Theory and Applications of Periodic Solutions and Almost Periodic SolutionsView this Special Issue
Stochastically Perturbed Epidemic Model with Time Delays
We investigate a stochastic epidemic model with time delays. By using Liapunov functionals, we obtain stability conditions for the stochastic stability of endemic equilibrium.
In , Zhen et al. introduced a deterministic SIRS model where is the number of susceptible population, is the number of infective members and is the number of recovered members. is the rate at which population is recruited, is the death rate for classes , , and , is the disease-induced death rate, is the transmission rate, is the recovery rate, and 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. is the limit superior of incubation time periods in the vector population. The is usually nonnegative and continuous and is the distribution function of incubation time periods among the vectors and .
To be more general, the following model is formulated: The positive constants , , and represent the death rates of susceptibles, infectives, and recovered, respectively. It is natural biologically to assume that . If , model (1.2) was considered in [2–5]. For and fixed delay, the global asymptotic stability of (1.2) was considered in .
The basic reproduction number for (1.2) is If , the system (1.2) has just one disease-free equilibrium ; otherwise, if , the disease-free equilibrium is still present, but there is also a unique positive endemic equilibrium , given by , , .
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 from , respectively. Then the system (1.2) will be reduced to the following form: Here, , , and are constants, and represents a three-dimensional standard Wiener processes.
This system has the same equilibria as system (1.2). We assume that ; we discuss the stability of the endemic equilibrium of (2.1). The stochastic system (2.1) can be centered at its endemic equilibrium 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 Let be the probability space, the family of -algebra, , the space of -adapted functions , , , the -dimensional -adapted Wiener process, the -dimensional vector, and the -dimensional matrix, both defined for . We assume that (2.3) has a unique global solution and that . Then, (2.3) has the trivial solution corresponding to the initial condition .
Definition 2.1. The trivial solution of (2.3) is said to be stochastically stable if, for every and , there exists a such that for any initial condition satisfying .
Definition 2.2. The trivial solution of (2.3) is said to be mean square stable if, for every , there exists a such that for any provided that .
Definition 2.3. The trivial solution of (2.3) is said to be asymptotically mean square stable if it is mean square stable and .
The differential operator associated to (2.3) is defined by the formula where , is the solution of (2.3) with initial condition , and is a functional defined for .
If , , we can define the function , , , . Let us define as a class of function so that for almost all , the first and second derivatives with respect to of are continuous, and the first derivative with respect to is continuous and bounded. Then the generating operator of (2.3) is defined by
The following theorems  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 functional such that for , . Then, the trivial solution of (2.3) is asymptotically mean square stable.
Theorem 2.5. Let there exist a functional such that for , and for any such that , where is sufficiently small. Then, the trivial solution of (2.3) is stochastically stable.
Consider the linear part of (2.2)
Theorem 2.6. Assume that and the parameters of system (2.2) satisfy conditions where . Then, the trivial solution of system (2.9) is asymptotically mean square stable.
for some and . Let be the generating operator of the system (2.9), then
Since , it means that . By using the inequality and , we find that
We now choose the functional to eliminate the term with delay Then for functional , we obtain If the first condition of (2.10) holds, then . Set , and if the second condition of (2.10) is true, then , thus has one positive root , for any , . From (2.10), there exists a , such that Therefore, there exists a such that , where . 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 , then the system (2.1) becomes an SIR model, which has been discussed in . The conditions (2.10) of Theorem 2.6 reduce to The constant in the proof of Theorem 2.6 is with . The first two conditions in (2.18) are the same as those in Theorem 7 of . Since for , we use different inequality to zoom up the term , then the third condition in (2.18) is different from that in Theorem 7 of .
Theorem 2.8. Assume that 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.
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 . Our main results extend the corresponding results in paper , which discussed an SIR epidemic model.
This research was partially supported by the National Natural Science Foundation of China (nos. 11001215, 11101323) and the Scientific Research Program Funded by Shaanxi Provincial Education Department (no. 12JK0859).
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