# Asymptotic Behavior and Stationary Distribution of a Nonlinear Stochastic Epidemic Model with Relapse and Cure

Authors: Jiying Ma; Qing Yi
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, by introducing environmental perturbation, we extend an epidemic model with graded cure, relapse, and nonlinear incidence rate from a deterministic framework to a stochastic differential one. The existence and uniqueness of positive solution for the stochastic system is verified. Using the Lyapunov function method, we estimate the distance between stochastic solutions and the corresponding deterministic system in the time mean sense. Under some acceptable conditions, the solution of the stochastic system oscillates in the vicinity of the disease-free equilibrium if the basic reproductive numberR0≤1, while the random solution oscillates near the endemic equilibrium, and the system has a unique stationary distribution if R0>1. Moreover, numerical simulation is conducted to support our theoretical results.


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## Body

## 1. Introduction

Mathematical models can improve our understanding of the dynamics of infectious diseases, predict the transmission trend, and help us formulate preventive measures. The classicalSIR and SIS epidemic models established by Kermack and McKendrick are one of the most important models in epidemiology [1, 2]. From then on, a large number of researchers have proposed and investigated more accurate epidemic models, taking into account different forms of incidence rate, intervention strategies, random perturbation, and other factors. In particular, the human body has certain immune mechanisms to keep itself healthy, and individuals recovered from some diseases may relapse and become reinfected [3–6]. Therefore, it is natural to consider immune effects in mathematical models. Recently, to explore infectious diseases in which infected individuals may be permanently rehabilitated or reinfected, Lahrouz et al. [7] proposed a nonlinear SIR epidemic model with relapse and graded cure as follows:(1)S˙t=μ−μ1St−βStfIt+εIt+γRt,I˙t=βStfIt−μ2+ε+λIt+αRt,R˙t=λIt−μ3+γ+αRt,where St, It, and Rt denote the numbers of the population that are susceptible, infective, and recovered with temporary immunity, respectively. The parameter μ is the growth rate of S; μii=1,2,3 denotes the death rates of S, I, and R, respectively; λ is the recovery rate of I; α, ε, and γ denote the relapse rate, the temporary immunity, and cure rate, respectively; β is the transmission rate from S to I. All the parameters are assumed to be positive, and it is biologically meaningful to suppose that μ1≤minμ2,μ3. The nonlinear incidence rate βStfIt in model (1) reflects the heterogeneous mixing of susceptible and infective population, and the force of infection fI is a function of C∞ on 0,∞ such that(2)f0=0,0<fI<f′0I,∀I>0,S0,I0,R0∈ℝ+3.Lahrouz et al. [7] have studied the global dynamics of system (1). The basic reproduction number is computed as(3)R0=βμf′0μ1μ2+ε+λ−αλ/μ3+γ+α,and the unique disease-free equilibrium E0=μ/μ1,0,0 is globally asymptotically stable if R0<1. Under some additional conditions, system (1) has a unique endemic equilibrium E∗ and it is globally asymptotically stable if R0>1.A large amount of research studies have found that the spread of diseases is naturally subject to random environmental perturbation, such as unpredictable human exposure and meteorological factors [8, 9]. Hence, an increasing number of stochastic epidemic models including environmental noise have been developed [10–13]. In the present paper, motivated by the approach in [14, 15], we introduce system (1) environmental noise which is directly proportional to S, I, and R and establish the following stochastic system:(4)dSt=μ−μ1St−βStfIt+εIt+γRtdt+σ1StdB1t,dIt=βStfIt−μ2+ε+λIt+αRtdt+σ2ItdB2t,dRt=λIt−μ3+γ+αRtdt+σ3RtdB3t.Throughout the paper, we letΩ,ℱ,ℱtt≥0,P be a complete probability space with a filtration ℱtt≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while ℱ0 contains all P-null sets), Biti=1,2,3 denotes a scalar Brownian motion defined on the complete probability space Ω, and ℝ+3=xi>0,i=1,2,3.The rest of the paper is organized as follows. In Section2, we prove the existence and uniqueness of the positive solution for system (4), and some long time behavior of the solution is discussed. In Section 3, we analyze the asymptotic behavior of system (4) near the disease-free equilibrium and estimate the distance between stochastic solutions and the corresponding deterministic system in the time mean sense. In Section 4, asymptotic behavior near endemic equilibrium is analyzed, and we also obtain sufficient conditions for the existence of stationary distribution and persistence of diseases. In Section 5, numerical simulation is displayed to support our theoretical results. A brief conclusion is given in the last section.


## 2. Existence and Uniqueness of the Positive Solution

In this section, we present two main results. The first theorem guarantees the existence and uniqueness of the positive solution for system (4), and the second one shows some long time behavior of the solution.Theorem 1.
For any initial valueS0,I0,R0∈ℝ+3, there exists a unique positive solution St,It,Rt for system (4) on t≥0 and the solution will remain in ℝ+3 with probability one.Proof.
Since the coefficients of system (4) are locally Lipschitz continuous, then for any initial value S0,I0,R0∈ℝ+3 there is a unique local solution St,It,Rt on t∈0,τe, where τe is the explosion time. Moreover, the unique local solution to model (4) is positive by Itô’s formula [16]. Therefore, it suffices to verify that the solution is global, i.e., τe=∞ a.s. Let n0>0 be sufficiently large such that S0,I0, and R0 lie within the interval 1/n0,n0. For each integer n≥n0, define the stopping times:(5)τn=inft∈0,τe:minSt,It,Rt≤1normaxSt,It,Rt≥n.
Setinf∅=∞ (∅ represents the empty set). Note that τn is increasing as n⟶∞. Let τ∞=limn⟶∞τn, then τ∞≤τe a.s. Now, we state that τ∞=∞ a.s. If this statement is violated, then there exists a constant T>0 and ε∈0,1 such that ℙτ∞≤T>ε. As a consequence, there exists an integer n1≥n0 such that(6)ℙτn≤T≥ε,for alln≥n1.
Define a nonnegativeC2-function V:ℝ+3⟶ℝ+ as(7)Vt=VS,I,R=S−C−ClnSC+I−1−lnI+R−1−lnR,where C>0 is a positive constant determined later, and the nonnegativity can be obtained from u−1−lnu≥0for anyu>0.
Letn≥n0 and T>0 be arbitrary. Applying Itô’s formula to V, we obtain that(8)dVS,I,R=ℒVdt+σ1S1−CSdB1t+σ2I1−1IdB2t+σ3R1−1RdB3t,where(9)ℒV=μ+Cμ1+μ2+μ3+ε+λ+γ+α+CβfI+Cσ12+σ22+σ322−μ1S−CμS−CIεS−CRγS−μ2I−βSfII−αRI−μ3R−λIR≤μ+Cμ1+μ2+μ3+ε+λ+γ+α+Cσ12+σ22+σ322+Cβf′0−μ2I≤μ+Cμ1+μ2+μ3+ε+λ+γ+α+Cσ12+σ22+σ322≔K.
Here, we chooseC such that Cβf′0−μ2≤0. Then, substituting the inequality into (8) yields that(10)dVS,I,R≤Kdt+σ1S1−CSdB1t+σ2I1−1IdB2t+σ3R1−1RdB3t.
Furthermore,(11)∫0τn∧TdVSr,Ir,Rr≤∫0τn∧TKdt+∫0τn∧Tσ1Sr1−CSrdB1r+∫0τn∧Tσ2Ir1−1IrdB2r+∫0τn∧Tσ3Rr1−1RrdB3r,where τn∧T=minτn,T. Taking the expectation of the above inequality yields(12)EVSτn∧T,Iτn∧T,Rτn∧T≤VS0,I0,R0+KT.
SetΩn=τn≤T for n≥n1, then ℙΩk≥ε due to (6). Note that, for every ω∈Ωn, at least one of Sτn,ω,Iτn,ω,Rτn,ω equals to either n or 1/n. Hence,(13)VSτn∧T,Iτn∧T,Rτn∧T≥n−1−lnn∧1n−1−ln1n.
Following from (12), we obtain(14)VS0,I0,R0+KT≥EIΩnωVSτn,Iτn,Rτn≥εn−1−lnn∧1n−1−ln1n,where IΩnω is the indicator function of Ωn. Taking n⟶∞, we have ∞>+∞, which is a contradiction. The conclusion is confirmed.Theorem 2.
For any initial valueS0,I0,R0∈ℝ+3, the solution St,It,Rt of system (4) has the following properties:(15)limt⟶∞Stt=0,limt⟶∞Itt=0,limt⟶∞Rtt=0,a.s.,limt⟶∞∫0tSrdB1rt=0,limt⟶∞∫0tIrdB2rt=0,limt⟶∞∫0tRrdB3rt=0,a.s.
The proof is standard, and we omit it here.


## 3. Asymptotic Behavior around the Disease-Free Equilibrium

For the deterministic system (1), the unique disease-free equilibrium E0=μ/μ1,0,0 is globally asymptotically stable if the reproduction number R0<1. The following theorem shows the asymptotic behavior of the stochastic system (4) near E0.Theorem 3.
LetSt,It,Rt be the solution of system (4) with any initial value S0,I0,R0∈ℝ+3. If R0≤1,μ1>2σ12∨σ32, and M1>0,M2>0, then(16)limsupt⟶∞1tE∫0t34μ1−σ12Sr−μμ12+M1I2r+M2R2rdr≤σ12μ2μ12+μ1+μ2+λ2μμ12μ2+λμ1−2σ12,where M1=μ2+λ/8−σ22/2−μ1+μ2+λ2ε2/μ1μ2+λμ1−2σ12,M2=μ1−σ32μ1μ2+λ/8λ−2μ1+μ2+λ2γ+α2/μ1μ2+λμ1−2σ12.Proof.
DefineC2 functions as follows:(17)V1S=S−μ/μ122,V2I,R=μμ1I+αμ3+γ+αR,V3S,I=S−μ/μ1+I22,V4R=R22.
Making use of Itô’s formula, we obtain(18)dV1=S−μμ1μ−μ1S−βSfI+εI+γR+σ12S22dt+σ1SS−μμ1dB1t≔ℒV1dt+σ1SS−μμ1dB1t,where(19)ℒV1=−μ1S−μμ12−βfIS−μμ12−βμfIμ1S−μμ1+S−μμ1εI+γR+σ12S22≤−μ1S−μμ12−βμfIμ1S−μμ1+ε2I2μ1+μ14S−μμ12+γ2R2μ1+μ14S−μμ12+σ12S−μμ12+σ12μ2μ12=−μ12−σ12S−μμ12−βμfIμ1S−μμ1+ε2I2μ1+γ2R2μ1+σ12μ2μ12.
AsR0≤1, we have(20)ℒV2=βμSfIμ1−μμ2+ε+λIμ1+μαλIμ1μ3+γ+α=βμfIμ1S−μμ1+βμ2fIμ12−μμ2+ε+λIμ1+μαλIμ1μ3+γ+α≤βμfIμ1S−μμ1+βμ2f′0Iμ12−μμ2+ε+λIμ1+μαλIμ1μ3+γ+α=βμfIμ1S−μμ1+μμ2+ε+λ−αλ/μ3+γ+αμ1R0−1I≤βμfIμ1S−μμ1.
Similarly,(21)ℒV3=−μ1S−μμ12+γ+αS−μμ1R−μ1+μ2+λS−μμ1I+γ+αRI−μ2+λI2+S2σ122+I2σ222≤−μ1S−μμ12+μ14S−μμ12+γ+α2R2μ1+μ2+λ2I2+μ1+μ2+λ22μ2+λS−μμ12−μ2+λI2+I2σ222+γ+α2R2μ2+λ+μ2+λ4I2+σ12S−μμ12+σ12μ2μ12=−3μ14+σ12+μ1+μ2+λ22μ2+λS−μμ12−μ2+λ4−σ222I2+γ+α2μ1+μ2+λμ1μ2+λR2+σ12μ2μ12,(22)μ1μ2+λ4λℒV4=μ1μ2+λIR4−μ1μ2+λμ3+γ+αR24λ+σ32μ1μ2+λR28λ≤μ2+λ4I22+μ12R22−μ12μ2+λR24λ+σ32μ1μ2+λR28λ≤μ2+λI28−μ12μ2+λR28λ+σ32μ1μ2+λR28λ=μ2+λI28−μ1−σ32μ1μ2+λR28λ.
In inequalities (19)–(22), we have used the inequality 2ab≤a2+b2,a+b2≤2a2+2b2 for any a,b∈ℝ, and the young inequality −μ1+μ2+λS−μ/μ1I≤μ2+λ/2I2+μ1+μ2+λ2/2μ2+λS−μ/μ12.
Define a nonnegativeC2 function V:ℝ+3⟶ℝ+ as follows:(23)V=V3+μ1+μ2+λ22μ2+λμ1/2−σ12V1+V2+μ1μ2+λ4λV4.
Together with (19)–(22), we obtain(24)LV≤−3μ14−σ12S−μμ12−μ2+λ8−σ222−μ1+μ2+λ2ε2μ1μ2+λμ1−2σ12I2−μ1μ2+λμ1−σ328λR2+γ+α2μ1+μ2+λμ1μ2+λ+μ1+μ2+λ2γ+α2μ1μ2+λμ1−2σ12R2+σ12μ2μ12+σ12μ1+μ2+λ2μ2μ12μ2+λμ1−2σ12≤−3μ14−σ12S−μμ12−μ2+λ8−σ222−μ1+μ2+λ2ε2μ1μ2+λμ1−2σ12I2−μ1μ2+λμ1−σ328λ−2γ+α2μ1+μ2+λ2μ1μ2+λμ1−2σ12R2+σ12μ2μ12+σ12μ1+μ2+λ2μ2μ12μ2+λμ1−2σ12=−3μ14−σ12S−μμ12−M1I2−M2R2+σ12μ2μ12+σ12μ1+μ2+λ2μ2μ12μ2+λμ1−2σ12.
Integrating both sides of (24) and taking expectation, we obtain(25)EVt−V0≤−E∫0t3μ14−σ12Sr−μμ12dr−E∫0tM1I2rdr−E∫0tM2R2rdr+σ12μ2μ12+σ12μ1+μ2+λ2μ2μ12μ2+λμ1−2σ12t.
Hence,(26)limsupt⟶∞1tE∫0t3μ14−σ12Sr−μμ12+M1I2r+M2R2rdr≤σ12μ2μ12+μ1+μ2+λ2μ2μ12μ2+λμ1−2σ12.
The above theorem shows that the solution of system (4) oscillates near the disease-free equilibrium E0 in the time mean sense if R0≤1, and the magnitude of the oscillation is proportional to the intensity of noise. From the perspective of biology, the disease will be controlled in a small range if the intensity of noise is sufficiently small.


## 4. Stationary Distribution and Asymptotic Behavior around the Endemic Equilibrium

In this section, we turn to the case when the reproduction numberR0>1 and discuss sufficient conditions for the persistence of disease. We first recall some general results. Consider ℓ-dimension stochastic equation:(27)dXt=fXtdt+∑r=1kgrXdBrt,where Xt is a homogeneous Markov process in ℓ-dimension Euclidean space ℝℓ. The diffusion matrix is defined as follows:(28)Ax=aijx,aijx=∑r=1kgrixgrjx.Lemma 1.
(see [17]). The Markov process X(t), the solution of system (27), has a unique ergodic stationary distribution μ⋅, if there exists a bounded domain D⊂ℝℓ with regular boundary and(i)
There exists a positive numberM such that ∑i,j=1laijxξiξj≥Mξ2, x∈D, ξ∈Rℓ.(ii)
There exists a nonnegative functionV:Dc⟶ℝ such that V is twice continuously differentiable and that for some θ>0, ℒVx≤−θ, for any x∈Dc. Then,(29)ℙxlimT⟶∞1T∫0TfXtdt=∫ℝnfxμdx=1,for all x∈ℝℓ, where f:ℝℓ⟶ℝ be a function integrable with respect to the measure μ.
For the deterministic system (1), there exists at least one positive equilibrium E∗=S∗,I∗,R∗ if R0>1. Moreover, assume the condition(30)I−I∗fII−fI∗I∗≤0,for allI≥0,holds; then, the equilibrium E∗ is unique and globally stable according to Theorem 5.1 in [7].Theorem 4.
LetR0>1, and assume conditions (2) and (30) hold. If minm1S∗2,m2I∗2,m3R∗2>δ>0,mi>0,i=1,2,3, then for any initial value S0,I0,R0∈ℝ+3, system (4) has a unique stationary distribution and the ergodicity hold. Especially, we have(31)limt⟶∞1tE∫0tm1Sr−S∗2+m2Ir−I∗2+m3Rr−R∗2dr≤δ,Here, m1=μ1/2−σ12,m2=μ2+λ/4−σ22,m3=μ2+λμ3+γ+α/λ2μ3+γ+α/2−σ32−α+γ2/2μ1−α+γ+αμ1+μ2+λ/βfI∗2/μ2+λ, and δ=σ12S∗2+σ22I∗2+σ22I∗2μ1+μ2+λ/2βfI∗+μ2+λμ3+γ+α/λ2σ32R∗2.Proof.
In order to prove the existence of a unique ergodic stationary distribution, it suffices to verify conditions (i) and (ii) in Lemma1. To begin with, it is easy to see that E∗=S∗,I∗,R∗ satisfies(32)μ=μ1S∗+βS∗fI∗−εI∗−γR∗,βS∗fI∗=μ2+ε+λI∗−αR∗,λI∗=μ3+γ+αR∗.
Define nonnegativeC2 functions as follows:(33)V1S,I=S−S∗+I−I∗22,V2I=I−I∗−I∗logII∗,V3R=R−R∗22.
By a standard calculation, we obtain(34)ℒV1=−μ1S−S∗2−μ1+μ2+λS−S∗I−I∗+α+γS−S∗R−R∗−μ2+λI−I∗2+α+γI−I∗R−R∗+S2σ12+I2σ222≤−μ12S−S∗2−μ2+λI−I∗2−μ1+μ2+λS−S∗I−I∗+α+γI−I∗R−R∗+α+γ22μ1R−R∗2+S2σ12+I2σ222,(35)ℒV2=I−I∗βSfII−fI∗I∗+βfI∗I∗S−S∗−αRII∗I−I∗+αI∗R−R∗+σ22I∗2=βSI−I∗fII−fI∗I∗+βfI∗I∗S−S∗I−I∗−αRI∗II−I∗2+αI∗R−R∗I−I∗+σ22I∗2≤βfI∗I∗S−S∗I−I∗+αI∗R−R∗I−I∗+σ22I∗2,(36)ℒV3=R−R∗λI−μ3+γ+αR−λI∗+μ3+γ+αR∗+σ32R22=λI−I∗R−R∗−μ3+γ+αR−R∗2+σ32R22≤λ22μ3+γ+αI−I∗2+μ3+γ+α2R−R∗2−μ3+γ+αR−R∗2+σ32R−R∗2+σ32R∗2=λ22μ3+γ+αI−I∗2−μ3+γ+α2−σ32R−R∗2+σ32R∗2.
Then, define a nonnegativeC2 function V:ℝ+3⟶ℝ+:(37)Vt=VS,I,R=V1S,I+I∗μ1+μ2+λβfI∗V2I+μ2+λμ3+γ+αλ2V3R.
According to (34)–(36), it implies(38)ℒV≤−μ12S−S∗2−μ2+λ2I−I∗2−μ2+λμ3+γ+αλ2μ3+γ+α2−σ32−α+γ22μ1R−R∗2+S2σ122+I2σ222+α+γ+αμ1+μ2+λβfI∗I−I∗R−R∗+σ22I∗2μ1+μ2+λ2βfI∗+μ2+λμ3+γ+ασ32R∗2λ2≤−μ12−σ12S−S∗2−μ2+λ4−σ22I−I∗2−μ2+λμ3+γ+αλ2μ3+γ+α2−σ32−α+γ22μ1−α+γ+αμ1+μ2+λ/βfI∗2μ2+λR−R∗2+σ22I∗2μ1+μ2+λ2βfI∗+μ2+λμ3+γ+αλ2σ32R∗2+σ12S∗2+σ22I∗2=−m1S−S∗2−m2I−I∗2−m3R−R∗2+δ≔Ft.
Denote an ellipsoidΣ=S,I,R:m1S−S∗2+m2I−I∗2+m3R−R∗2≤δ, then the ellipsoid Σ lies entirely in ℝ+3 if 0<δ<minm1S∗2,m2I∗2,m3R∗2. Take D to be any neighborhood of Σ with D¯⊆ℝ+3, then there exists some C>0 such that ℒV≤−C for any x∈ℝ+3∖D. That is, condition (ii) holds.
The diffusion matrix of system (4) is given by(39)A=σ12S2000σ22I2000σ32R2.
ChooseM=minS,I,R∈D¯⊂ℝ+3σ12S2,σ22I2,σ32R2, we have(40)∑i,j=13∑k=13aikS,I,RajkS,I,Rξiξjξk=σ12S2ξ12+σ22I2ξ22+σ32R2ξ32≥Mξ2,ξ1,ξ2,ξ3∈ℝ+3.
Then, conditioni in Lemma 1 is satisfied. Therefore, according to Lemma 1, the stochastic system (4) has a unique stationary distribution and it is ergodic. Moreover, the dynamical behavior around the endemic equilibrium E∗ satisfies(41)limt⟶∞1tE∫0tm1Sr−S∗2+m2Ir−I∗2+m3Rr−R∗2dr≤δ.
The proof is completed.
Theorem4 shows that, under certain conditions, the solution of system (4) will oscillate around the endemic equilibrium E∗ for the long time. Furthermore, the following theorem indicates that the disease will be almost surely persistent in the time mean sense.Theorem 5.
Suppose the conditions in Theorem4 hold; then, the solution of system (4) has the property that(42)liminft⟶∞1t∫0tSrdr≥m1S∗2−δ2m1S∗,liminft⟶∞1t∫0tIrdr≥m2I∗2−δ2m2I∗,liminft⟶∞1t∫0tRrdr≥m3R∗2−δ2m3R∗,a.s.Proof.
According to the proof process of Theorem4, we have(43)dVt≤Ftdt+S−S∗+I−I∗σ1SdB1t+σ2IdB2t+σ2I∗μ1+μ2+λI−I∗βfI∗dB2t+σ3μ2+λμ3+γ+αR−R∗Rλ2dB3t.
Integrating the above inequality from 0 tot, we obtain(44)Vt−V0≤∫0tFrdr+Mt,where(45)Mt=∫0tSr−S∗+Ir−I∗σ1SrdB1r+σ2IrdB2r+σ2I∗μ1+μ2+λβfI∗∫0tIr−I∗dB2r+σ3μ2+λμ3+γ+αλ2∫0tRr−R∗RrdB3r.
From the strong law of large numbers for local martingales, we havelimt⟶∞Mt/t=0a.s. Thus, liminft⟶∞∫0tFrdr/t≥0a.s., which implies that(46)limsupt⟶∞1t∫0tm1Sr−S∗2+m2Ir−I∗2+m3Rr−R∗2dr≤δ,a.s.
Furthermore, we have(47)limsupt⟶∞1t∫0tm1Sr−S∗2dr≤δ,a.s.,limsupt⟶∞1t∫0tm2Ir−I∗2dr≤δ,a.s.,limsupt⟶∞1t∫0tm3Rr−R∗2dr≤δ,a.s.
Due to2S∗2−2S∗S=2S∗S∗−S≤S∗2+S∗−S2, it deduces(48)S≥S∗2−S∗−S22S∗.
Substituting (48) into (47), we obtain(49)liminft⟶∞1t∫0tSrdr≥S∗2−δ2m1S∗=m1S∗2−δ2m1S∗,a.s.,liminft⟶∞1t∫0tIrdr≥m2I∗2−δ2m2I∗,a.s.,liminft⟶∞1t∫0tRrdr≥m3R∗2−δ2m3R∗,a.s.
Therefore, the disease is persistent in the sense of time mean. The proof is completed.


## 5. Numerical Simulations

In this section, we choose the saturation incidence functionfI=1/1+θI as in [7]. Using Milstein’s higher order method [18], we obtain the following discrete equations with respect to system (4):(50)Sk+1=Sk+μ−μ1Sk−βSkIk1+θIk+εIk+γRk△t+σ1Skξ1,k△t+σ122Skξ1,k2△t−△t,Ik+1=Ik+βSkIk1+θIk−μ2+ε+λIk+αRk△t+σ2Ikξ2,k△t+σ222Ikξ2,k2△t−△t,Rk+1=Rk+λIk−μ3+γ+αRk△t+σ3Rkξ3,k△t+σ322Rkξ3,k2△t−△t.Here, the time increment△t>0, and ξ1,k, ξ2,k, and ξ3,k,k=1,2,…,n, are independent Gaussian random variables N0,1, and σk, k=1,2,3, are intensities of white noise. Choose the initial value S0,I0,R0=20,10,5, and the value of parameters as μ=0.99,μ1=0.16,μ2=0.3,μ3=0.2,β=0.04,θ=0.1,ε=0.002,γ=0.0007,λ=0.007,andα=0.0001. The selection of these parameters satisfies the condition of Theorem 3, and numerical simulation is displayed in Figure 1. Moreover, we can see that when the noise intensity decreases, and the vibration of the solution for system (4) also decreases.Figure 1
(a) The solution of deterministic system (1). (b) The solutions of stochastic system (4), where σ1=0.28,σ2=0.256,andσ3=0.396. (c) The solution of stochastic system (4), where σ1=0.1,σ2=0.15,andσ3=0.25.
(a)(b)(c)Similarly, take the value of parameters asμ=0.99,μ1=0.18,μ2=0.1999,μ3=0.5,β=0.4,θ=0.1,ε=0.002,γ=0.007,λ=0.07,andα=0.0001. It is easy to compute that R0>1. The numerical simulation is shown in Figure 2, from which one can see that the solution of stochastic system vibrates around the endemic equilibrium E∗. Moreover, Figure 3 shows the histograms of St,It,andRt, and it indicates the existence of a unique stationary distribution for system (4), and the disease will be almost surely persistent in the time mean sense.Figure 2
(a) The solution of deterministic system (1). (b) The solutions of stochastic system (4), where σ1=0.29,σ2=0.25,andσ3=0.5. (c) The solution of stochastic system (4), where σ1=0.2,σ2=0.2,andσ3=0.3.
(a)(b)(c)Figure 3
The histograms of the solutionSt,It,Rt of stochastic system (4) with the environmental white noise intensity σ1=0.2,σ2=0.2,andσ3=0.3. (a) The density function of S(t). (b) The density function of I(t). (c) The density function of R(t).
(a)(b)(c)


## 6. Conclusion

Since most systems in the real world are disturbed by random and unpredictable perturbation, we introduce environmental noise of white noise type into the transmission of disease and study a stochastic version of a nonlinearSIRS epidemic model with relapse and cure. The reproduction number R0 is a threshold parameter. If the conditions of Theorems 3 and 4 hold, according to R0≤1 or R0>1, we prove that the solution of the stochastic model oscillates in the vicinity of the disease-free equilibrium and the endemic equilibrium, respectively, and the fluctuation intensity is proportional to the white noise intensity. Throughout the paper, we use numerical simulations to illustrate our theoretical results. In particular, we also prove that the stochastic SIR model has a unique ergodic stationary distribution, and the disease will be prevalent in the sense of time mean under some conditions. However, based on our theoretical analysis and numerical simulations, it is not clear that the disease can be eradicated.



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