# Stochastically Perturbed Epidemic Model with Time Delays

Authors: Tailei Zhang
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

We investigate a stochastic epidemic model with time delays. By using Liapunov functionals, we obtain stability conditions for the stochastic stability of endemic equilibrium.


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## Body

## 1. Introduction

In [1], Zhen et al. introduced a deterministic SIRS model
(1.1)S˙(t)=b-μS(t)-βS(t)∫0hf(s)I(t-s)ds+αR(t),I˙(t)=βS(t)∫0hf(s)I(t-s)ds-(μ+c+λ)I(t),R˙(t)=λI(t)-(μ+α)R(t),
where S(t) is the number of susceptible population, I(t) is the number of infective members and R(t) is the number of recovered members. b is the rate at which population is recruited, μ is the death rate for classes S(t), I(t), and R(t), c is the disease-induced death rate, β is the transmission rate, λ is the recovery rate, and α is the loss of immunity rate. Equation (1.1) represents an SIRS model with epidemics spreading via a vector, whose incubation time period is a distributed parameter over the interval[0,h]. h∈ℝ+ is the limit superior of incubation time periods in the vector population. The f(s) is usually nonnegative and continuous and is the distribution function of incubation time periods among the vectors and ∫0hf(s)ds=1.To be more general, the following model is formulated:(1.2)S˙(t)=b-μ1S(t)-βS(t)∫0hf(s)I(t-s)ds+αR(t),I˙(t)=βS(t)∫0hf(s)I(t-s)ds-(μ2+λ)I(t),R˙(t)=λI(t)-(μ3+α)R(t).
The positive constants μ1, μ2, and μ3 represent the death rates of susceptibles, infectives, and recovered, respectively. It is natural biologically to assume that μ1<min{μ2,μ3}. If α=0, model (1.2) was considered in [2–5]. For α=0 and fixed delay, the global asymptotic stability of (1.2) was considered in [6].The basic reproduction number for (1.2) is
(1.3)R0=βbμ1(μ2+λ).
If R0≤1, the system (1.2) has just one disease-free equilibrium E0=(b/μ1,0,0); otherwise, if R0>1, the disease-free equilibrium E0 is still present, but there is also a unique positive endemic equilibrium E*=(S*,I*,R*), given by S*=(μ2+λ)/β, I*=(b(μ3+α)(R0-1))/(R0[μ2(μ3+α)+μ3λ]), R*=(λ/(μ3+α))I*.


## 2.  Stability Analysis of the Atochastic Delay Model

Since environmental fluctuations have great influence on all aspects of real life, then it is natural to study how these fluctuations affect the epidemiological model (1.2). We assume that stochastic perturbations are of white noise type and that they are proportional to the distances of S,I,R from S*,I*,R*, respectively. Then the system (1.2) will be reduced to the following form:
(2.1)S˙(t)=b-μ1S(t)-βS(t)∫0hf(s)I(t-s)ds+αR(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+α)R(t)+σ3(R(t)-R*)w˙3(t).
Here, σ1,  σ2, and σ3 are constants, and w(t)=(w1(t),w2(t),w3(t)) represents a three-dimensional standard Wiener processes.This system has the same equilibria as system (1.2). We assume that R0>1; we discuss the stability of the endemic equilibrium E* of (2.1). The stochastic system (2.1) can be centered at its endemic equilibrium E* by the changes of variables x1=S-S*, x2=I-I*, x3=R-R*. By this way, we obtain
(2.2)x˙1=-(βI*+μ1)x1-βx1∫0hf(s)x2(t-s)ds-βS*∫0hf(s)x2(t-s)ds+αx3+σ1x1w˙1(t),x˙2=βI*x1-βS*x2+βx1∫0hf(s)x2(t-s)ds+βS*∫0hf(s)x2(t-s)ds+σ2x2w˙2(t),x˙3=λx2-(μ3+α)x3+σ3x3w˙3(t).In order to investigate the stability of endemic equilibrium of system (2.1), we study the stability of the trivial solution of system (2.2).First, consider the stochastic functional differential equation(2.3)dy(t)=h(t,yt)dt+g(t,yt)dw(t),t≥0,y0=φ∈H.
Let {Ω,σ,P} be the probability space, {ft,t≥0} the family of σ-algebra, ft∈σ, H the space of f0-adapted functions φ(s)∈Rn, s≤0, ∥φ∥=sups≤0|φ(s)|, w(t) the  m-dimensional ft-adapted Wiener process, h(t,yt) the n-dimensional vector, and g(t,yt) the n×m-dimensional matrix, both defined for t≥0. We assume that (2.3) has a unique global solution y(t;φ) and that h(t,0)=g(t,0)≡0. Then, (2.3) has the trivial solution y(t)≡0 corresponding to the initial condition y0=0.Definition 2.1.
The trivial solution of (2.3) is said to be stochastically stable if, for every ε∈(0,1) and r>0, there exists a δ>0 such that
(2.4)P{|y(t;φ)|>r,t≥0}≤ε
for any initial condition φ∈H satisfying P{∥φ∥≤δ}=1.Definition 2.2.
The trivial solution of (2.3) is said to be mean square stable if, for every ε>0, there exists a δ>0 such that E|y(t;φ)|2<ε for any t≥0 provided that sups≤0E|φ(s)|2<δ.Definition 2.3.
The trivial solution of (2.3) is said to be asymptotically mean square stable if it is mean square stable and limt→∞E|y(t;φ)|2=0.The differential operator associated to (2.3) is defined by the formula
(2.5)LV(t,φ)=limsupΔ→0Et,φV(t+Δ,yt+Δ)-V(t,φ)Δ,
where y(s), s≥t is the solution of (2.3) with initial condition yt=φ∈H, and V(t,φ) is a functional defined for t≥0.IfV(t,φ)=V(t,φ(0),φ(s)), s<0, we can define the function Vφ(t,y)=V(t,φ)=V(t,yt)=V(t,y,y(t+s)), s<0, φ=yt, y=φ(0)=y(t). Let us define C1,2 as a class of function V(t,φ) so that for almost all t≥0, the first and second derivatives with respect to y of Vφ(t,y) are continuous, and the first derivative with respect to t is continuous and bounded. Then the generating operator L of (2.3) is defined by
(2.6)LV(t,yt)=∂Vφ(t,y)∂t+hT(t,yt)∂Vφ(t,y)∂y+12trace[gT(t,yt)∂2Vφ(t,y)∂y2g(t,yt)].The following theorems [7] contain conditions under which the trivial solution of (2.3) is asymptotically mean square stable and stochastically stable.Theorem 2.4.
If there exist a functionalV(t,φ)∈C1,2 such that
(2.7)c1E|y(t)|2≤EV(t,yt)≤c2sups≤0E|y(t+s)|2,ELV(t,yt)≤-c3E|y(t)|2
for ci>0,  i=1,2,3. Then, the trivial solution of (2.3) is asymptotically mean square stable.Theorem 2.5.
Let there exist a functionalV(t,φ)∈C1,2 such that
(2.8)c1|y(t)|2≤V(t,yt)≤c2sups≤0|y(t+s)|2,LV(t,yt)≤0
for ci>0, i=1,2 and for any φ∈H such that P{∥φ∥≤δ}=1, where δ>0 is sufficiently small. Then, the trivial solution of (2.3) is stochastically stable.Consider the linear part of (2.2)
(2.9)y˙1=-(βI*+μ1)y1-βS*∫0hf(s)y2(t-s)ds+αy3+σ1y1w˙1(t),y˙2=βI*y1-βS*y2+βS*∫0hf(s)y2(t-s)ds+σ2y2w˙2(t),y˙3=λy2-(μ3+α)y3+σ3y3w˙3(t).Theorem 2.6.
Assume thatR0>1 and the parameters of system (2.2) satisfy conditions
(2.10)0≤σ12<2μ1-α(1+q)q,0≤σ22<q(2βS*-α)1+q=q[2(μ2+λ)-α]1+q,0≤σ32<2μ3+α-λ,2αq2μ3+α-λ-σ32<min{(2μ1-σ12)q-α(1+q)βS*,p*},
where p*=(-βS*+((βS*)2-4λ[(1+q)σ22-2qβS*+qα]))/2λ. Then, the trivial solution of system (2.9) is asymptotically mean square stable.Proof.
Set(2.11)V1=py12+y22+p2y32+q(y1+y2)2
for some p>0 and q>0. Let L be the generating operator of the system (2.9), then
(2.12)LV1=[-(βI*+μ1)y1-βS*∫0hf(s)y2(t-s)ds+αy3][2py1+2q(y1+y2)]+[βI*y1-βS*y2+βS*∫0hf(s)y2(t-s)ds][2y2+2q(y1+y2)]+2p2y3[λy2-(μ3+α)y3]+(p+q)σ12y12+2qσ1σ2y1y2+(1+q)σ22y22+p2σ32y32=[(σ12-2μ1)(p+q)-2pβI*]y12+(1+q)(σ22-2βS*)y22+p2[σ32-2(μ3+α)]y32+2α(p+q)y1y3+2(qα+p2λ)y2y3+2[(σ1σ2-βS*-μ1)q+βI*]y1y2+2βS*(y2-py1)∫0hf(s)y2(t-s)ds.
Let
(2.13)q=βI*βS*+μ1-σ1σ2.
Since σ1σ2≤(σ12+σ22)/2<μ1+βS*, it means that q>0. By using the inequality 2|uv|≤u12+u22 and 2αpy1y3≤αp(y12/p+py32)=αy12+αp2y32, we find that
(2.14)LV1≤[(σ12-2μ1)q+α(1+q)+pβS*]y12+[(1+q)(σ22-2βS*)+qα+p2λ+βS*]y22+[p2(σ32-2μ3-α)+2αq+p2λ]y32+(1+p)βS*∫0hf(s)y22(t-s)ds.
We now choose the functionalV2 to eliminate the term with delay
(2.15)V2=(1+p)βS*∫0hf(s)∫t-sty22(τ)dτds.
Then for functional V=V1+V2, we obtain
(2.16)LV≤[(σ12-2μ1)q+α(1+q)+pβS*]y12+[p2λ+pβS*+(1+q)σ22-2qβS*+qα]y22+[p2(σ32-2μ3-α+λ)+2αq]y32.
If the first condition of (2.10) holds, then (σ12-2μ1)q+α(1+q)<0. Set F(p)=p2λ+pβS*+(1+q)σ22-2qβS*+qα, and if the second condition of (2.10) is true, then F(0)<0, thus F(p)=0 has one positive root p*=(-βS*+((βS*)2-4λ[(1+q)σ22-2qβS*+qα]))/2λ, for any 0<p<p*, F(p)<0. From (2.10), there exists a p>0, such that
(2.17)2αq2μ3+α-λ-σ32<p<min{(2μ1-σ12)q-α(1+q)βS*,p*}.
Therefore, there exists a c>0 such that LV≤-c|y|2, where y=(y1,y2,y3). From Theorem 2.4, we can conclude that the zero solution of system (2.9) is asymptotically mean square stable. The theorem is proved.Remark 2.7.
Ifα=0, then the system (2.1) becomes an SIR model, which has been discussed in [8]. The conditions (2.10) of Theorem 2.6 reduce to
(2.18)0≤σ12<2μ1,0≤σ22<2q(μ2+λ)1+q,0≤σ32<2μ3-λ.
The constant p in the proof of Theorem 2.6 is 0<p<min{((2μ1-σ12)q)/(βS*),p1*} with p1*=(-βS*+((βS*)2-4λ[(1+q)σ22-2qβS*]))/2λ. The first two conditions in (2.18) are the same as those in Theorem 7 of [8]. Since for α>0, we use different inequality to zoom up the term 2(qα+p2λ)y2y3, then the third condition in (2.18) is different from that in Theorem 7 of [8].Theorem 2.8.
Assume thatR0>1 and that conditions (2.10) are satisfied. Then the trivial solution of system (2.2) is stochastically stable.The proof is omitted because of the fact that the initial system (2.2) has a nonlinearity order more than one, then the conditions sufficient for asymptotic mean square stability of the trivial solution of the linear part of this system are sufficient for stochastic stability of the trivial solution of the initial system [9, 10]. Thus, if the conditions (2.10) hold, then the trivial solution of system (2.2) is stochastically stable.


## 3. Conclusions

In this paper, we have extended the well-known SIRS epidemic model with time delays by introducing a white noise term in it. We want to examine how environmental fluctuations affect the stability of system (1.2). By constructing Liapunov functional, we obtain sufficient conditions for the stochastic stability of the endemic equilibrium E*. Our main results extend the corresponding results in paper [8], which discussed an SIR epidemic model.



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