# Stability of SIRS Epidemic Model with Stochastic Perturbation and Distributed Delays

Authors: Ramziya Rifhat; Xamxinur Abdurahman; Ahmadjan Muhammadhaji
Journal: ISRN Applied Mathematics
 (2013)

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## Abstract

A class of SIRS epidemic model with stochastic perturbation and distributed delays is proposed and discussed. Some sufficient conditions on the stability of the zero solution are established. Finally, concluding that, the white noise is favorable for the stability of zero solution and the distributed time delays have no impact on the stability of zero solution.


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## Body

## 1. Introduction

As an important class of mathematical models, stochastic differential equations have been widely used in automatic control, biological, chemical reaction engineering, medicine, economics, demography, and many other scientific fields. For better application, people have done a lot of work in the stochastic differential equations and various particular equations proposed for the study of stochastic differential equations. Subsequently, a lot of the literature related to this topic was published and extensive research results were obtained. However, many of the problems have not been fully investigated and deserve further study. As it is well known, in recent years, the stochastic epidemic systems are extensively studied. Many important and influential results have been established and can be found in many articles and books. Particularly, the stability of zero solutions for various type stochastic epidemic systems has been extensively studied in [1–7] and the references are cited therein.In [8], the authors studied the following classical SIRS epidemiological model with distributed time delay:
(1)S˙(t)=-βS(t)∫0h‍f(s)I(t-s)ds-μ1S(t)+b+kR(t),I˙(t)=βS(t)∫0h‍f(s)I(t-s)ds-(μ2+λ)I(t),R˙(t)=λI(t)-(μ3+k)R(t).
Individual are susceptible, then infectious, recovered with temporary immunity, and then susceptible again when the immunity is lost. Then parameter b>0 is the rate of susceptible individuals recruited into the population (either by birth or immigration) and μ1>0 is the natural death rate. The parameter λ>0 describes the rate at which the infectious population becomes recovered, and k>0 denotes the rate at which the recovered population loses immunity. The positive constant β>0 is the average number of contacts per infective per day. The nonnegative constant h is the time delay. f(s) is the fraction of vector population for which the time taken to become infectious, and it is assumed to be a nonnegative function on [0,h] as density function and satisfies∫0h‍f(s)ds=1. For detailed biological background, we refer to [8, 9].In [8], the authors derived the disease-free equilibrium P0(S0,0,0) and the basic reproductive number R0=(βb/μ1(λ+μ2)) of system (1), where S0=(b/μ1), β is the transmission coefficient, and 1/(λ+μ2) is the average residence time in the infectious individuals class. By straightforward computations, it can be seen that, for R0>1, system (1) has a unique endemic equilibrium state P*(S*,I*,R*) in the interior of ℜ+3, where
(2)S*=S0R0=λ+μ2β,I*=k+μ3μ1(k+λ+μ2)(b-μ1S*),R*=λI*k+μ3.On the other hand, white noise stochastic perturbations around the positive endemic equilibrium of epidemic models were considered in [1, 3]. In [1], the authors studied the following stochastic differential equation model:
(3)S˙(t)=-βS(t)∫0h‍f(s)I(t-s)ds-μ1S(t)+b+σ1(S(t)-S*)w˙1(t),I˙(t)=βS(t)∫0h‍f(s)I(t-s)ds-(μ2+λ)I(t)+σ2(I(t)-I*)w˙2(t),R˙(t)=λI(t)-μ3R(t)+σ3(R(t)-R*)w˙3(t),
where S*,I*, andR* are the positive points of equilibrium for the corresponding deterministic system (1), constants σi(i=1,2) are the intensity of white noise, and wi(t)(i=1,2,3) are the standard Wiener processes. In [1], the authors used the Lyapunov functionals method and the sufficient conditions for the stability of the zero solution are obtained for system (3).Likewise, in [3], the authors studied the same type of stochastic perturbations and extended the deterministic model for the epidemics induced by virulent phages on bacteria in marine environment, allowing random fluctuations around the positive equilibrium, and concluded that the solution of the corresponding stochastic model for phage bacteria interaction was asymptotically mean square stable both analytically and numerically.Motivated by the aforementioned works, in this paper,we consider the following SIRS epidemic model with stochastic perturbation and distributed delays:(4)S˙(t)=-βS(t)∫0h‍f(s)I(t-s)ds-μ1S(t)+b+kR(t)+σ1(S(t)-S*)w˙1(t),I˙(t)=βS(t)∫0h‍f(s)I(t-s)ds-(μ2+λ)I(t)+σ2(I(t)-I*)w˙2(t),R˙(t)=λI(t)-(μ3+k)R(t)+σ3(R(t)-R*)w˙3(t),
where the constants σ1,σ2, and σ3 are the intensity of white noise w˙1(t),w˙2(t),w˙3(t) and w1(t),w2(t),w3(t) are standard Wiener process defined on a complete probability space (Ω,ℱ,{ℱt}t≥0,P). The detailed definition of Wiener process can be found in [2].Letu1=S-S*,u2=I-I*,u3=R-R*, where N*=S*+I*+R*. Then, system (4) is rewritten in the following form:
(5)u˙1=-(βI*+μ1)u1-βS*∫0h‍f(s)u2(t-s)ds-βu1×∫0h‍f(s)u2(t-s)ds+ku3+σ1u1w˙1,u˙2=βI*u1-βS*u2+βS*∫0h‍f(s)u2(t-s)ds+βu1×∫0h‍f(s)u2(t-s)ds+σ2u2w˙2,u˙3=λu2-(μ3+k)u3+σ3u3w˙3.It is easy to see that the stability of zero solution in system (4) is equivalent to the stability of zero solution in system (5). In order to obtain the sufficient conditions for the stability of zero solution in system (5), we will consider the following linear part of system (5):
(6)z˙1=-(βI*+μ1)z1-βS*∫0h‍f(s)z2(t-s)ds+kz3+σ1z1w˙1,z˙2=βI*z1-βS*z2+βS*∫0h‍f(s)z2(t-s)ds+σ2z2w˙2,z˙3=λz2-(μ3+k)z3+σ3z3w˙3,
and the auxiliary system without delays
(7)y˙1=-(βI*+μ1)y1+ky3+σ1y1w˙1,y˙2=βI*y1-βS*y2+σ2y2w˙2,y˙3=λy2-(μ3+k)y3+σ3y3w˙3.


## 2. Preliminaries

Consider the following stochastic differential equation [2]:
(8)dx(t)=a(t,xt)dt+b(t,xt)dw(t),x0=φ∈H,
where function a(t,xt) defined in [t0,∞)×Rnand b(t,xt) is an n×m matrix, xt(s)=x(t+s) with s≤0, w(t) ism-dimensional Wiener process. We assume that x=0 is a trivial solution of system (8), that is, a(t,0)=0, b(t,0)=0 for allt≥t0. Denote with H the space of f0-adapted random variables φ, with φ(s)∈ℜn, s≤0 and ∥φ∥0=sups≤0|φ(s)|, ∥φ∥12=sups≤0𝔼|φ(s)|2 (𝔼 is the mathematical expectation). Let V:[0,∞)×H→R be a functional defined for ≥0,φ∈H.Generating operatorL of (8) is defined [2] by formula
(9)LV(t,φ)=limΔ→0𝔼t,φV(t+Δ,yt+Δ)-V(t,φ)Δ,
where a scalar functional V(t,φ) is defined by t≥0, φ∈H, and x(s) is the solution of (8) by s≥t with initial function xt=φ∈H. Let us describe one class of functionals V(t,φ) for which the operator L can be calculated in final form. We reduce the arbitrary functional V(t,φ), t≥0, φ∈H, to the form V(t,φ)=V(t,φ(0),φ(s)), s<0 and define the function
(10)Vφ(t,x)=V(t,φ)=V(t,x,x(t+s)),s<0,φ=xt,x=φ(0)=x(t).
Let D be the class of functionals V(t,φ) for which functionalVφ(t,x) has two continuous derivations with respect to x and one bounded derivative with respect to t for almost all t≥0. For functionals from D the generating operator L of (8) is defined and is equal to the following:
(11)LV(t,xt)=∂Vφ(t,x)∂t+a′(t,xt)∂Vφ(t,x)∂x+12Tr[b′(t,xt)∂2Vφ(t,x)∂x2b(t,xt)].Definition 1.
The zero solution of (8) is called mean square stable if for any ϵ>0, there exists a δ>0 such that 𝔼|x(t)|2<ϵ for any t≥0 provided that ∥φ∥12<δ.Definition 2.
The zero solution of (8) is called asymptotically mean square stable if it is mean square stable and limt→∞𝔼|x(t)|2=0.Definition 3.
The zero solution of (8) is called stable in probability if for any ϵ1>0 and ϵ2>0, there exists δ>0 such that solution x(t)=x(t,φ) of (8) satisfies
(12)ℙ{x(t,φ)>ϵ1}<ϵ2
for any initial function φ∈H such that ℙ{|φ|≤δ}=1. Here ℙ{·} is the probability of the event enclosed in braces.Theorem 4 (see [2]).
Let there exist the functionalV(t,φ)∈D such that
(13)c1𝔼|x(t)|2≤𝔼V(t,xt)≤c2∥xt∥12,𝔼LV(t,xt)≤-c3𝔼|x(t)|2,ci>0.
Then zero solution of (8) is asymptotically mean square stable.Theorem 5 (see [2]).
Let there exist the functionalV(t,φ)∈D such that
(14)c1|x(t)|2≤V(t,xt)≤c2∥xt∥02,LV(t,xt)≤0,ci>0
for any function φ∈H such that P{|φ|≤δ}=1, where δ>0 is sufficiently small. Then the zero solution of (8) is stable in probability.


## 3. Main Results

To prove the zero solution of system (5) is stable in probability, firstly, we will consider the linear part of system (5) and the auxiliary system without delays. Next, we will consider the auxiliary system without delays of system (5) and we will prove that the zero solution of auxiliary system (7) is asymptotic mean square stable.Theorem 6.
Suppose that(15)σ12<2μ1,σ22<2μ2,(1+qp2)σ32<2μ3;
then zero solution of system (7) is asymptotic mean square stable.Proof.
Consider the following Lyapunov function:(16)V1=py12+y22+p2y32+q(y1+y2+y3)2,
where p, q are real positive constants to be chosen in the following. From the Itô formula (11), we have the following:
(17)LV1=2py1y˙1+2py2y˙2+2p2y3y˙3+2q(y1+y2+y3)×(y˙1+y˙2+y˙3)+(p+q)σ12y12+2(1+q)σ22y22+2(p2+q)σ32y32=2(py1+q(y1+y2+y3))×(-(βI*+μ1)y1+ky3)+2(y2+q(y1+y2+y3))×(βI*y1-βS*y2)+2(p2y3+q(y1+y2+y3))×(λy2-μ3y3-ky3)+(p+q)σ12y12+(1+q)σ22y22+(p2+q)σ32y32+2qσ1σ2y1y2+2qσ2σ3y2y3≤+2qσ1σ3y1y3-2py12(βI*+μ1)-2q(y1+y2+y3)×(μ1y1+βS*y2-λy2+μ3y3)+2βI*y1y2-2βS*y22-2p2μ3y32+(p+q)σ12y12+(1+q)σ22y22+p2λ(y22p+py32)+(p2+q)σ32y32+2qσ1σ2y1y2+2qσ2σ3y2y3+2qσ1σ3y1y3=[-2pβI*-2pμ1-2qμ1+(p+q)σ12]+y22[-2qβS*-2βS*+(1+q)σ22]+y32[-2qμ3-2p2μ3+p2λp-2p2k+p2σ32]+2y1y2[-qμ1-qβS*+qλ+βI*+qσ1σ2]+2y2y3[-qμ3-qβS*+qλ+qσ2σ3]+2y1y3[-qμ3-qμ1+pk+qσ1σ3]+qσ32y32≤y12(p+q)(σ12-2μ1)+y22[(1+q)(σ22-2βS*)+pλ+2qλ]+p2y32(pλ-2μ3+σ32)+2y1y2[βI*+q(σ1σ2-μ1-βS*+λ)]+2y1y3[q(-μ3-μ1+σ1σ3)+pk]+qσ32y32+2y2y3[q(-μ3-βS*+λ+σ2σ3)].
Set
(18)q=min{pkμ1+μ3-σ1σ3,βI*μ1+βS*-λ-σ1σ2},
and σ32<(1+(q/p2))σ32<2μ3. From the condition (15), it follows
(19)|σ1σ2|≤σ12+σ222<μ1+βS*-λ,|σ2σ3|≤σ22+σ322<μ3+βS*-λ,|σ1σ3|≤σ12+σ322<μ1+μ3.
By this way, we obtain the following:
(20)LV1≤-y12(p+q)(2μ1-σ12)-y22[(1+q)(2βS*-σ22)-pλ-2qλ]-p2y32(2μ3-(1+qp2)σ32-pλ).
Choose constant p such that
(21)p=1λmin{2μ3-(1+qp2)σ32,(1+q)(2βS*-σ22-2qλ)(1+qp2)}.
From (15), it follows that there exists c>0 such that LV1≤-c|y|2, where y=(y1,y2,y3). From Theorem 4, we have that the zero solution of system (7) is asymptotic mean square stable. This completes the proof.In the second place, we will consider the linear part of system (5) and will prove that the zero solution of system (6) is asymptotic mean square stable.Theorem 7.
Suppose that(22)σ12<2μ1,σ22<2qμ21+q,(1+qp2)σ32<2μ3,
where q is defined in (18); then the zero solution of the system (6) is asymptotic mean square stable.Proof.
Consider the following Lyapunov function;(23)V2=pz12+z22+p2z32+q(z1+z2+z3)2.
Let L be the generating operator [1] of the system (6). We compute the following:
(24)LV2=2(pz1+q(z1+z2+z3))×(-(βI*+μ1)z1+kz3-βS*∫oh‍f(s)z2(t-s)ds)+2(z2+q(z1+z2+z3))×(βI*z1-βS*z2+βS*∫oh‍f(s)z2(t-s)ds)+2(p2z3+q(z1+z2+z3))(λz2-μ3z3-kz3)+(p+q)σ12z12+(1+q)σ22z22+(p2+q)σ32z32+2qσ1σ2z1z2+2qσ2σ3z2z3+2qσ1σ3z1z3≤-z12(p+q)(2μ1-σ12)-z22[(1+q)(2βS*-σ22)-pλ-2qλ]-p2z32(2μ3-σ32-pλ)+qσ32z32+2z1z2×[βI*+q(σ1σ2-μ1-βS*+λ)]+2z1z3[q(-μ3-μ1+σ1σ3)+pk]+2z2z3[q(-μ3-βS*+λ+σ2σ3)]+2βS*(z2-pz1)∫0h‍f(s)z2(t-s)ds.
Using (18), we obtain the following:
(25)LV2≤-z12q(2μ1-σ12)-z22[(1+q)(2βS*-σ22)-pλ-2qλ]-p2z32(2μ3-σ32-pλ)+qσ32z32+pβS*[z12+∫0h‍f(s)z22(t-s)ds]+βS*[z22+∫0h‍f(s)z22(t-s)ds]=-z12[q(2μ1-σ12)-pβS*]-z22[(1+q)(2βS*-σ22)-pλ-2qλ-βS*]-p2z32(2μ3-σ32-pλ)+qσ32z32+βS*(1+p)×∫0h‍f(s)z22(t-s)ds,
and we define a Lyapunov function as follows:
(26)V3=βS*(1+p)∫0h‍f(s)∫t-st‍z22(τ)dτds.
With the application of the multidimensional Itô’s formula [1], we obtain the following:
(27)LV3=βS*(1+p)z22(t)-βS*(1+p)∫0h‍f(s)z22(t-s)ds.
Finally, we define a Lyapunov function
(28)V4=V2+V3.
By (22), (25), and (27), we have the following:
(29)LV4≤-z12[q(2μ1-σ12)-pβS*]-z22[2q(βS*-λ)-(1+q)σ22-p(λ+βS*)]-p2z32(2μ3-(1+qp2)σ32-pλ),
where
(30)p=min{q(2μ1-σ12)βS*,2q(βS*-λ)-(1+q)σ22λ+βS*,2μ3-(1+(q/p2))σ32λ}.
Therefore, there exists a c>0 such that LV4≤-c|z|2, where z=(z1,z2,z3). From Theorem 4, we have that the zero solution of system (6) is asymptotic mean square stable. This completes the proof.Theorem 8.
Suppose that the conditions of Theorem7 hold; then the zero solution of the system (5) is stable in probability.Proof.
Consider the following Lyapunov function:(31)V=pu12+u22+p2u32+q(u1+u2+u3)2+βS*(1+p)∫0h‍f(s)∫t-st‍u22(τ)dτds.
Then, we obtain the following:
(32)LV=2(pu1+q(u1+u2+u3))×(-(βI*+μ1)u1+ku3-βS*∫0h‍f(s)u2(t-s)ds-βu1∫0h‍f(s)u2(t-s)ds)+2(u2+q(u1+u2+u3))×(βI*u1-βS*u2+βS*∫0h‍f(s)u2(t-s)ds+βu1∫0h‍f(s)u2(t-s)ds)+2(p2u3+q(u1+u2+u3))×(λu2-μ3u3-ku3)+(p+q)σ12u12+(1+q)σ22u22+(p2+q)σ32u32+2qσ1σ2u1u2+2qσ2σ3u2u3+2qσ1σ3u1u3+(1+p)βS*u22-(1+p)βS*∫0h‍f(s)u22(t-s)ds≤-u12[q(2μ1-σ12)-pβS*]-u22[2q(βS*-λ)-(1+q)σ22-p(λ+βS*)]-p2u32(2μ3-σ32-pλ)+2βu1(u2-pu1)×∫0h‍f(s)u2(t-s)ds.
From Theorem 5, there exists a sufficiently small δ>0 such that ℙ{|u2(s)|<δ}=1. Then
(33)2β|u1(u2-pu1)∫0h‍f(s)u2(t-s)ds|≤|2βu1u2∫0h‍f(s)u2(t-s)ds|+|2βpu12∫0h‍f(s)u2(t-s)ds|≤2βu1u2δ+2βpu12δ=βδ(2u1u2+2pu12)≤βδ(u12(1+2q)+u22).
Therefore,
(34)LV≤-u12[q(2μ1-σ12)-pβS*-βδ(1+2q)]-u22[2q(βS*-λ)-(1+q)σ22-p(λ+βS*)-βδ]-p2u32(2μ3-σ32-pλ).
Hence, for sufficiently small δ>0, we obtain LV≤0. From Theorem 5, we have that the zero solution of system (5) is asymptotic mean square stable. This completes the proof.


## 4. Conclusion

In this paper, we considered the SIRS epidemic model with stochastic perturbation and distributed delays and some sufficient conditions on the stability of the zero solution are established. Further, we obtain that the white noiseσi(i=1,2,3) is favorable for the stability of zero solution and the distributed time delays have no impact on the stability of zero solution. The results obtained in this paper indicate that suitable stochastic perturbation can maintain the stability of zero solution.



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