1,σi2T≤dlai2TβiTAuaiT,i=1,2,hold, the two infectious diseases of system (3) go to extinction a.s.; that is, (67)limt→+∞⁡Iit=0,i=1,2.Proof.
Applying Itô’s formula to system (3), we have(68)dln⁡Iit=βitStait+Iit-dt+αit+rit-σi2tS2t2ait+Iit2dt+σitStait+IitdBit,i=1,2.Case (i). Integrating (68) from 0 to t and dividing t on both sides, we obtain(69)ln⁡Iitt=-σi2t2t∫0tSτaiτ+Iiτ-βiτσi2τ2dτ-1t∫0tdτ+αiτ+riτdτ+1t∫0tβi2τ2σi2τdτ+Mitt+ln⁡Ii0t≤-d+αi+ri-βi22σi2t+Mitt+ln⁡Ii0t.Case (ii). Integrating (68) from 0 to t first and then dividing by t on both sides yield(70)ln⁡Iitt=1t∫0tβiτSτaiτ+Iiτ-dτ+αiτ+riτ-σi2τS2τ2aiτ+Iiτ2dτ+Mitt+ln⁡Ii0t≤βiAuaidl-d+αi+ri-σi2A2u2ai2d2lt+Mitt+ln⁡Ii0t≤d+αi+riTRi-1+Mitt+ln⁡Ii0t,where Mi(t)=∫0tσi(τ)S(τ)/(ai(τ)+Ii(τ))dBi(τ),i=1,2, which is a local continuous martingale with Mi(0)=0. By Lemma 3, we have (71)limt→+∞⁡Mitt=0,i=1,2.Taking the limit superior of both sides of (69) leads to (72)limsupt→+∞⁡ln⁡Iitt≤-d+αi+ri-βi22σi2T<0,which implies limt→+∞⁡Ii(t)=0.
Taking the superior limit of both sides of (70) leads to (73)limsupt→+∞⁡ln⁡Iitt≤d+αi+riTRi-1<0,which implies limt→+∞Ii(t)=0,i=1,2. This completes the proof.Remark 7.
Theorem6 shows that the two diseases will die out if the white noise disturbance is large or the white noise disturbance is not large and Ri<1. When σi2T>βi2T/2dl+αil+rilT, the two infectious diseases of system (3) die out almost surely; that is to say, large white noise stochastic disturbance can lead to the two epidemics being extinct.


## 6. Numerical Simulations

Now we introduce some numerical simulations examples which illustrate our theoretical results.Example 8.
In model (3), let (74)At=0.5+0.1sin⁡πt,a1t=0.3+0.1sin⁡πt,a2t=0.31+0.1sin⁡πt,β1t=0.62+0.1sin⁡πt,β2t=0.62+0.1sin⁡πt,r1t=0.2+0.1sin⁡πt,r2t=0.35+0.1sin⁡πt,dt=0.2+0.1sin⁡πt,α1t=0.2+0.1sin⁡πt,α2t=0.25+0.1sin⁡πt,σ1t=0.1+0.05sin⁡πt,σ2t=0.1+0.05sin⁡πt.Note that R>1, a1(t)r1(t)/(β1(t)-a1(t)d(t))>Au/dl, and a2(t)r2(t)/(β2(t)-a2(t)d(t))>Au/dl hold; that is, the conditions of Theorem 5 hold. Hence, system (3) has a positive periodic solution with T=1. Figure 1(a) shows the periodicity of the nonautonomous stochastic model (3) with σ1=0 and σ2=0. Figure 1(b) shows that solution of the nonautonomous stochastic model (3) with the initial value (S(t),I1(t),I2(t))=(0.3,0.15,0.15) tends to a periodic orbit in the sense of joint distribution.Figure 1
The solution(S(t),I1(t),I2(t))=(1,0.2,0.2) to the nonautonomous stochastic model (3).
(a)
(b)Example 9.
Choose the parameters in model (3) as follows:(75)At=0.2+0.2sin⁡πt,a1t=0.23+0.2sin⁡πt,a2t=0.25+0.2sin⁡πt,β1t=0.2+0.2sin⁡πt,β2t=0.3+0.2sin⁡πt,r1t=0.2+0.2sin⁡πt,r2t=0.3+0.2sin⁡πt,dt=0.25+0.2sin⁡πt,α1t=0.2+0.2sin⁡πt,α2t=0.3+0.2sin⁡πt,σ1t=0.4+0.2sin⁡πt,σ2t=0.5+0.2sin⁡πt.Note that σi2T>βi2T/2dl+αil+rilT. Therefore, conditions (i) of Theorem 6 hold. Then the two infectious diseases will go to extinction. Figure 2(a) shows that one of two diseases in the deterministic SIS epidemic model is extinct and the other is persistent. Figure 2(b) shows that the two diseases will die out under the large white noise disturbance of model (3).Figure 2
The solution(S(t),I1(t),I2(t))=(0.3,0.25,0.25) to the nonautonomous stochastic model (3).
(a)
(b)Example 10.
Choose the parameters in model (3) as follows: (76)At=0.2+0.2sin⁡πt,a1t=0.23+0.2sin⁡πt,a2t=0.25+0.2sin⁡πt,β1t=0.2+0.2sin⁡πt,β2t=0.3+0.2sin⁡πt,r1t=0.2+0.2sin⁡πt,r2t=0.3+0.2sin⁡πt,dt=0.25+0.2sin⁡πt,α1t=0.2+0.2sin⁡πt,α2t=0.3+0.2sin⁡πt,σ1t=0.2+0.2sin⁡πt,σ2t=0.2+0.2sin⁡πt.
Note thatR1<1, R2<1, and σi2T≤dlai2Tβi2T/AuaiT. That is, conditions (ii) of Theorem 6 hold. Then the two infectious diseases will go to extinction. Figure 3(a) shows that one of two diseases in the deterministic SIS epidemic model is extinct and the other is persistent without the white noises. Figure 3(b) shows that the two diseases will die out under a small white noise disturbance of model (3).Figure 3
The solution(S(t),I1(t),I2(t))=(0.3,0.25,0.25) to the nonautonomous stochastic model (3).
(a)
(b)


## 7. Discussion and Conclusions

This paper explores the existence of nontrivial positive T-periodic solution of a nonautonomous stochastic SIS epidemic model with nonlinear growth rate and double epidemic hypothesis. By constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of nontrivial positiveT-periodic solution of system (3). Furthermore, the sufficient conditions for the extinction of the two diseases are obtained. Our results are given as follows:(1)
Ifa1(t)r1(t)/(β1(t)-a1(t)d(t))>Au/dl, a2(t)r2(t)/(β2(t)-a2(t)d(t))>Au/dl, and R>1 hold, the SIS model has at least one nontrivial positive T-periodic solution, where (77)R=∑i=12AαiβiTd+αi+ri+σi2A2u/2ai2d2lTd+σ12/2+σ22/2TA+a1α1+a2α2T.(2)
Ifσi2T>βi2T/2dl+αil+rilT, the two infectious diseases go extinct.(3)
Ifσi2T≤dlai2Tβi2TAuaiT and Ri<1, the two infectious diseases also go extinct, where (78)Ri=AuβiTdlaiTd+αi+riT-A2uσi2T2d2lai2Td+αi+riT.Some interesting questions deserve further investigation. On the one hand, we may explore some realistic but complex models, such as considering the effects of impulsive or delay perturbations on system (3). On the other hand, we can concern the dynamics of a nonautonomous stochastic SIS epidemic model with two infectious diseases driven by Lévy jumps. What is more, we can also investigate the nonautonomous stochastic SIS epidemic model with two infectious diseases by a continuous time Markov chain. We will investigate these cases in our future work.



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