In this section, we consider the stochastic version of the deterministic MSIRS model. Under the assumption that and is irreducible, we know from Section 4 that there exists a unique positive endemic equilibrium in . Furthermore, we assume stochastic perturbations on the , , are of white-noise type, which are directly proportional to deviations , , and from the values of , , , respectively. Thus, system (2.1) results in
where , , and are independent standard Brownian motions and represent the intensities of (), respectively. Obviously, the stochastic system (5.1) has the same equilibrium points as system (2.1). Next, let us now proceed to discuss asymptotic stability of system (5.1). In this paper, unless otherwise specified, let be a complete probability space with a filtration satisfying the usual conditions (i.e. it is increasing and right continuous while contains all P-null sets). Let be the Brownian motions defined on this probability space. If , then the stochastic system (5.1) can be centered at its endemic equilibrium , by the change of variables
we obtain the following system:
It is clear that the stability of equilibrium of system (5.1) is equivalent to the stability of zero solution of system (5.2). Considering the d-dimensional stochastic differential equation [38, 39]
If the assumptions of the existence-and-uniqueness theorem are satisfied, then, for any given initial value , (5.3) has a unique global solution denoted by . For the purpose of stability we assume in this section and for all . So (5.3) admits a solution , which is called the trivial solution or the equilibrium position.
Let κ denote the family of all continuous nondecreasing functions such that and if . For and , denote the family of all nonnegative functions on which are continuously twice differentiable in x and once in t. Define the differential operator L associated with (5.3) by
If L acts on a function , then
The trivial solution of (5.3) is said to be stochastically stable or stable in probability if for every pair of and , there exists a such that
whenever . Otherwise, it is said to be stochastically unstable.
The trivial solution is said to be stochastically asymptotically stable if it is stochastically stable and for every , there exists a such that
The trivial solution is said to be stochastically asymptotically stable in the large if it is stochastically stable and for all
Definition 5.2 A continuous nonnegative function is said to be decrescent if for some :
Before presenting the main theorem we put forward a lemma from .
Lemma 5.3 
If there exists a positive-definite decrescent function such that is negative-definite, then the trivial solution of (5.3) is stochastically asymptotically stable.
From the above lemma, we can obtain the stochastically asymptotically stability of equilibrium as follows.
Theorem 5.4 Assume that is irreducible and . Then, if the following condition is satisfied:
the endemic equilibrium is stochastically asymptotically stable.
Proof It is easy to see that we only need to prove the zero solution of (5.1) is stochastically asymptotically stable. Let , and . We define the Lyapunov function as follows:
where , , are real positive constants to be chosen later. Then it can be described as the quadratic form
is a symmetric positive-definite matrix. So it is obviously that is positive-definite decrescent. For the sake of simplicity, (5.5) may be divided into four functions: , where
Using Itô’s formula, we compute
Similarly, from Itô’s formula, we obtain
Let , so . It follows from and that
Hence inequality (5.7) becomes
Then we compute
We can choose , , such that
and the proofs above show that if the condition (5.4) is satisfied, then , , are positive constants. Let
then . From (5.8), one sees that
where , and is an infinitesimal of higher order of for . Hence is negative-definite in a sufficiently small neighborhood of for . According to Lemma 5.3, we therefore conclude that the zero solution of (5.2) is stochastically asymptotically stable. The proof is complete. □