# Stability of a Nonlinear Stochastic Epidemic Model with Transfer from Infectious to Susceptible

Authors: Yanmei Wang; Guirong Liu
Journal: Complexity
 (2020)

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## Abstract

We investigate a stochastic SIRS model with transfer from infectious to susceptible and nonlinear incidence rate. First, using stochastic stability theory, we discuss stochastic asymptotic stability of disease-free equilibrium of this model. Moreover, if the transfer rate from infectious to susceptible is sufficiently large, disease goes extinct. Then, we obtain almost surely exponential stability of disease-free equilibrium, which implies that noises can lead to extinction of disease. By the Lyapunov method, we give conditions to ensure that the solution of this model fluctuates around endemic equilibrium of the corresponding deterministic model in average time. Furthermore, numerical simulations show that the fluctuation increases with increase in noise intensity. Finally, these theoretical results are verified by numerical simulations. Hence, noises play a vital role in epidemic transmission. Our results improve and extend previous related results.


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## Body

## 1. Introduction

Mathematical models have become a crucial tool in understanding dynamics of population growth [1–3]. In recent decades, some realistic mathematical models have been established to investigate dynamics of epidemic [4–10]. In order to simulate epidemic transmission process, many dynamic models have been established, such as SIS, SEIR, and SIRS models [11–13]. In these models, the incidence rate is crucial. Classical disease transmission models adopt the standard or bilinear incidence rate. However, in the course of epidemic propagation, nonlinear incidence may be more realistic than other incidence rates [14]. In addition, infected individuals may recover after a period of treatment or become susceptible individuals directly due to transient antibody. In [15], a deterministic SIRS model with transfer from infectious to susceptible and nonlinear incidence can be modeled as follows:(1)S˙=Λ+γ1I+δR−βSfI−μS,I˙=βSfI−α+γ1+γ2+μI,R˙=γ2I−δ+μR.Here,S, I, and R denote numbers of susceptible, infectious, and recovered individuals, respectively. Λ is the recruitment rate of susceptible; β denotes the disease propagation coefficient; μ and α denote, respectively, the natural death rate and mortality caused by the disease; δ denotes the immunity loss rate; γ1 represents the transfer rate from infectious to susceptible; γ2 denotes the recovery rate of infectious individuals. In addition, Λ>0, μ>0, γ1≥0, γ2≥0, δ≥0, and α≥0.From [15], (1) has disease-free equilibrium E0Λ/μ,0,0 which is globally asymptotic stable in S,I,R∈ℝ+3:S+I+R≤Λ/μ if ℛ0=βΛ/μγ1+γ2+μ+α<1. If ℛ0>1, there exists a globally asymptotic stable endemic equilibrium E∗S∗,I∗,R∗.However, dynamics of epidemic is often disturbed by some random factors. Hence, stochastic epidemic models are more realistic and have attracted much attention [16–19]. In [20], the authors discussed threshold behavior for a stochastic SIS model. In [21], asymptotic properties of a stochastic SIR model were considered. In [22, 23], the authors investigated persistence and extinction for a stochastic SIRS model. In [24], the authors studied stability of a stochastic SIRS model. Fatini et al. [25] considered stochastic stability and instability for a stochastic SIR model. Recently, Wang et al. [26] established a stochastic SIRS epidemic model:(2)dS=Λ+γ1I+δR−βSfI−μSdt−σSfIdBt,dI=βSfI−α+γ1+γ2+μIdt+σSfIdBt,dR=γ2I−μ+δRdt,with initial values S0>0,I0>0,andR0>0. Here, Bt represents Brownian motion on Ω,ℱ,ℙ which is a complete probability space. σ2 denotes the intensity of Bt. Other parameters are defined as (1). Model (2) covers many stochastic models as particular cases (see, for example, [15, 22, 27]). In [26], extinction and persistence are obtained.As is well known, stability of the dynamic system means that solutions are insensitive to small changes of initial value. Hence, stability is one of the important topics encountered in applications. However, because of the complexity of stochastic dynamics, there are not many results on stability of stochastic differential equations.Motivated by the above work, we consider (2) and obtain stochastic stability of disease-free equilibrium and asymptotic behavior around endemic equilibrium of corresponding deterministic model (1).Throughout this paper, we give the following hypotheses:H1f is locally Lipschitz on 0,∞;fI> 0 for I>0; f0=0H2limI⟶0+fI/I=1 and fI/I is nonincreasing on 0,∞FromH1, (2) has disease-free equilibrium E0Λ/μ,0,0. By H2, if I∈0,∞, then(3)I≥fI.


## 2. Preliminaries

We will give some definitions and lemmas. Consider(4)dx=Fx,tdt+Gx,tdWt,t≥t0.Here,FandG are, respectively, ℝd−valued and ℝd×n−valued functions defined on ℝd×t0,∞ and ℝd×t0,∞. Wt denotes n-dimensional Brownian motion on Ω,ℱ,ℱtt≥0,ℙ. Assume that existence-and-uniqueness theorem is fulfilled. For t≥t0, G0,t=0 and F0,t=0. Denote ℝ¯+=ℝ+∪0 and K=μ∈Cℝ¯+,ℝ¯+:μis nondecreasing;μ0=0;μr>0ifr>0. Set Sh=x∈ℝd:x<h.Definition 1.
([[28], p.108]).(i)
Assume thatV is continuous on Sh×t0,∞ and V0,t≡0. If there is μ∈K such that, for x,t∈Sh×t0,∞,(5)Vx,t≥μx,thenV is positive-definite. In addition, V is negative-definite if −V is positive-definite.(ii)
Assume thatV is nonnegative and continuous on Sh×t0,∞. If there is μ∈K, satisfying for x,t∈Sh×t0,∞,(6)Vx,t≤μx,then V is decrescent.Definition 2.
([[28], p.110]).(i)
If for anyr>0 and ε∈0,1, there is δ=δε,r,t0>0, satisfying for any x0∈ℝd with x0<δ,(7)ℙxt;t0,x0<rfort≥t0≥1−ε,then trivial solution to (4) is stochastically stable.(ii)
The trivial solution to (4) is stochastically asymptotically stable if it is stochastically stable, and for any ε∈0,1, there is δ0=δ0ε,t0>0 satisfying(8)ℙlimt⟶∞xt;t0,x0=0≥1−ε,wheneverx0<δ0.(iii)
If for anyx0∈G⊂ℝd, limsupt⟶∞lnxt;t0,x0/t<0 a.s., then trivial solution to (4) is almost surely exponentially stable in G.Lemma 1.
([[28], p.112]). If V∈C2,1Sh×t0,∞;ℝ¯+ is positive-definite and decrescent, and LVx,t is negative-definite, then trivial solution to (4) is stochastically asymptotically stable.
By Theorem 1 and Remark1 in [26], the following result holds.Lemma 2.
(see [26]). For S0,I0,R0∈ℝ+3, there is a unique global positive solution to (2). Moreover,(9)D=S,I,R∈ℝ+3:S+I+R≤Λμis positively invariant.


## 3. Stability of Disease-Free Equilibrium

In epidemiology, stability has important practical significance.Theorem 1.
Ifℛ0<1, σ2Λ2/2μ2<μ+γ1+γ2+α1−ℛ0, then disease-free equilibrium E0 to (2) is stochastically asymptotically stable in D.Proof.
Denotex=x1,x2,x3=−S+Λ/μ,I,R. Define Lyapunov function(10)V2x=x12+bx22+x32for S,I,R∈D, where b>0 is to be chosen later. Clearly, V2 is positive-definite. Note that V2x≤1∨bx2=:μx. From Definition 1 (ii), it follows that V2 is decrescent. Now, we show that LV2 is negative-definite.
From Ito^ formula, for any S,I,R∈D,(11)LV2x=−2Λμ−SΛ−μS−βSfI+δR+γ1I+2bIβSfI−μ+α+γ1+γ2I+2Rγ2I−δ+μR+1+bσSfI2≤−2μΛμ−S2+2Λμ−SβS−γ1I+2bβS−2bμ+γ1+γ2+α+1+bσ2S2I2+2γ2IR−2δΛμ−SR−2δ+μR2≤−2μΛμ−S2+2Λμ−SβΛμ−γ1I+2bβΛμ−2bμ+γ1+γ2+α+1+bσ2Λ2μ2I2+2γ2IR−2δI+RR−2δ+μR2.
Obviously, we have(12)−μΛμ−S2+Λμ−SβΛμ−γ1I=−μΛμ−S−12μ2βΛ−γ1μI2+14μ3βΛ−γ1μ2I2,(13)−μ+2δR2+γ2−δIR=−μ+2δR−γ2−δ2μ+2δI2+γ2−δ24μ+2δI2.
Substituting (12) and (13) into (11) yields(14)LV2x≤−2μΛμ−S−12μ2βΛ−γ1μI2+12μ3βΛ−γ1μ2I2+γ2−δ22μ+2δI2+σ2Λ2μ2I2−2bμ+γ1+γ2+α−βΛμ−σ2Λ22μ2I2−2μ+2δR−γ2−δ2μ+2δI2≤−2μΛμ−S−12μ2βΛ−γ1μI2−2μ+2δR−γ2−δ2μ+2δI2−2bμ+γ1+γ2+α−βΛμ−σ2Λ22μ2−12μ3βΛ−γ1μ2−σ2Λ2μ2−γ2−δ22μ+2δI2.
Noteσ2Λ2/2μ2<α+γ1+γ2+μ1−ℛ0. Then, α+γ1+γ2+μ−βΛ/μ−σ2Λ2/2μ2>0. Take(15)b>1/2μ3βΛ−γ1μ2+σ2Λ2/μ2+γ2−δ2/2μ+2δ2μ+γ1+γ2+α−βΛ/μ−σ2Λ2/2μ2.
This yields thatLV2 is negative-definite. From Lemma 1, E0 is stochastically asymptotically stable in D.Lemma 3.
For anyS0,I0,R0∈D, solution S,I,R of (2) satisfies the following:(i)
Ifσ2>βμ/Λ, then(16)lim supt⟶∞1tlnI+R+Λμ−S≤β2−2μσ22σ2.(ii)
Ifσ2≤βμ/Λ, then(17)lim supt⟶∞1tlnI+R+Λμ−S≤βΛμ−μ−Λ2σ22μ2.Proof.
Obviously,S,I,R∈D for t≥0. Define(18)V3S,I,R=lnI+R+Λμ−S.
Then,(19)LV3=−Λ+2βSfI+μS−γ1I−δR−μ+α+γ1+γ2I−δ+μR+γ2II+R+Λ/μ−S−2σSfII+R+Λ/μ−S2=2βSfII+R+Λ/μ−S+−γ1I−δR−γ1+αI−δRI+R+Λ/μ−S−μ−2σ2SfIR+I+Λ/μ−S2≤−μ−2σ2SfIR+I+Λ/μ−S2+2βSfII+R+Λ/μ−S.
LetZ=SfI/I+R+Λ/μ−S and ΨZ=−2σ2Z2+2βZ−μ. From S+I+R≤Λ/μ and (3),(20)Z≤SII+R+Λ/μ−S≤SI2I+R≤S2≤Λ2μ.
Letr0=supZ∈0,Λ/2μΨZ. Then,(21)dV3=LV3dt+2σSfII+R+Λ/μ−SdBt≤ΨZdt+2σZdBt≤r0dt+2σZdBt,which yields(22)lnIt+Rt+Λμ−St≤lnI0+R0+Λμ−S0+r0t+∫0t2σZsdBs.
From the strong law of large numbers,(23)limt⟶∞1t∫0t2σZsdBs=0a.s.
Then,(24)lim supt⟶∞1tlnRt+It+Λμ−St≤r0.
Obviously, ifσ2>βμ/Λ, then r0=Ψβ/2σ2=β2−2μσ2/2σ2; if σ2≤βμ/Λ, then r0=ΨΛ/2μ=βΛ/μ−μ−Λ2σ2/2μ2. Lemma 3 holds.
By Lemma3, the following result holds.Theorem 2.
Assume that(i)
σ2>maxβμ/Λ,β2/2μor(ii)
2βμ/Λ1−μ2/βΛ<σ2≤βμ/Λ.Then, disease-free equilibriumE0 of (2) is almost surely exponentially stable in D.Remark 1.
(i)
Ifℛ0<1 and σ2=0, then σ2Λ2/2μ2<μ+γ1+γ2+α1−ℛ0 holds. From Theorem 1, if ℛ0<1, then disease-free equilibrium E0 of (1) is asymptotically stable in D. Hence, Theorem 1 extends Theorem 2.1 in [15].(ii)
From Theorem2, if ℛ0<1,βΛ<μ2, then disease-free equilibrium E0 of (1) is exponentially stable in D. Hence, Theorem 2 partially improves Theorem 2.1 in [15].(iii)
From Theorem1, if γ1>σ2Λ2/2μ2+βΛ/μ−μ−γ2−α, then disease-free equilibrium E0 of (2) is stochastically asymptotically stable in D.Remark 2.
(i)
Assume that2βμ/Λ1−μμ+γ1+γ2+α/Λβ>β2/2μ and ℛ0>2. From condition i in Theorem 2, if σ2>maxβμ/Λ,β2/2μ=β2/2μ, then disease-free equilibrium E0 of (2) is almost surely exponentially stable in D. However, Theorem 2 in [26] implies that disease of (2) will become extinct if C3 of Theorem 2 in [26] holds, i.e., σ2>maxβμ/Λ⋅ℛ0/2,2βμ/Λ1−μμ+γ1+γ2+α/Λβ.(ii)
Assume thatβ2/2μ<βμ/Λ<2βμ/Λ1−μγ1+γ2+α+μ/Λβ and ℛ0<2. From Theorem 2i, E0 is almost surely exponentially stable in D if σ2>maxβμ/Λ,β2/2μ=βμ/Λ, whereas disease will become extinct with probability one if σ2>maxβμ/Λ,2βμ/Λ1−μμ+γ1+γ2+α/Λβ in [26].
Obviously, conditioni of Theorem 2 is weaker than condition C3 of Theorem 2 in [26].Remark 3.
Letβ2/2γ1+γ2+μ+α>2βμ/Λ1−μ2/βΛ. By Theorem 2, E0 is almost surely exponentially stable in D if condition (ii) holds. However, disease will become extinct if maxβ2/2γ1+γ2+μ+α,2βμ/Λ1−μγ1+γ2+μ+α/Λβ<σ2<βμ/Λ in [26]. Thus, condition (ii) of Theorem 2 is weaker than condition C2 of Theorem 2 in [26].Remark 4.
From Remarks2 and 3, Theorem 2 partially improves Theorem 2 in [26].


## 4. Asymptotic Properties around Endemic Equilibrium

In studying epidemic dynamics, we have interest in persistence of epidemic. We consider the behavior of solutions to (2) around endemic equilibrium E∗S∗,I∗,R∗ of corresponding deterministic model (1). Denote(25)a1=αγ2,a2=2μ2μ+γ2+α+2μδ+αδI∗βδfI∗,a3=2μδ.Theorem 3.
Ifℛ0>1 and σ2<μa3+1/a2I∗, then(26)lim supt⟶∞1t∫0tη1S−S∗2+η2I−I∗2+η3R−R∗2ds≤a2I∗S∗2σ2,where S,I,R be the solution of (2) with S0,I0,R0∈ℝ+3 and(27)η1=a3+1μ−a2I∗σ2,η2=a3μ+γ2+α+μ+α,η3=a1μ+δ+μ.Proof.
DefineV4:ℝ+3⟶ℝ+ by(28)V4S,I,R=a1W1R+a2W2I+a3W3S,I+W4S,I,R,where(29)W1R=12R−R∗2,W2I=I−I∗−I∗lnII∗,W3S,I=12S−S∗+I−I∗2,W4S,I,R=12S−S∗+R−R∗+I−I∗2.
From Ito^ formula, (3), and H2,(30)LW1=R−R∗γ2I−μ+δR=−δ+μR−R∗2+γ2I−I∗R−R∗,(31)LW2=I−I∗−γ1+γ2μ+α+βSfI1I+12I∗σSfI1I2=I−I∗βS−S∗fI∗1I∗+βSfI1I−fI∗1I∗+12I∗σ2S2fI1I2≤βI−I∗S−S∗fI∗1I∗+βSI−I∗fI1I−fI∗1I∗+σ2I∗S−S∗2+σ2I∗S∗2≤βI−I∗S−S∗fI∗1I∗+σ2I∗S−S∗2+σ2I∗S∗2,(32)LW3=S−S∗+I−I∗Λ+δR−μS−γ2+μ+αI=S−S∗+I−I∗−μS−S∗+δR−R∗−γ2+μ+αI−I∗=−μS−S∗2−γ2+α+2μS−S∗I−I∗−γ2+α+μI−I∗2+δR−R∗S−S∗+I−I∗,(33)LW4=R+I+S−R∗+I∗+S∗Λ−μS−μ+αI−μR=R−R∗+S−S∗+I−I∗−μS−S∗−μR−R∗−α+μI−I∗=−μR−R∗2−μS−S∗2−α+μI−I∗2−α+2μI−I∗S−S∗−2μR−R∗S−S∗−α+2μR−R∗I−I∗.
From (25)–(33),(34)LV4≤a1γ2R−R∗I−I∗−a1δ+μR−R∗2+a2βfI∗I∗S−S∗I−I∗+a2I∗σ2S−S∗2+a2I∗σ2S∗2−a3μS−S∗2−a32μ+α+γ2I−I∗S−S∗−a3μ+α+γ2I−I∗2+a3δS−S∗R−R∗+a3δR−R∗I−I∗−μS−S∗2−μR−R∗2−α+μI−I∗2−α+2μI−I∗S−S∗−2μR−R∗S−S∗−α+2μR−R∗I−I∗=−a3μ+μ−a2I∗σ2S−S∗2−μ+α+a3γ2+α+μI−I∗2−μ+a1δ+μR−R∗2+a2I∗S∗2σ2=−η1S−S∗2−η2I−I∗2−η3R−R∗2+a2I∗S∗2σ2.
Hence, we have(35)dVS,I,R≤a2I∗S∗2σ2−η1S−S∗2−η2I−I∗2−η3R−R∗2dt+σSI−I∗fI1IdBt.
It follows from (35) that(36)VSt,It,Rt−VS0,I0,R0≤∫0t−η2Is−I∗2−η1Ss−S∗2−η3Rs−R∗2ds+a2I∗S∗2σ2t+∫0tσfIsIsSsIs−I∗dBs.
From (36),(37)VSt,It,Rt−VS0,I0,R0≤∫0t−η2Is−I∗2−η1Ss−S∗2−η3Rs−R∗2ds+a2I∗S∗2σ2t.
Consequently,(38)limsupt⟶∞1t∫0tη1Ss−S∗2+η2Is−I∗2+η3Rs−R∗2ds≤a2I∗S∗2σ2.Remark 5.
Theorem3 shows that if ℛ0>1, σ is small enough and then solution to (2) fluctuates around E∗; that is, disease will persist. Furthermore, if σ=0, then (34) becomes(39)LV4≤−η1S−S∗2−η2I−I∗2−η3R−R∗2,which yields that for (1), E∗ is globally asymptotically stable in ℝ+3. This is consistent with Corollary 2.3 in [15]. Hence, Theorem 3 generalizes Corollary 2.3 in [15].


## 5. Numerical Simulations

By numerical simulation, we analyze the asymptotic behavior of model (2) so that readers can better understand our results. Let fI=I/1+I. Then, I≥fI for I≥0. Let(40)Λ=0.6,μ=0.32,γ2=0.1,α=0.1,δ=0.4,S0,I0,R0=1,0.8,0.Example 1.
Takeβ=0.3,γ1=0.57, and σ2=0.2. By a simple computation, we obtain ℛ0≈0.516<1, 0.352≈σ2Λ2/2μ2<μ+γ1+γ2+α1−ℛ0≈0.5276, βΛ−γ1μ=−0.0024<0, and γ2−δ=−0.3<0. Hence, the conditions of Theorem 1 hold. Furthermore, for (2), E01.875,0,0 is stochastically asymptotically stable. Figure 1 supports the result.Figure 1
Trajectories of deterministic model (1) and stochastic model (2) with (a) β=0.3, (b) γ1=0.57, and (c) σ2=0.2.
(a)(b)(c)Example 2.
Takeβ=0.3,γ1=0.01, and σ2=0.3. Hence, ℛ0≈1.0613>1 and maxβμ/Λ,β2/2μ=0.16<σ2. Then, according to conclusion (i) in Theorem 2, solutions of (2) will tend almost surely exponentially to E01.875,0,0. However, from Corollary 2.3 in [15], the solution of deterministic model (1) will converge to E∗1.8261,0.0337,0.0047. This demonstrates that noises can result in extinction of disease. Figure 2 clearly supports these results.Figure 2
Trajectories of deterministic SIRS model (1) and stochastic SIRS model (2) with (a) β=0.3, (b) γ1=0.01, and (c) σ2=0.3.
(a)(b)(c)Example 3.
Letβ=0.3,γ1=0.01, and σ2=0.15 such that ℛ0≈1.0613>1 and 2βμ/Λ1−μ2/βΛ≈0.1380<σ2<βμ/Λ=0.16. Then, according to conclusion (ii) in Theorem 2, solutions of (2) will tend almost surely exponentially to E01.875,0,0. However, from Corollary 2.3 in [15], endemic equilibrium E∗1.8261,0.0337,0.0047 of (1) is globally asymptotically stable in ℝ+3. This represents the extinction of disease due to noise. Figure 3 clearly supports these results.Figure 3
Trajectories of deterministic SIRS model (1) and stochastic SIRS model (2) with (a) β=0.3, (b) γ1=0.01, and (c) σ2=0.15.
(a)(b)(c)Example 4.
Takeβ=0.5,γ1=0.01. Then, ℛ0≈1.7689>1 and σ2<μa3+1/a2I∗≈0.4644. By Theorem 3, solutions of (2) fluctuate around endemic equilibrium E∗1.404,0.3245,0.0451 of deterministic model (1) in time average, which can be verified by using Figure 4. In addition, Figure 4 shows that the fluctuation increases with increase in σ2.Figure 4
Trajectories of deterministic SIRS model (1) and stochastic SIRS model (2) for different σ2 with β=0.5andγ1=0.01.
(a)(b)(c)Example 5.
Takeγ1=0.01. Figure 5 plots the average in time of infected 1/t∫0tIsds for different β in (a) and (b), respectively. From Figure 5, the smaller the β is, the smaller the number of infected cases is. In addition, when β tends to 0, the number of infected cases will tend to 0. This result can also be derived from Theorem 2.Figure 5
Trajectories of the average in time of infected1/t∫0tIsds for stochastic SIRS model (2) for different β with γ1=0.01. (a) σ2=0.02. (b) σ2=0.04.
(a)(b)Example 6.
Takeβ=0.5. Figure 6 plots the average in time of infected 1/t∫0tIsds for different γ1 in (a) and (b), respectively. Figure 6 shows that the larger the γ1 is, the smaller the number of infected cases is. Furthermore, when γ1 is sufficiently large, the number of infected cases tends to 0. This result can be derived from Remark 1 (iii).Figure 6
Trajectories of the average in time of infected1/t∫0tIsds for stochastic SIRS model (2) for different γ1 with β=0.5. (a) σ2=0.02. (b) σ2=0.1.
(a)(b)Example 7.
Takeβ=0.3. Figure 7 plots the number of infected cases for different σ2 in (a) and (b), where γ1=0.01andℛ0≈1.0613>1 in (a) and γ1=0.02andℛ0≈1.0417>1 in (b). From Corollary 2.3 in [15], endemic equilibrium of deterministic model (1) is globally asymptotically stable in ℝ+3. Figure 7 shows that σ2 has a significant effect on both extinction and persistence of disease.Figure 7
Trajectories of infected of stochastic SIRS model (2) for different σ2 with β=0.3. (a) γ1=0.01. (b) γ1=0.02.
(a)(b)


## 6. Conclusions

Stability is one of the important topics encountered in applications. However, because of the complexity of stochastic dynamics, there are not many results on stability analysis of stochastic differential equations.Based on this, we investigate stochastic stability of a stochastic SIRS model. To begin with, using stochastic stability theory, we study stochastic asymptotic stability of disease-free equilibrium of (2), which generalizes Theorem 2.1 in [15]. Moreover, if the transfer rate from infectious to susceptible is sufficiently large, disease goes extinct. Then, exponential stability of disease-free equilibrium is obtained. This result partially improves Theorem 2.1 in [15] and Theorem 2 in [26] and demonstrates that noises can result in extinction of the disease. Furthermore, by the Lyapunov method, we give conditions to ensure that solution of (2) fluctuates around endemic equilibrium of (1) in time average. This generalizes Corollary 2.3 in [15]. At last, numerical simulations are presented to confirm theoretical results and find new properties.Figure4 shows that if ℛ0>1 and σ2<μa3+1/a2I∗, then the solution of (2) fluctuates around endemic equilibrium of (1). Moreover, Figure 4 also shows that the fluctuation increases with increase in noise intensity. From Figure 5, the smaller the β is, the smaller the number of infected individuals will be. In addition, when β tends to 0, the number of infected individuals will tend to 0. This result can also be derived from Theorem 2. Figure 6 shows that the larger the γ1 is, the smaller the number of infected will be. Furthermore, when γ1 is sufficiently large, the number of infected tends to 0. This result can be derived from Remark 1 (iii). Figure 7 shows that noise intensity has a significant effect on both extinction and persistence of the disease. Hence, noises play a vital role in epidemic transmission.For deterministic SIRS model (1), ℛ0 is the basic reproduction number. However, for stochastic SIRS model (2), ℛ0 is not a threshold parameter. From Theorem 2, no matter what the value of ℛ0 is, the disease could go extinct. This can also be verified by the examples in this paper.Although there are important findings revealed by the above investigation, the results still have some limitations. One may consider stochastic asymptotic stability inℝ+3. In addition, our numerical simulation results show that the disease goes extinct as long as ℛ0<1. Regrettably, our theoretical results do not lead to this conclusion.



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