# Stationary Distribution and Dynamic Behaviour of a Stochastic SIVR Epidemic Model with Imperfect Vaccine

Authors: Driss Kiouach; Lahcen Boulaasair
Journal: Journal of Applied Mathematics
 (2018)

---

## Abstract

We consider a stochastic SIVR (susceptible-infected-vaccinated-recovered) epidemic model with imperfect vaccine. First, we obtain critical condition under which the disease is persistent in the mean. Second, we establish sufficient conditions for the existence of an ergodic stationary distribution to the model. Third, we study the extinction of the disease. Finally, numerical simulations are given to support the analytical results.


---

## Body

## 1. Introduction

Mathematical models have been an unavoidable tool in analyzing the mechanisms of infectious diseases. Most modelers were inspired by the works in [1–3]. For controlling the spread of diseases, vaccination is considered the most effective way to reduce both the morbidity and mortality of individuals (see [4–6]). In this paper, we consider the deterministic SIVR (susceptible-infected-vaccinated-recovered) model:(1)dS=μ-μ+ϕS-βSIdt,dI=βS+ηVI-μ+γIdt,dV=ϕS-ηβVI-μVdt,dR=γI-μRdt, where S, I, and R denote the densities of susceptible, infected, and recovered individuals, respectively. V denotes the density of individuals who are immune to an infection as result of vaccination.The parameters involved in the system are described below:β: the average number of contacts per infected per unit timeμ: the recruitment rate and the death rateγ: recovery rate of infected individualsϕ: the rate at which susceptible individuals are moved into the vaccination processη: a positive factor satisfying 0≤η≤1, and η=0 means that the vaccine is perfectly effective and η=1 means that the vaccine has no effectIn model (1), the fundamental parameter that governs the spread of the disease into a population is the basic reproduction number denoted by R0. It can be thought of as the number of cases one case generates on average over the course of its infectious period in an otherwise uninfected population (see [7]).Let us denote byR0=β/(μ+γ) the basic reproduction number of model (1) and by Rϕ=R0(μ+ηϕ/μ+ϕ) the basic reproduction number in a population in which a proportion ϕ had been vaccinated. It is known that, in the absence of the disease, there is a unique disease-free equilibrium S0,I0,V0,R0=μ/(μ+ϕ),0,ϕ/(μ+ϕ),0 which is globally asymptotically stable if Rϕ<1. If Rϕ>1 for some parameters values, the model exhibits a backward bifurcation leading to the existence of multiple endemic equilibria and news subthreshold, which may be important when it comes to designing vaccination strategies (see [8, 9]).It is well known that epidemic models are inevitably affected by the environmental noise that influences the dynamic behaviour of the epidemic models (see [10–14]). For this, inspired by the works in [15–17], Tornatore et al. [18] have formulated and studied a stochastic version of model (1) by replacing the contact rate β in system (1) by β+σ(dWt/dt), where W(t) denotes the standard Brownian motion and σ2 denotes the intensity of the white noise. Then, system (1) becomes as follows:(2)dS=μ-μ+ϕS-βSIdt-σSIdWt,dI=βS+ηVI-μ+γIdt+σS+ηVIdWt,dV=ϕS-ηβVI-μVdt-σηVIdWt,dR=γI-μRdt.Dianli Zhao and Sanling Yuan [19] have obtained conditions ensuring the persistence and extinction of model (2) and have found a threshold RϕS=(1/μ+γ)[β(μ+ηϕ/μ+ϕ)-(σ2/2)((μ+ηϕ)/(μ+ϕ))2] whose value below 1 or above 1 can determine the extinction and persistence of the epidemic under mild extra conditions.In this paper, we assume that the multiplicative noise sources are linear inS(t),I(t) and V(t), according to [20]. Note that recovered population has no effect on the dynamics of S,I, and V. Then, following this approach, we obtain the following reduced stochastic SIVR model:(3)dS=μ-μ+ϕS-βSIdt+σ1SdB1t,dI=βS+ηVI-μ+γIdt+σ2IdB2t,dV=ϕS-ηβVI-μVdt+σ3VdB3t, where B1(t),B2(t), and B3(t) are standard independent Brownian motions and σi2 is a positive constant, for all i∈{1,2,3}.This paper is organized as follows. In Section2, we present some lemmas concerning the existence of a global positive solution and ergodic stationary distribution. In Section 3, we prove that the disease is persistent under one condition. In Section 4, we establish sufficient conditions for the existence of a unique ergodic stationary distribution. In Section 5, we determine a condition under which the disease goes to extinction. In the last section, we introduce some examples and numerical simulations to confirm our results.


## 2. Preliminaries

Throughout this paper, let(Ω,F,(Ft)t≥0,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). Moreover, let R+d=(x1,…,xd)∈Rd:xi>0,i=1,…,d. If f is a bounded function on R+, define f∨=supt∈[0,∞)f(t) and f∧=inft∈[0,∞)f(t). We define also 〈h〉=(1/t)∫0th(s)ds.The following theorem concerns the existence and uniqueness of the global positive solution. Since the proof is almost the same as that in [21, Lemma 2.1 and Lemma 2.2], we omit it here.Theorem 1.
For any initial value(S(0),I(0),V(0))∈R+3, there is a unique solution (S(t),I(t),v(t))∈R+3 of system (3) on t≥0 and the solution will remain in R+3 with probability one. The solution (S(t),I(t),v(t)) has the following properties: (4)limt→∞Stt=0,limt→∞Itt=0,limt→∞Vtt=0,a.s.,limsupt→∞log⁡Stt≤0,limsupt→∞log⁡Itt≤0,limsupt→∞log⁡Vtt≤0,a.s. Furthermore, if μ>max⁡σ12,σ22,σ32/2, then (5)limt→∞∫0tSrdB1rt=0,limt→∞∫0tIrdB2rt=0,limt→∞∫0tVrdB3rt=0,a.s.Now, we consider thed-dimensional stochastic differential equation:(6)dxt=fxt,tdt+gxt,tdBtfor  allt≥t0,with initial value x(0)=x0∈Rd. B(t) denotes an n-dimensional standard Brownian motion defined on the complete probability space (Ω,F,{Ft}t≥0,P). Let us denote by C2,1(Rd×[t0,∞];R+) the family of all nonnegative functions V(x,t) defined on Rd×[t0,∞]Rd×[t0,∞] such that they are continuously twice differentiable in x and once in t. The differential operator L of (6) is defined by [22] (7)L=∂∂t+∑i=1dfix,t∂∂xi+12∑i,j=1dgTx,tgx,tij∂2∂xi∂xj. If L acts on a function V∈C2,1(Rd×[t0,∞];R+), then (8)LVx,t=Vtx,t+Vxx,tfx,t+12tracegTx,tVxxx,tgx,t, where (9)Vt=∂V∂t,Vx=∂V∂x1,…,∂V∂xd,Vxx=∂2V∂xi∂xjd×d. By Itô’s formula, we get (10)dVxt,t=LVxt,tdt+Vxxt,tgxt,tdBt.Next, we shall present a lemma that gives a criterion for the existence of an ergodic stationary distribution to system (3).LetX(t) be a homogeneous Markov process in Ed (Ed denotes d-dimensional Euclidian space) and it is described by the following stochastic differential equation: (11)dXt=hXtdt+∑r=1kgrXtdBrt The diffusion matrix is defined as follows: (12)AX=aijx,aijX=∑r=1kgriXgrjX.Lemma 2 (see [23], Chapter 4).
The Markov processX(t) has a unique ergodic stationary distribution π(.) if there exists a bounded domain D⊂Ed with regular boundary Γ and
(H.1): there is a positive numberλ such that ∑i,j=1daij(X)ξiξj≥λξ2,X∈D,ξ∈Rd,
(H.2): there exists a nonnegativeC2-function V such that LV is negative for any Ed∖D. Then (13)Pxlimt→∞1t∫0tΦXsds=∫EdΦxπxdx=1, for all x∈Ed, where Φ(.) is a function integrable with respect to the measure π.Now, we consider the following one-dimensional homogeneous Markov process:(14)dXt=bXtdt+σXtdBt,Xt0=X0,for  allt≥t0,where b(X) and σ(X) are measurable functions from R to R and B(t) is the Brownian motion. It is assumed that the functions b(X), σ(X), and 1/σ(X) are locally bounded.(H.3): the functionsb(X) and σ(X) are such that(15)Yx=∫0xexp⁡-2∫0sbuσ2ududs→±∞asx→±∞,C=∫-∞+∞1σ2xexp⁡2∫0xbuσ2ududx<∞.Lemma 3 (see [24], Theorem 1.13).
If (H.3) is satisfied, then the stochastic process (14) has ergodic properties with the density given by (16)πx=1Cσ2xexp⁡2∫0xbuσ2udu.


## 3. Persistence

Definition 4.
System (3) is said to be persistent in the mean, if (17)liminft→∞It>0a.s.Theorem 5.
IfR0S>1, then the disease will be persistent in mean; that is, (18)liminft→∞βa+b+ηcIt≥μ+γ+σ222R0S-1,a.s., where(19)a=βμμ+ϕ+σ12/22,b=ηβμϕμ+ϕ+σ12/22μ+σ32/2,c=ηβμϕμ+ϕ+σ12/2μ+σ32/22,R0S=βμμ+ϕ+σ12/2μ+γ+σ22/2+ηβμϕμ+ϕ+σ12/2μ+γ+σ22/2μ+σ32/2.Proof.
Set(20)V1=-log⁡I-a+blog⁡S-clogV,where positive constants a,b, and c will be determined later.
We apply Itô’s formula onV1; then we get (21)dV1=-aμS-bμS+a+bμ+ϕ+βa+bI-βS-ηβV+μ+γ-cϕSV+cηβI+μc+a2σ12+b2σ12+σ222+c2σ32dt-σ1a+bdB1+σ2dB2+cσ3dB3=-βS-aμS-bμS-ηβV-cϕSV+μ+γ+σ222+βa+b+ηcI+aμ+ϕ+σ122+bμ+ϕ+σ122+cμ+σ322dt-σ1a+bdB1+σ2dB2+cσ3dB3≤-2aβμ-3bcηβμϕ3+βa+b+ηcI+μ+γ+σ222+aμ+ϕ+σ122+bμ+ϕ+σ122+cμ+σ322dt-σ1a+bdB1+σ2dB2+cσ3dB3
Let(22)a=βμμ+ϕ+σ12/22,b=ηβμϕμ+ϕ+σ12/22μ+σ32/2,c=ηβμϕμ+ϕ+σ12/2μ+σ32/22. Then (23)dV1≤-βμμ+ϕ+σ12/2-ηβμϕμ+ϕ+σ12/2μ+σ32/2+μ+γ+σ222+βa+b+ηcIdt-σ1a+bdB1+σ2dB2+cσ3dB3. By integrating the last inequality, we obtain (24)V1St,It,Vt-V1S0,I0,V0t≤-μ+γ+σ222R0S-1+βa+b+ηcI-1t∫0tσ1a+bdB1s+σ2dB2s2+cσ3dB3s. By applying Theorem 1 and the strong law of large numbers for martingales [22], we get the desired result.


## 4. Stationary Distribution

In this section, by using the theory of Has’minski [18], we prove the existence of a unique ergodic stationary distribution, which indicates that the disease is persistent.Theorem 6.
IfR0S>1, then system (3) has a unique stationary distribution and it has the ergodic property.Proof.
To prove Theorem6, it is sufficient to verify assumptions (H.1) and (H.2).
(i) To verify(H.2), we show that there exists a neighborhood D⊂R+3 and a nonnegative C2-function V such that LV(X) is negative for any X∈R+3∖D. To this end, we define a C2-function in the form (25)WS,I,V=MV1S,I,V+V2S,I,V-log⁡S-log⁡V, where V1 is the same function defined in Section 3, V2≕(1/1+θ)(S+I+V)1+θ, and M,θ are positive constants that satisfy (26)-Mμ+γ+σ222R0S-1+A+2μ+ϕ+σ122+σ322<0,μ>θ2max⁡σ12,σ22,σ32, where (27)A=supS,I,V∈R+3μS+I+Vθ-ρ2S+I+Vθ+1. It is easy to see that (28)limS2+I2+V2→∞WS,I,V=limS2+I2+V2→0WS,I,V=+∞. Moreover, W(S,I,V) is a continuous function. Hence, W(S,I,V) must have a minimum point (S-,I-,V-) in the interior of R+3. Then we define a nonnegative C2-function V as (29)VS,I,V=WS,I,V-WS-,I-,V-. From the proof of Theorem 5, we have (30)LV1S,I,V≤-μ+γ+σ222R0S-1+βa+b+ηcIThe operator L defined in Section 2 acts on V2, -log⁡S, and -log⁡V as follows: (31)L-log⁡S=-μS+μ+ϕ+βI+σ122,L-log⁡V=-ϕSV+μ+ηβI+σ322,LV2S,I,V=S+I+Vθμ-μS+I+V-γI+θ2S+I+Vθ-1σ12S2+σ22I2+σ32V2≤μS+I+Vθ-μS+I+Vθ+1+θ2σ12∨σ22∨σ32S+I+Vθ+1=μS+I+Vθ-μ-θ2σ12∨σ22∨σ32S+I+Vθ+1=μS+I+Vθ-ρS+I+Vθ+1≤A-ρ2S1+θ+I1+θ+V1+θ,where (32)ρ=μ-θ2σ12∨σ22∨σ32.Therefore (33)LVS,I,V≤fS+gI-ρ2V1+θ-ϕSV-Mμ+γ+σ222R0S-1+A+2μ+ϕ+σ122+σ322,where (34)fS=-μS-ρ2S1+θ,gI=βMa+b+ηc+1+ηI-ρ2I1+θ.
Next, we construct the following compact subset:(35)D=ϵ≤S≤1ϵ,ϵ2≤V≤1ϵ2,ϵ≤I≤1ϵ,where ϵ is a sufficiently small positive number, satisfying the following inequalities:(36)g∨-μϵ<0,(37)f∨+βMa+b+ηc+1+ηϵ<0,(38)f∨+g∨-ϕϵ<0,(39)g∨-ρ21ϵ1+θ<0,(40)f∨+B-ρ41ϵ1+θ<0,(41)f∨+g∨-ρ21ϵ21+θ<0, where the constant B will be determined later.
Then(42)R+3∖D=D1c∪D2c∪D3c∪D4c∪D5c∪D6c, with (43)D1c=S,I,V∈R+30<S<ϵ,D2c=S,I,V∈R+30<I<ϵ,D3c=S,I,V∈R+30<V<ϵ2,S>ϵ,D4c=S,I,V∈R+3S>1ϵ,D5c=S,I,V∈R+3I>1ϵ,D6c=S,I,V∈R+3V>1ϵ2.Now, we will show that LV is negative for any (S,I,V)∈R+3∖D.
Case 1. If (S,I,V)∈D1c, we obtain from (36) that(44)LVS,I,V≤-Mμ+γ+σ222R0S-1+A+2μ+ϕ+σ122+σ322+g∨-μϵ<0.
Case 2. If (S,I,V)∈D2c, (37) implies that (45)LVS,I,V≤-Mμ+γ+σ222R0S-1+A+2μ+ϕ+σ122+σ322+f∨+βMa+b+ηc+1+ηϵ<0.
Case 3. If (S,I,V)∈D3c, from (38) it follows that (46)LVS,I,V≤-Mμ+γ+σ222R0S-1+A+2μ+ϕ+σ122+σ322+f∨+g∨-ϕϵ<0.
Case 4. If (S,I,V)∈D4c, (39) implies that (47)LVS,I,V≤-Mμ+γ+σ222R0S-1+A+2μ+ϕ+σ122+σ322+g∨-ρ21ϵ1+θ<0.
Case 5. If (S,I,V)∈D5c, from (40) we get (48)LVS,I,V≤-Mμ+γ+σ222R0S-1+A+2μ+ϕ+σ122+σ322+f∨+B-ρ41ϵ1+θ<0, where (49)B=supI∈0,∞βMa+b+ηc+1+ηI-ρ4I1+θ.
Case 6. If (S,I,V)∈D6c, from (41) we obtain (50)LVS,I,V≤-Mμ+γ+σ222R0S-1+A+2μ+ϕ+σ122+σ322+f∨+g∨-ρ21ϵ21+θ<0.
From the previous discussion, we have(51)LVS,I,V<0,S,I,V∈R+3∖D.
(ii) Now, we verify assumption(H.1). The diffusion matrix of system (3) is (52)A∽=σ12S2000σ22I2000σ32V2. There is λ=min⁡σ12S2,σ22I2,σ32V2>0 such that (53)∑i,j=13aij∽Xξiξj=σ12S2ξ12+σ22SI2ξ22+σ32V2ξ32≥λξ2,for (S,I,V)∈D- and ξ∈R+3. That is to say, assumption (H.1) holds.
Consequently, system (3) has an ergodic stationary distribution.


## 5. Extinction

Theorem 7.
We assume thatμ>max⁡σ12,σ22,σ32/2. Let S(t),I(t),V(t)∈R+3 be the solution of system (3). If R0S∽<1, then the disease dies out with probability one.
The thresholdR0S∽ is defined as follows: (54)R0S∽=βμμ+ϕμ+γ+σ22/2+ηβϕμ+ϕμ+γ+σ22/2Proof.
By integrating system (3), we obtain (55)St-S0t+It-I0t+Vt-V0t=μ-μS-μ+γI-μV+1tσ1∫0tSrdB1r+σ2∫0tIrdB2r+σ3∫0tvrdB3r. It follows that(56)V=1-S-μ+γμI+ψ1t, where (57)ψ1t=-St-S0t+It-I0t+Vt-V0t+1μtσ1∫0tSrdB1r+σ2∫0tIrdB2r+σ3∫0tvrdB3r. From the first equation of system (3), we obtain(58)S=μμ+ϕ-βμ+ϕSI+ψ2t,where (59)ψ2t=-St-S0μ+ϕt+σ1μ+ϕt∫0tSsdB1s. Applying Itô’s formula to system (3), one obtains(60)log⁡It-log⁡I0t=βS+ηβV-μ+γ-σ222+σ2B2tt.By injecting (56) and (58) into (60), we have(61)log⁡It-log⁡I0t=βμ+ηϕμ+ϕ-μ+γ+σ222+β1-ηψ1t+ηβψ2t-β21-ημ+ϕSI-βημ+γμI+σ2B2tt≤βμ+ηϕμ+ϕ-μ+γ+σ222+β1-ηψ1t+ηβψ2t+σ2B2tt.From the strong law of large numbers for martingales [22], we get (62)limt→∞B2tt=0,a.s.By Theorem 1, we get (63)limt→∞ψ1t=limt→∞ψ2t=0. By taking the superior limit of both sides of (61), we obtain (64)limsupt→∞log⁡Itt≤βμμ+ϕ+ηβϕμ+ϕ-μ+γ+σ222=μ+γ+σ222R0S∽-1. This finishes the proof.Theorem 8.
We assume thatσ1=σ3=σ and B1(t)=B3(t)=B(t). Let S(t),I(t),V(t)∈R+3 be the solution of system (3). If σ2>2ηϕ, then (65)limsupt→∞log⁡Itt≤μ+γβμ+γ∫0∞xπxdx-1a.s. If (β/μ+γ)∫0∞xπ(x)dx<1, then (66)limItt→∞=0a.s., and the distribution of S(t)+ηV(t) converges weakly to the measure that has the density given by (67)πx=Qσ2x-21-ηϕ/σ2e-2μ/σ2x,x∈0.∞, where (68)Q=σ22μσ21-2ηϕ/σ2Γ-11-2ηϕσ2.Proof.
We assume thatσ1=σ3=σ and B1(t)=B3(t)=B(t).
From system (3), we have(69)dSt+ηVt≤μ+ηϕSt+ηVtdt+σ1StdB1t+ηVtσ3dB3t,=dXt,where(70)dXt=μ+ηϕXtdt+σXtdBt,with the initial value X(0)=S(0)+ηV(0).
We compute that(71)∫μ+ηϕxσ2x2dx=1σ2∫μx2+ηϕxdx=1σ2-μx+ηϕln⁡x+B, where B is a constant.
One can see that(72)∫0∞x-21-ηϕ/σ2e-2μ/σ2xdx=2μσ22ηϕ/σ2-1∫0∞y1-2ηϕ/σ2-1e-ydy=2μσ22ηϕ/σ2-1Γ1-2ηϕσ2, where Γ is the Gamma function defined by (73)ΓZ=∫0∞xz-1e-xdx. If σ2>2ηϕ, then (74)∫0∞x-21-ηϕ/σ2e-2μ/σ2xdx<∞. Thus, condition (H.3) in Lemma 3 is verified. So, system (70) has the ergodic property and the invariant density given by (75)πx=Qσ2x-21-ηϕ/σ2e-2μ/σ2x,x∈0,∞. Let us compute Q.
According to(76)∫0∞πxdx=1, one has (77)Q=σ22μσ21-2ηϕ/σ2Γ-11-2ηϕσ2. From the ergodic theorem, it follows that (78)limt→∞1t∫0tXrdr=∫0∞xπxdxa.s. On the other hand, by applying Itô’s formula to log⁡I(t) and then integrating, one has(79)log⁡It-log⁡I0t=βt∫0tSr+ηVrdr-μ+γ-σ22∫0tSr+ηVr2dr+1tMt,where M(t)=σ∫0t(S(r)+ηV(r))dB(r) whose quadratic variation is (80)M,Mt=σ2∫0tSr+ηVr2dr.In view of the exponential martingales inequality [22], for any positive constants T,α, and ν, we have (81)Psup0≤t≤TMt-α2M,Mt>ν≤e-αν. Choosing T=k,α=1 and ν=2log⁡k, one has (82)Psup0≤t≤kMt-12M,Mt>2log⁡k≤1k2. Applying the Borel-Cantelli Lemma [22] leads to the fact that, for almost all ω∈Ω, there exists a random integer k0=k0(ω) such that, for any k≥k0, we obtain (83)sup0≤t≤kMt-12M,Mt≤2log⁡k. That is, (84)Mt≤12M,Mt+2log⁡kfor  allt≤k.Substituting this inequality into (79) yields (85)log⁡It-log⁡I0t≤βt∫0tSr+ηVrdr-μ+γ+2log⁡ktfor  allt≤k.Applying the comparison theorem of stochastic differential equations on inequality (69) gives (86)St+ηVt≤Xta.s.It follows that (87)log⁡It-log⁡I0t≤βt∫0tXrdr-μ+γ+2log⁡kt,for  allt≤k.Taking the superior limit on both sides of the previous inequality leads to (88)limsupt→∞log⁡It-log⁡I0t≤β∫0∞xπxdx-μ+γa.s.If (β/μ+γ)∫0∞xπ(x)dx<1, we conclude that (89)limt→∞It=0a.s.As a result, for any small ϵ>0, there exist t0 and a set Ωϵ⊂Ω such that P(Ωϵ)>1-ϵ and β(S+η2V)I≤ϵ(S+η2V)≤ϵ(S+ηV) for all t≥t0 and ω∈Ωϵ. From (90)μ-μSt+ηVt-ϵSt+ηVtdt+σSt+ηVtdBt≤dSt+ηVt≤μ-μSt+ηVtdt+σSt+ηVtdBt, it follows that the distribution of the process S(t)+ηV(t) converges weakly to the measure that has the density π.
This completes the proof.


## 6. Numerical Simulations

In this section, we present the numerical simulations to support the above analytical results, illustrating persistence in mean and extinction of the disease.In the two following examples, we choose the initial value asS(0),I(0),V(0)=0.2,0.2,0.2.Example 9.
In model (3), we choose the parameters as follows: (91)σ1=0.2,σ2=0.2,σ3=0.2,β=10,μ=1,ϕ=15,γ=4,η=0.6.We compute that (92)R0S=1.2215>1.Then, according to Theorem 5, the disease is persistent (see Figure 1). Furthermore, by Theorem 6, system (3) has a unique stationary distribution (see Figure 2).Figure 1
Simulation of the pathsS(t),I(t), and V(t) for model (3).Figure 2
Distribution ofS(300),I(300), and V(300).Example 10.
We choose the parameters as follows:(93)σ1=0.6,σ2=1,σ3=0.8,β=8,μ=1,ϕ=15,γ=4,η=0.6.Then (94)R0S∽=0.9091<1Theorem 7 implies that system (3) has disease extinction (see Figure 3).Figure 3
Simulation of the pathsS(t),I(t), and V(t) for model (3).Example 11.
We choose the parameters as follows:(95)σ1=σ3=σ=1.15,β=10,μ=1,ϕ=1,γ=4,η=0.2.Then (96)σ2>2ηϕ,βμ+γ∫0∞xπxdx<1.According to Theorem 8, we get(97)limt→∞It=0a.s.It follows that the distribution of the process S(t)+ηV(t) converges weakly to the measure that has the density π (see Figure 4).Figure 4
Distribution ofS(t)+ηV(t) at time t =10 000.


## 7. Conclusion

This paper is concerned with the dynamics of a stochastic SIVR epidemic model with multiplicative noise sources. By constructing a convenient positive function, we establish sufficient conditions for the existence of a unique ergodic stationary distribution to model (3) and we prove that the disease will be permanent. In addition, we also establish sufficient conditions for extinction of the disease.Many works have been done to study the continuous stochastic models by using the white noise. Sometimes, population systems may suffer sudden environmental perturbations, such as SARS, floods, and toxic pollutants. These phenomena cannot be modeled by stochastic continuous models. During the last years, there was an increasing interest in Lévy noise that has non-Gaussian statistics. A lot of works have been realized and have shown the effectiveness of the Lévy noise in studying such phenomena (see [25–27]). We investigate in our future works the impact of the Lévy noise on the dynamic of complicated population systems.



---

*Source: 1291402-2018-07-18.xml*