Stochastically Perturbed Epidemic Model with Time Delays

Tailei Zhang School of Science, Chang’an University, Xi’an 710064, China Correspondence should be addressed to Tailei Zhang, t.l.zhang@126.com Received 3 November 2012; Accepted 4 December 2012 Academic Editor: Junli Liu Copyright q 2012 Tailei Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate a stochastic epidemic model with time delays. By using Liapunov functionals, we obtain stability conditions for the stochastic stability of endemic equilibrium.


Introduction
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 h 0 f s ds 1.
To be more general, the following model is formulated:
The basic reproduction number for 1.2 is

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:
This system has the same equilibria as system 1.2 .We assume that R 0 > 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 By this way, we obtain 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 dy t h t, y t dt g t, y t dw t , t ≥ 0, y 0 ϕ ∈ H.

2.3
Let {Ω, σ, P} be the probability space, {f t , t ≥ 0} the family of σ-algebra, f t ∈ σ, H the space of f 0 -adapted functions ϕ s ∈ R n , s ≤ 0, ϕ sup s≤0 |ϕ s |, w t the m-dimensional f t -adapted Wiener process, h t, y t the n-dimensional vector, and g t, y t 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 y 0 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

2.9
Theorem 2.6.Assume that R 0 > 1 and the parameters of system 2.2 satisfy conditions
for some p > 0 and q > 0. Let L be the generating operator of the system 2.9 , then

2.14
We now choose the functional V 2 to eliminate the term with delay

2.15
Then for functional

2.17
Therefore, there exists a c > 0 such that LV ≤ −c|y| 2 , where y y 1 , y 2 , y 3 .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.If α 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
The constant p in the proof of Theorem 2.6 is 0 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 p 2 λ y 2 y 3 , then the third condition in 2.18 is different from that in Theorem 7 of 8 .
Theorem 2.8.Assume that R 0 > 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.

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.
Definition 2.3.The trivial solution of 2.3 is said to be asymptotically mean square stable if it is mean square stable and lim t → ∞ E|y t; ϕ | 2 0. so that for almost all t ≥ 0, the first and second derivatives with respect to y of V ϕ 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 y t; ϕ > r, t ≥ 0 ≤ ε 2.4 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 sup s≤0 E|ϕ s| 2 < δ.If V t, ϕ V t, ϕ 0 , ϕ s , s < 0, we can define the function V ϕ t, y V t, ϕ V t,y t V t, y, y t s , s < 0, ϕ y t , y ϕ 0 y t .Let us define C 1,2 as a class of function V t, ϕ i > 0, i 1, 2, 3.Then, the trivial solution of 2.3 is asymptotically mean square stable.Theorem 2.5.Let there exist a functional V t, ϕ ∈ C 1,2 such that c 1 y t 2 ≤ V t, y t ≤ c 2 sup s≤0 y t s 2 , LV t, y t ≤ 0 2.8 for c i > 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 ẏ1 − βI * μ 1 y 1 − βS * h 0 f s y 2 t − s ds αy 3 σ 1 y 1 ẇ1 t , ẏ2 βI * y 1 − βS * y 2 βS * h 0 f s y 2 t − s ds σ 2 y 2 ẇ2 t , ẏ3 λy 2 − μ 3 α y 3 σ 3 y 3 ẇ3 t .