# Analysis of a Deterministic and a Stochastic SIS Epidemic Model with Double Epidemic Hypothesis and Specific Functional Response

Authors: Mouhcine Naim; Fouad Lahmidi
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

The purpose of this paper is to investigate the stability of a deterministic and stochastic SIS epidemic model with double epidemic hypothesis and specific nonlinear incidence rate. We prove the local asymptotic stability of the equilibria of the deterministic model. Moreover, by constructing a suitable Lyapunov function, we obtain a sufficient condition for the global stability of the disease-free equilibrium. For the stochastic model, we establish global existence and positivity of the solution. Thereafter, stochastic stability of the disease-free equilibrium in almost sure exponential andpth moment exponential is investigated. Finally, numerical examples are presented.


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## Body

## 1. Introduction

Epidemiology is the study of the spread of infectious diseases with the objective to trace factors that are responsible for or contribute to their occurrence. Mathematical modeling has become an important tool in analyzing the epidemiological characteristics of infectious diseases and can provide useful control measures (see, for example, [1–5]).In classical epidemic models, the susceptible individuals can be infected with only a disease. In the real world, the susceptible individuals can be infected by two or more kinds of diseases at the same time such as HBV coinfection with HCV and HDV and HIV coinfection with HBV, HCV, and TB. Recently, the authors of [6–9] investigated the epidemic model SIS (where infection with the disease does not confer permanent immunity against reinfection so that those who survived the infection revert to the class of wholly susceptible individuals [10]) with double epidemic hypothesis which has two epidemic diseases caused by two different viruses. In this paper, we consider a deterministic SIS model with double epidemic hypothesis described by the following differential system:(1)S˙t=A−μSt−β1StI1t1+α1St+γ1I1t+μ1StI1t−β2StI2t1+α2St+γ2I2t+μ2StI2t+r1I1t+r2I2t,I˙1t=β1StI1t1+α1St+γ1I1t+μ1StI1t−μ+a1+r1I1t,I˙2t=β2StI2t1+α2St+γ2I2t+μ2StI2t−μ+a2+r2I2t,where St represents the number of susceptible at time t, I1t and I2t are the total population of the infected with virus V1 and V2 at time t, respectively, A represents the recruitment rate of the population, μ is the natural death rate of the population, ri is the treatment cure rate of the disease caused by virus Vi, ai is the disease-related death rate, and βi is the infection coefficient, i=1,2. The incidence rate of disease is modeled by the specific functional response βiSIi/1+αiS+γiIi+μiSIi, where αi,γi,μi are saturation factors measuring the psychological or inhibitory effect. This specific functional response was introduced by Hattaf et al. [11], and here, it becomes to be a bilinear incidence rate if αi=γi=μi=0, a saturated incidence rate if αi=μi=0 or γi=μi=0, a Beddington–DeAngelis functional response [12, 13] if μi=0, and a Crowley–Martin functional response [14] if αiγi=μi, i=1,2.In the reality, epidemic systems are inevitably affected by environmental white noise. Therefore, it is necessary to study how the noise influences the epidemic models. Consequently, many authors have studied stochastic epidemic models, see, e.g., [15–17]. For this, we consider the case in which the rates βi (i=1,2) are subject to random fluctuations, namely, βidt is replaced by βidt+σidBit, where Bit (i=1,2) are independent standard Brownian motions, and σi>0 represents the intensity of Bit for i=1,2. Therefore, the corresponding stochastic system to (1) can be described by the following Itô equations:(2)dSt=A−μSt−β1StI1tf1St,I1t−β2StI2tf2St,I2t+r1I1t+r2I2tdt−σ1StI1tf1St,I1tdB1t−σ2StI2tf2St,I2tdB2t,dI1t=β1StI1tf1St,I1t−μ+a1+r1I1tdt+σ1StI1tf1St,I1tdB1t,dI2t=β2StI2tf2St,I2t−μ+a2+r2I2tdt+σ2StI2tf2St,I2tdB2t,with fiS,Ii=1+αiS+γiIi+μiSIi, i=1,2.The rest of this paper is organized in the following manner. In Section2, we present a local stability analysis of the equilibria and a global stability analysis of the disease-free equilibrium for the deterministic model (1). In Section 3, we prove that the stochastic model (2) has a unique global positive solution, and we give sufficient conditions for the almost sure exponential stability and the pth moment exponential stability of the disease-free equilibrium. Numerical examples will be presented in Section 4. Finally, we close the paper with a brief conclusion.


## 2. Deterministic SIS Epidemic Model

For biological reasons, we assume that the initial conditions of system (1) satisfy(3)S0≥0,I10≥0,I20≥0.Thus, system (1) is positive [18], that is, St≥0,I1t≥0, and I2t≥0 for all t≥0. In fact, by Proposition 2.1 in [19], we have(4)S˙=A+r1I1+r2I2≥0,forS=0andI1,I2≥0,I˙1=0≥0,forI1=0andS,I2≥0,I˙2=0≥0,forI2=0andS,I1≥0.By summing all the equations of system (1), we find that the total population size Nt=St+I1t+I2t satisfies the inequality(5)N˙t=A−μNt−a1I1t−a2I2t≤A−μNt,which ensures that N˙t<0 if Nt>A/μ. The standard comparison theorem [20] can be used to deduce that(6)Nt≤Aμ−Aμ−N0e−μt.Thus, the feasible solution set of the system equation of model (1) enters and remains in the region(7)Γ=S,I1,I2∈ℝ+3:S+I1+I2≤Aμ.Therefore, model (1) is well posed epidemiologically and mathematically [21]. Hence, it is sufficient to study the dynamics of model (1) in Γ.It is easy to see that system (1) has a disease-free equilibrium state E0=A/μ,0,0. Therefore, the basic reproduction number is(8)R0=maxR01,R02,where(9)R01=β1Aμ+α1Aμ+a1+r1,R02=β2Aμ+α2Aμ+a2+r2.We mention that the expressions ofR01 and R02 can also be obtained by applying the next generation matrix method provided by van den Driessche and Watmough [22].Now, we investigate the local stability of the disease-free equilibriumE0. The Jacobian matrix of system (1) at the equilibrium E0 is as follows:(10)JE0=−μ−β1Aμ+α1A+r1−β2Aμ+α2A+r20β1Aμ+α1A−μ+a1+r1000β2Aμ+α2A−μ+a2+r2.The three eigenvalues ofJE0 are λ1=−μ<0, λ2=μ+a1+r1R01−1, and λ3=μ+a2+r2R02−1. Hence, the equilibrium E0 will be locally asymptotically stable if R0<1 and unstable when R0>1.The following theorem discusses the global stability of the disease-free equilibriumE0.Theorem 1.
IfR0≤1, then the disease-free equilibrium E0 of (1) is globally asymptotically stable in Γ.Proof.
LetU be the Lyapunov function defined as(11)Ut=I1t+I2t.
DifferentiatingU with respect to t along the positive solutions of system (1), we get(12)U˙t=β1S1+α1S+γ1I1+μ1SI1−μ+a1+r1I1+β2S1+α2S+γ2I2+μ2SI2−μ+a2+r2I2.
We have(13)βiS1+αiS+γiIi+μiSIi≤βiS1+αiS,i=1,2.
SinceS≤A/μ and the functions fi:x∈ℝ+↦βix/1+αix are increasing, then fiS≤fiA/μ=βiA/μ+αiA. Thus,(14)U˙t≤μ+a1+r1R01−1I1+μ+a2+r2R02−1I2.
Therefore,R0≤1 ensures that U˙t≤0. Suppose that S,I1,I2 is a solution of (1) contained entirely in the set Δ=S,I1,I2∈Γ:U˙t=0. Then, I˙1+I˙2=0. We discuss four cases:Case 1.
IfR01<1 and R02<1, then(15)Xi=βiS1+αiS+γiIi+μiSIi−μ+ai+ri≤βiAμ+αiA−μ+ai+ri=μ+ai+riR0i−1<0,i=1,2.
From the second and third equations of (1), we have X1I1+X2I2=0, which implies, according to (15), that I1=I2=0. On the other hand, solutions of (1) contained in the plane I1=I2=0 satisfy S˙=A−μS, which implies that S⟶A/μ as t⟶∞.Case 2.
IfR01<1 and R02=1, then X1<0 and(16)X2=β2S1+α2S+γ2I2+μ2SI2−β2Aμ+α2A=β2Sμ−A−β2AI2γ2+μ2S1+α2S+γ2I2+μ2SI2μ+α2A≤0.
Then,X1I1+X2I2=0 implies that I1=0 and consequently X2I2=0. Suppose that I2>0; then, X2=0. Hence, Sμ−A=AI2γ2+μ2S>0; then, S>A/μ which is a contradiction. Then, I1=I2=0.Case 3.
The caseR01=1 and R02<1 is analogue to the previous case.Case 4.
IfR01=1 and R02=1, then X1I1+X2I2=0 such that Xi=βiSμ−A−βiAIiγi+μiS/1+αiS+γiIi+μiSIiμ+αiA≤0, i=1,2. Hence, X1I1=X2I2=0, and by the same analysis in Case 2, we obtain that I1=I2=0.Hence, by LaSalle’s invariance principle [23], every solution to equations of system (1), with initial conditions in Γ, approaches E0 as t⟶∞. Thus, E0 is globally asymptotically stable.Now, ifR01>1, then system (1) has the disease-free equilibrium for I2, E1∗=S1∗,I¯1∗,0, where(17)S1∗=A−μ+a1I¯1∗μ,I¯1∗=2ϖ1μ+α1AR01−1μ+a1β1−α1ϖ1+ϖ1γ1μ+μ1A+Δ1,with ϖ1=μ+a1+r1 and(18)Δ1=μ+a1β1−α1ϖ1+ϖ1γ1μ+μ1A2−4μ1μ+a1ϖ1β1A−μ+α1Aϖ1=μ+a1β1−α1ϖ1+ϖ1γ1μ−μ1A2+4μ1μϖ12μ+a1+γ1A.Theorem 2.
IfR01>1 and R02<1, then the equilibrium E1∗ is locally asymptotically stable.Proof.
The Jacobian matrix of system (1) at the equilibrium E1∗ is determined by(19)JE1∗=−m1−m2+r1−m3+r2m4m2−m5000m3−m6,where(20)m1=μ+β1I¯1∗1+γ1I¯1∗1+α1S1∗+γ1I¯1∗+μ1S1∗I¯1∗2,m2=β1S1∗1+α1S1∗1+α1S1∗+γ1I¯1∗+μ1S1∗I¯1∗2,m3=β2S1∗1+α2S1∗,m4=β1I¯1∗1+γ1I¯1∗1+α1S1∗+γ1I¯1∗+μ1S1∗I¯1∗2,m5=μ+a1+r1,m6=μ+a2+r2.
Clearly,λ1=β2S1∗/1+α2S1∗−μ+a2+r2 is an eigenvalue of JE1∗. Since S1∗<A/μ because A−μS1∗=μ+a1I¯1∗>0 and the function f2:x∈ℝ+↦β2x/1+α2x is increasing, then λ1<f2A/μ−μ+a2+r2=β2A/μ+α2A−μ+a2+r2=μ+a2+r2R02−1. Hence, λ1<0 if R02<1. The other two eigenvalues of JE1∗ are determined by the following equation:(21)λ2+a1λ+a0=0,where(22)a1=m1+m5−m2,a0=μ+a1m4+μm5−m2.
Sincem5−m2=β1S1∗I¯1∗γ1+μ1S1∗/1+α1S1∗+γ1I¯1∗+μ1S1∗I¯1∗2>0, then a1>0 and a0>0. Thus, by the Routh–Hurwitz criterion, the eigenvalues λjj=2,3 of JE1∗ have negative real part. Therefore, the equilibrium E1∗ of system (1) is asymptotically stable if R01>1 and R02<1.
Furthermore, ifR02>1, then system (1) has the disease-free equilibrium for I1, E2∗=S2∗,0,I¯2∗, where(23)S2∗=A−μ+a2I¯2∗μ,I¯2∗=2ϖ2μ+α2AR02−1μ+a2β2−α2ϖ2+ϖ2γ2μ+μ2A+Δ2,with ϖ2=μ+a2+r2 and(24)Δ2=μ+a2β2−α2ϖ2+ϖ2γ2μ+μ2A2−4μ2μ+a2ϖ2β2A−μ+α2Aϖ2=μ+a2β2−α2ϖ2+ϖ2γ2μ−μ2A2+4μ2μϖ22μ+a2+γ2A.Theorem 3.
IfR01<1 and R02>1, then the equilibrium E2∗ is locally asymptotically stable.Proof.
It is analogue to the previous proof.
Next, we investigate the local stability of system (1) at both-endemic equilibrium E∗=S∗,I1∗,I2∗. To obtain conditions for the existence of the equilibrium E∗, system (1) is rearranged to get I1∗ and I2∗ which gives(25)I1∗=β1−α1ϖ1S∗−ϖ1ϖ1γ1+μ1S∗,I2∗=β2−α2ϖ2S∗−ϖ2ϖ2γ2+μ2S∗.
We haveIi∗>0 if βi−αiϖi>0 for i=1,2, and S∗>maxi=1,2ϖi/βi−αiϖi. In addition, S∗ is given by the following cubic equation:(26)C0S∗3+C1S∗2+C2S∗−C3=0,where(27)C0=μμ1μ2ϖ1ϖ2>0,C1=ϖ1ϖ2μγ1μ2+γ2μ1−Aμ1μ2+μ1ϖ1μ+a2β2−α2ϖ2+μ2ϖ2μ+a1β1−α1ϖ1,C2=ϖ1ϖ2μγ1γ2−Aγ1μ2+γ2μ1+ϖ1μ+a2γ1β2−α2ϖ2−μ1ϖ2+ϖ2μ+a1γ2β1−α1ϖ1−μ2ϖ1,C3=ϖ1ϖ2Aγ1γ2+γ1μ+a2+γ2μ+a1>0.
With the help of Descartes’ rule of signs [24], equation (26) has a unique positive real root S∗ if any one of the following holds:(i)
C1>0 and C2>0(ii)
C1>0 and C2<0(iii)
C1<0 and C2<0
Hence, system (1) has a unique positive equilibrium E∗ if βi−αiϖi>0 for i=1,2, one of the conditions i, ii, and iii hold true, and S∗>maxi=1,2ϖi/βi−αiϖi.
The Jacobian matrix of system (1) at the equilibrium E∗ is determined by(28)JE∗=−p1−p2+r1−p3+r2p4p2−p50p60p3−p7,where(29)p1=μ+β1I1∗1+γ1I1∗1+α1S∗+γ1I1∗+μ1S∗I1∗2+β2I2∗1+γ2I2∗1+α2S∗+γ2I2∗+μ2S∗I2∗2,p2=β1S∗1+α1S∗1+α1S∗+γ1I1∗+μ1S∗I1∗2,p3=β2S∗1+α2S∗1+α2S∗+γ2I2∗+μ2S∗I2∗2,p4=β1I1∗1+γ1I1∗1+α1S∗+γ1I1∗+μ1S∗I1∗2,p5=μ+a1+r1,p6=β2I2∗1+γ2I2∗1+α2S∗+γ2I2∗+μ2S∗I2∗2,p7=μ+a2+r2.Theorem 4.
The endemic equilibriumE∗ is locally asymptotically stable if it exists.Proof.
The characteristic equation of Jacobian matrixJE∗ can be written as(30)λ3+Q2λ2+Q1λ+Q0=0,where(31)Q2=p1+p5−p2+p7−p3,Q1=μ+p4p7−p3+μ+p6p5−p2+p5−p2p7−p3+μ+a1p4+μ+a2p6,Q0=μp5−p2p7−p3+μ+a1p7−p3p4+μ+a2p5−p2p6.
Note that(32)p5−p2=β1S∗I1∗γ1+μ1S∗1+α1S∗+γ1I1∗+μ1S∗I1∗2>0,p7−p3=β2S∗I2∗γ2+μ2S∗1+α2S∗+γ2I2∗+μ2S∗I2∗2>0.
Then, it is easy to show thatQ2>0, Q1>0, Q0>0, and Q2Q1>Q0. Thus, by the Routh–Hurwitz criterion, all roots λii=1,2,3 of (30) have negative real part. Therefore, the equilibrium E∗ of system (1) is asymptotically stable.


## 3. Stochastic SIS Epidemic Model

LetΩ,ℱ,ℱtt≥0,ℙ be a complete probability space with filtration ℱtt≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while ℱ0 contains all ℙ-null sets). We consider the following stochastic differential system:(33)dxt=fxt,tdt+gxt,tdBt,t≥0,where xt∈ℝn, x0=x0 represents the initial value, and f:ℝn×0,+∞↦ℝn and g:ℝn×0,+∞↦ℝn×m are locally Lipschitz functions in x. Btt≥0 is an m-dimensional standard Wiener process defined on the above probability space.Let us suppose thatf0,t=g0,t=0 for all t≥0 so that zero of ℝn is an equilibrium point of system (33).Definition 1.
(see [25]). The trivial solution x=0 of system (33) is said to be almost surely exponentially stable if for all x0∈ℝn, we have(34)limsupt⟶∞1tlnxt<0a.s..Denote byC2,1ℝn×0,+∞;ℝ+ the family of all nonnegative functions Vx,t defined on ℝn×0,+∞ such that they are continuously twice differentiable in x and once in t. Denote by EX the mathematical expectation of a random variable X. If ℒ acts on a function V∈C2,1ℝn×0,+∞;ℝ+, then(35)ℒVx,t=Vtx,t+Vxx,tfx,t+12tracegx,tTVxxx,tgx,t,where Vtx,t=∂V/∂t, Vxx,t=∂V/∂x1,…,∂V/∂xn, and Vxxx,t=∂2V/∂xi∂xj.By Itô’s formula, we have(36)dVx,t=ℒVx,tdt+Vxx,tgx,tdBt.Lemma 1.
(see [26]). Suppose there exists a function V∈C2,1ℝn×0,+∞;ℝ+ satisfying the inequalities(37)K1xp≤Vx,t≤K2xp,ℒVx,t≤−K3xp,where p>0 and Kii=1,2,3 are positive constants. Then, the equilibrium of system (33) is pth moment exponentially stable. When p=2, it is usually said to be exponentially stable in mean square, and the equilibrium x=0 is globally asymptotically stable.


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

The following theorem shows that the solution of our system (2) is global and positive.Theorem 5.
For any initial valueS0,I10,I20∈Γ, there is a unique solution St,I1t,I2t to (2) on t≥0, and this solution remains in Γ with probability one.Proof.
LetS0,I10,I20∈Γ. The total population in system (2) verifies the equation(38)dNt=A−μNt−a1I1t−a2I2tdt.
IfSs,I1s,I2s∈ℝ+3 for all s∈0,t (a.s.), then we get(39)dNs≤A−μNsdsa.s..
Hence, by integration, we have(40)Ns≤Aμ−Aμ−N0e−μsa.s..
Then,Ns≤A/μ (a.s.), so(41)Ss,I1s,I2s∈0,Aμfor alls∈0,ta.s..
Since the coefficients of system (2) are locally Lipschitz continuous, then by the work of Mao [25] for any initial value S0,I10,I20∈Γ, there is a unique local positive solution St,I1t,I2t on 0,τe, where τe is the explosion time. To show that this solution is global, we only need to prove τe=∞ (a.s.).
Letε0>0 such that S0,I10,I20>ε0. For ε≤ε0, we define the stopping time:(42)τε=inft∈0,τe:St≤εorI1t≤εorI2t≤ε.
Then,(43)τ=limε⟶0τε=inft∈0,τe:St≤0orI1t≤0orI2t≤0.
Consider the functionU defined for S,I1,I2∈ℝ+3 by(44)US,I1,I2=−lnμAS−lnμAI1−lnμAI2.
Calculating the differential ofU along the solution trajectories of system (2) and using Itô’s formula, for all t≥0 and s∈0,t∧τε, we get(45)dUSs,I1s,I2s=ℒUds+σ1I1−Sf1S,I1dB1s+σ2I2−Sf2S,I2dB2s,where(46)ℒU=−A+r1I1+r2I2S+3μ+a1+r1+a2+r2+β1I1−Sf1S,I1+β2I2−Sf2S,I1+σ122I12+S2f1S,I12+σ222I22+S2f2S,I22≤3μ+a1+r1+a2+r2+β1I1f1S,I1+β2I2f2S,I1+σ122I12+S2f1S,I12+σ222I22+S2f2S,I22.
According to (41), we have Ss,I1s,I2s∈0,A/μ for all s∈0,t∧τε (a.s.). Hence,(47)IisfiSs,Iis≤Aμ,SsfiSs,Iis≤Aμ,i=1,2.
Therefore,(48)dU≤ℳds+σ1I1−Sf1S,I1dB1s+σ2I2−Sf2S,I2dB2s,where(49)ℳ=3μ+a1+r1+a2+r2+β1Aμ+β2Aμ+σ1Aμ2+σ2Aμ2.
Integrating both sides of (48) from 0 to t∧τε and after taking the expectation on both sides, we obtain that(50)EUSt∧τε,I1t∧τε,I2t∧τε≤US0,I10,I20+ℳt.
SinceUSt∧τε,I1t∧τε,I2t∧τε>0, then(51)EUSt∧τε,I1t∧τε,I2t∧τε≥EUSt∧τε,I1t∧τε,I2t∧τεχτε≤t≥EUSτε,I1τε,I2τεχτε≤t,where χτk≤t is the indicator function of τk≤t. Note that there are some components of Sτε,I1τε,I2τε equal to ε. Therefore,(52)USτε,I1τε,I2τε≥−lnμAε=lnAμε.
Thus,(53)EUSt∧τε,I1t∧τε,I2t∧τε≥lnAμεℙτε≤t.
By combining (50) and (53), we get that, for all t>0,(54)ℙτε≤t≤US0,I10,I20+ℳtlnA/με.
Extendingε to 0, we obtain for all t>0, ℙτ≤t=0. Hence, ℙτ=∞=1. As τe≥τ, then τ=τe=∞ (a.s.) which completes the proof.


### 3.2. Almost Sure Exponential Stability

The goal of this section is to establish a sufficient condition for the almost sure exponential stability of the disease-free equilibriumE0 in Γ. For this, we consider(55)Ψt=Aμ−St+I1t+I2t,VSt,I1t,I2t=lnΨt.Proposition 1.
Ψt almost surely converges exponentially to 0 if(56)limsupt⟶∞1t∫0tℒVSs,I1s,I2sds<0a.s..Proof.
By Itô’s formula, we have(57)dVSs,I1s,I2s=ℒVSs,I1s,I2sds+2σ1SsI1sΨsf1Ss,I1sdB1s+2σ2SsI2sΨsf2Ss,I2sdB2s.
Integrating both sides from 0 tot yields that(58)VSt,I1t,I2t=VS0,I10,I20+∫0tℒVSs,I1s,I2sds+M1t+M2t,where Mit=∫0t2σiSsIis/ΨsfiSs,IisdBis, i=1,2, are continuous local martingales with Mi0=0. Moreover, we have(59)Mi,Mit=∫0t2σiSsIisΨsfiSs,Iis2ds≤Cit,where Cii=1,2 are positive constants. Thus, the strong law of large numbers for local martingales [27] implies that(60)limt⟶∞Mitt=0a.s..
It follows that(61)limsupt⟶∞VSt,I1t,I2tt=limsupt⟶∞1t∫0tℒVSs,I1s,I2sdsa.s..
The proposition is proved.Then, we obtain the following theorem.Theorem 6.
Ifβ1σ22+β2σ12<μ/2σ1σ22, then the disease-free equilibrium E0 of stochastic system (2) is almost surely exponentially stable in Γ.Proof.
It suffices to prove thatΨt converges to 0 exponentially (a.s.). Then, by Proposition 1, it suffices to prove that(62)limsupt⟶∞1t∫0tℒVSs,I1s,I2sds<0a.s..
By Itô’s formula, we have(63)ℒVSs,I1s,I2s=1Ψ−A+μS+2β1SI1f1S,I1+2β2SI2f2S,I2−r1I1−r2I2−μ+a1+r1I1−μ+a2+r2I2−2σ1SI1Ψf1S,I12−2σ2SI2Ψf2S,I22=1Ψ2β1SI1f1S,I1+2β2SI2f2S,I2−μAμ−S−μ+a1+2r1I1−μ+a2+2r2I2−2σ1SI1Ψf1S,I12−2σ2SI2Ψf2S,I22.
Since(64)μAμ−S+μ+a1+2r1I1+μ+a2+2r2I2≥μAμ−S+μI1+μI2=μΨ,we have(65)ℒVSs,I1s,I2s≤2β1SI1Ψf1S,I1+2β2SI2Ψf2S,I2−μ−2σ1SI1Ψf1S,I12−2σ2SI2Ψf2S,I22.
SetX1=SI1/Ψf1S,I1 and X2=SI2/Ψf2S,I2. Then,(66)ℒVSs,I1s,I2s≤2β1X1−2σ12X12+2β2X2−2σ22X22−μ.
Since2βiXi−2σi2Xi2−μ/2=−2σi2βi/σi2−Xi2+4βi2−μσi2/2σi2, i=1,2, hence,(67)ℒVSs,I1s,I2s≤4β12−μσ122σ12+4β22−μσ222σ22=2β1σ22+2β2σ12−μσ1σ22σ1σ22.
Therefore,(68)limsupt⟶∞1t∫0tℒVSs,I1s,I2sds≤2β1σ22+2β2σ12−μσ1σ22σ1σ22<0a.s..
This completes the proof.


### 3.3. Moment Exponential Stability

In this section, we investigate thepth moment exponential stability of the disease-free equilibrium E0 in Γ of stochastic system (2).We use Lemma1 to prove the following theorem.Theorem 7.
Letp≥2. If(69)maxR01+p−1σ122μ+a1+r1Aμ+α1A2,R02+p−1σ222μ+a2+r2Aμ+α2A2<1,then the disease-free equilibrium E0 of stochastic system (2) is pth moment exponentially stable in Γ.Proof.
Letp≥2 and S0,I10,I20∈Γ. We define the Lyapunov function V as follows:(70)V=ωAμ−Sp+1p∑i=12Iip,where ω is a positive constant which will be determined later. By Itô’s formula, we have(71)ℒV=−ωpAμ−Sp−1A−μS−∑i=12βiSIifiS,Ii+∑i=12riIi+ωpp−12∑i=12σiSIifiS,Ii2Aμ−Sp−2+∑i=12βiSfiS,Ii−μ+ai+riIip+p−12∑i=12σiSfiS,Ii2Iip≤−ωμpAμ−Sp+ωp∑i=12βiAμ+αiAAμ−Sp−1Ii+ωpp−12∑i=12σi2Aμ+αiA2Aμ−Sp−2Ii2+∑i=12βiAμ+αiA−μ+ai+riIip+p−12∑i=12σ12Aμ+αiA2Iip.
From Young’s inequality, forε>0, we have(72)Aμ−Sp−1Ii≤p−1pεAμ−Sp+1pε1−pIip,Aμ−Sp−2Ii2≤p−2pεAμ−Sp+2pε2−p/pIip,i=1,2.
Then,(73)ℒV≤−Q0Aμ−Sp−∑i=12QiIip,where(74)Q0=ωμp−p−1∑i=12βiAμ+αiA+p−22∑i=12σi2Aμ+αiA2ε,Qi=μ+ai+ri−βiAμ+αiA−p−12σi2Aμ+αiA2−ωβiAμ+αiAε1−p+p−1σi2Aμ+αiA2ε2−p/p,i=1,2.
Now, we chooseε sufficiently small such that Q0>0. In view of condition (69), we have μ+ai+ri−βiA/μ+αiA−p−1/2σi2A/μ+αiA2>0 for i=1,2; hence, we can choose ω positive such that Qi>0 for i=1,2. According to Lemma 1, the proof is completed.Remark 1.
From Lemma1, Theorem 7, and the case p=2, we get that if(75)maxR01+σ122μ+a1+r1Aμ+α1A2,R02+σ222μ+a2+r2Aμ+α2A2<1,then the disease-free equilibrium E0 of stochastic system (2) is globally asymptotically stable in Γ.


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

The following theorem shows that the solution of our system (2) is global and positive.Theorem 5.
For any initial valueS0,I10,I20∈Γ, there is a unique solution St,I1t,I2t to (2) on t≥0, and this solution remains in Γ with probability one.Proof.
LetS0,I10,I20∈Γ. The total population in system (2) verifies the equation(38)dNt=A−μNt−a1I1t−a2I2tdt.
IfSs,I1s,I2s∈ℝ+3 for all s∈0,t (a.s.), then we get(39)dNs≤A−μNsdsa.s..
Hence, by integration, we have(40)Ns≤Aμ−Aμ−N0e−μsa.s..
Then,Ns≤A/μ (a.s.), so(41)Ss,I1s,I2s∈0,Aμfor alls∈0,ta.s..
Since the coefficients of system (2) are locally Lipschitz continuous, then by the work of Mao [25] for any initial value S0,I10,I20∈Γ, there is a unique local positive solution St,I1t,I2t on 0,τe, where τe is the explosion time. To show that this solution is global, we only need to prove τe=∞ (a.s.).
Letε0>0 such that S0,I10,I20>ε0. For ε≤ε0, we define the stopping time:(42)τε=inft∈0,τe:St≤εorI1t≤εorI2t≤ε.
Then,(43)τ=limε⟶0τε=inft∈0,τe:St≤0orI1t≤0orI2t≤0.
Consider the functionU defined for S,I1,I2∈ℝ+3 by(44)US,I1,I2=−lnμAS−lnμAI1−lnμAI2.
Calculating the differential ofU along the solution trajectories of system (2) and using Itô’s formula, for all t≥0 and s∈0,t∧τε, we get(45)dUSs,I1s,I2s=ℒUds+σ1I1−Sf1S,I1dB1s+σ2I2−Sf2S,I2dB2s,where(46)ℒU=−A+r1I1+r2I2S+3μ+a1+r1+a2+r2+β1I1−Sf1S,I1+β2I2−Sf2S,I1+σ122I12+S2f1S,I12+σ222I22+S2f2S,I22≤3μ+a1+r1+a2+r2+β1I1f1S,I1+β2I2f2S,I1+σ122I12+S2f1S,I12+σ222I22+S2f2S,I22.
According to (41), we have Ss,I1s,I2s∈0,A/μ for all s∈0,t∧τε (a.s.). Hence,(47)IisfiSs,Iis≤Aμ,SsfiSs,Iis≤Aμ,i=1,2.
Therefore,(48)dU≤ℳds+σ1I1−Sf1S,I1dB1s+σ2I2−Sf2S,I2dB2s,where(49)ℳ=3μ+a1+r1+a2+r2+β1Aμ+β2Aμ+σ1Aμ2+σ2Aμ2.
Integrating both sides of (48) from 0 to t∧τε and after taking the expectation on both sides, we obtain that(50)EUSt∧τε,I1t∧τε,I2t∧τε≤US0,I10,I20+ℳt.
SinceUSt∧τε,I1t∧τε,I2t∧τε>0, then(51)EUSt∧τε,I1t∧τε,I2t∧τε≥EUSt∧τε,I1t∧τε,I2t∧τεχτε≤t≥EUSτε,I1τε,I2τεχτε≤t,where χτk≤t is the indicator function of τk≤t. Note that there are some components of Sτε,I1τε,I2τε equal to ε. Therefore,(52)USτε,I1τε,I2τε≥−lnμAε=lnAμε.
Thus,(53)EUSt∧τε,I1t∧τε,I2t∧τε≥lnAμεℙτε≤t.
By combining (50) and (53), we get that, for all t>0,(54)ℙτε≤t≤US0,I10,I20+ℳtlnA/με.
Extendingε to 0, we obtain for all t>0, ℙτ≤t=0. Hence, ℙτ=∞=1. As τe≥τ, then τ=τe=∞ (a.s.) which completes the proof.


## 3.2. Almost Sure Exponential Stability

The goal of this section is to establish a sufficient condition for the almost sure exponential stability of the disease-free equilibriumE0 in Γ. For this, we consider(55)Ψt=Aμ−St+I1t+I2t,VSt,I1t,I2t=lnΨt.Proposition 1.
Ψt almost surely converges exponentially to 0 if(56)limsupt⟶∞1t∫0tℒVSs,I1s,I2sds<0a.s..Proof.
By Itô’s formula, we have(57)dVSs,I1s,I2s=ℒVSs,I1s,I2sds+2σ1SsI1sΨsf1Ss,I1sdB1s+2σ2SsI2sΨsf2Ss,I2sdB2s.
Integrating both sides from 0 tot yields that(58)VSt,I1t,I2t=VS0,I10,I20+∫0tℒVSs,I1s,I2sds+M1t+M2t,where Mit=∫0t2σiSsIis/ΨsfiSs,IisdBis, i=1,2, are continuous local martingales with Mi0=0. Moreover, we have(59)Mi,Mit=∫0t2σiSsIisΨsfiSs,Iis2ds≤Cit,where Cii=1,2 are positive constants. Thus, the strong law of large numbers for local martingales [27] implies that(60)limt⟶∞Mitt=0a.s..
It follows that(61)limsupt⟶∞VSt,I1t,I2tt=limsupt⟶∞1t∫0tℒVSs,I1s,I2sdsa.s..
The proposition is proved.Then, we obtain the following theorem.Theorem 6.
Ifβ1σ22+β2σ12<μ/2σ1σ22, then the disease-free equilibrium E0 of stochastic system (2) is almost surely exponentially stable in Γ.Proof.
It suffices to prove thatΨt converges to 0 exponentially (a.s.). Then, by Proposition 1, it suffices to prove that(62)limsupt⟶∞1t∫0tℒVSs,I1s,I2sds<0a.s..
By Itô’s formula, we have(63)ℒVSs,I1s,I2s=1Ψ−A+μS+2β1SI1f1S,I1+2β2SI2f2S,I2−r1I1−r2I2−μ+a1+r1I1−μ+a2+r2I2−2σ1SI1Ψf1S,I12−2σ2SI2Ψf2S,I22=1Ψ2β1SI1f1S,I1+2β2SI2f2S,I2−μAμ−S−μ+a1+2r1I1−μ+a2+2r2I2−2σ1SI1Ψf1S,I12−2σ2SI2Ψf2S,I22.
Since(64)μAμ−S+μ+a1+2r1I1+μ+a2+2r2I2≥μAμ−S+μI1+μI2=μΨ,we have(65)ℒVSs,I1s,I2s≤2β1SI1Ψf1S,I1+2β2SI2Ψf2S,I2−μ−2σ1SI1Ψf1S,I12−2σ2SI2Ψf2S,I22.
SetX1=SI1/Ψf1S,I1 and X2=SI2/Ψf2S,I2. Then,(66)ℒVSs,I1s,I2s≤2β1X1−2σ12X12+2β2X2−2σ22X22−μ.
Since2βiXi−2σi2Xi2−μ/2=−2σi2βi/σi2−Xi2+4βi2−μσi2/2σi2, i=1,2, hence,(67)ℒVSs,I1s,I2s≤4β12−μσ122σ12+4β22−μσ222σ22=2β1σ22+2β2σ12−μσ1σ22σ1σ22.
Therefore,(68)limsupt⟶∞1t∫0tℒVSs,I1s,I2sds≤2β1σ22+2β2σ12−μσ1σ22σ1σ22<0a.s..
This completes the proof.


## 3.3. Moment Exponential Stability

In this section, we investigate thepth moment exponential stability of the disease-free equilibrium E0 in Γ of stochastic system (2).We use Lemma1 to prove the following theorem.Theorem 7.
Letp≥2. If(69)maxR01+p−1σ122μ+a1+r1Aμ+α1A2,R02+p−1σ222μ+a2+r2Aμ+α2A2<1,then the disease-free equilibrium E0 of stochastic system (2) is pth moment exponentially stable in Γ.Proof.
Letp≥2 and S0,I10,I20∈Γ. We define the Lyapunov function V as follows:(70)V=ωAμ−Sp+1p∑i=12Iip,where ω is a positive constant which will be determined later. By Itô’s formula, we have(71)ℒV=−ωpAμ−Sp−1A−μS−∑i=12βiSIifiS,Ii+∑i=12riIi+ωpp−12∑i=12σiSIifiS,Ii2Aμ−Sp−2+∑i=12βiSfiS,Ii−μ+ai+riIip+p−12∑i=12σiSfiS,Ii2Iip≤−ωμpAμ−Sp+ωp∑i=12βiAμ+αiAAμ−Sp−1Ii+ωpp−12∑i=12σi2Aμ+αiA2Aμ−Sp−2Ii2+∑i=12βiAμ+αiA−μ+ai+riIip+p−12∑i=12σ12Aμ+αiA2Iip.
From Young’s inequality, forε>0, we have(72)Aμ−Sp−1Ii≤p−1pεAμ−Sp+1pε1−pIip,Aμ−Sp−2Ii2≤p−2pεAμ−Sp+2pε2−p/pIip,i=1,2.
Then,(73)ℒV≤−Q0Aμ−Sp−∑i=12QiIip,where(74)Q0=ωμp−p−1∑i=12βiAμ+αiA+p−22∑i=12σi2Aμ+αiA2ε,Qi=μ+ai+ri−βiAμ+αiA−p−12σi2Aμ+αiA2−ωβiAμ+αiAε1−p+p−1σi2Aμ+αiA2ε2−p/p,i=1,2.
Now, we chooseε sufficiently small such that Q0>0. In view of condition (69), we have μ+ai+ri−βiA/μ+αiA−p−1/2σi2A/μ+αiA2>0 for i=1,2; hence, we can choose ω positive such that Qi>0 for i=1,2. According to Lemma 1, the proof is completed.Remark 1.
From Lemma1, Theorem 7, and the case p=2, we get that if(75)maxR01+σ122μ+a1+r1Aμ+α1A2,R02+σ222μ+a2+r2Aμ+α2A2<1,then the disease-free equilibrium E0 of stochastic system (2) is globally asymptotically stable in Γ.


## 4. Numerical Examples

In this section, we give some numerical examples in order to illustrate our theoretical results in Theorem1 and Theorem 6.Example 1.
We consider the deterministic SIS system with parametersA=0.9, μ=0.3, β1=0.25, β2=0.2, a1=0.2, a2=0.3, r1=0.3, r2=0.2, α1=0.2, γ1=0.1, μ1=0.06, α2=0.15, γ2=0.2, and μ2=0.07. By calculation, we have R0=maxR01,R02=max0.5859,0.5172<1. Hence, according to Theorem 1, the disease-free equilibrium is globally asymptotically stable, which means that the disease dies out.Example 2.
In this example, we consider the stochastic SIS system with parameters the same as in Example1 and σ1=0.9, σ2=0.95, Then, we have 0.08880625=β1σ22+β2σ12<μ/2σ1σ22=0.10965375. Thus, from Theorem 6, we can conclude that the disease-free equilibrium is almost surely exponentially stable.
Now, we chooseσ1=0.4 and σ2=0.5. Then, we have 0.022025=β1σ22+β2σ12>μ/2σ1σ22=0.006. Therefore, the condition of Theorem 6 is not satisfied. Consequently, if the magnitude of the intensity of noise σii=1,2 is large, then the disease in the stochastic model will go extinct.


## 5. Conclusion

In this paper, we have proposed and analyzed a new stochastic SIS epidemic model with double epidemic hypothesis and specific functional response by introducing random perturbations of white noise. Firstly, in the absence of noise, we have derived sufficient conditions for local asymptotic stability of the equilibria; also, we have proved the global stability for disease-free equilibrium. Next, we have established global existence and positivity of the solution for our stochastic model. In addition, we have given a sufficient condition for the almost sure exponential stability andpth moment exponential stability of the disease-free equilibrium of model (2). It is shown that the magnitude of the intensity of noise σii=1,2 will have an effective impact on stochastic stability of E0.



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