Stability of SIRS Epidemic Model with Stochastic Perturbation and Distributed Delays

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.


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][2][3][4][5][6][7] and the references are cited therein.
In [8], the authors studied the following classical SIRS epidemiological model with distributed time delay: (1) Individual are susceptible, then infectious, recovered with temporary immunity, and then susceptible again when the immunity is lost. Then parameter > 0 is the rate of susceptible individuals recruited into the population (either by birth or immigration) and 1 > 0 is the natural death rate. The parameter > 0 describes the rate at which the infectious population becomes recovered, and > 0 denotes the rate at which the recovered population loses immunity. The positive constant > 0 is the average number of contacts per infective per day. The nonnegative constant ℎ is the time delay. ( ) 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 ∫ ℎ 0 ( )d = 1. For detailed biological background, we refer to [8,9].
In [8], the authors derived the disease-free equilibrium 0 ( 0 , 0, 0) and the basic reproductive number 0 = ( / 1 ( + 2 )) of system (1), where 0 = ( / 1 ), is the transmission coefficient, and 1/( + 2 ) is the average residence time in the infectious individuals class. By straightforward computations, it can be seen that, for 0 > 1, system (1) has a unique endemic equilibrium state * ( * , * , * ) in the interior of R 3 + , where * = 0 0 = + 2 , * = + 3 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: where * , * , and * are the positive points of equilibrium for the corresponding deterministic system (1), constants ( = 1, 2) are the intensity of white noise, and ( ) ( = 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: where the constants 1 , 2 , and 3 are the intensity of white noisė1( ),̇2( ),̇3( ) and 1 ( ), 2 ( ), 3 ( ) are standard Wiener process defined on a complete probability space (Ω, F, {F } ≥0 , ). The detailed definition of Wiener process can be found in [2]. (4) is rewritten in the following form: 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): and the auxiliary system without delayṡ
Definition 2. The zero solution of (8) is called asymptotically mean square stable if it is mean square stable and lim → ∞ E| ( )| 2 = 0.
for any initial function ∈ such that P{| | ≤ } = 1. Here P{⋅} is the probability of the event enclosed in braces.
Theorem 4 (see [2]). Let there exist the functional ( , ) ∈ such that Then zero solution of (8) is asymptotically mean square stable.

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
then zero solution of system (7) is asymptotic mean square stable.
Proof. Consider the following Lyapunov function: where , are real positive constants to be chosen in the following. From the Itô formula (11), we have the following: and 2 3 < (1 + ( / 2 )) 2 3 < 2 3 . From the condition (15), it follows By this way, we obtain the following: Choose constant such that From (15), it follows that there exists > 0 such that 1 ≤ − | | 2 , where = ( 1 , 2 , 3 ). 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
where is defined in (18); then the zero solution of the system (6) is asymptotic mean square stable.
Proof. Consider the following Lyapunov function; Let be the generating operator [1] of the system (6). We compute the following: Using (18), we obtain the following: and we define a Lyapunov function as follows: With the application of the multidimensional Itô's formula [1], we obtain the following: Finally, we define a Lyapunov function By (22), (25), and (27), we have the following: Therefore, there exists a > 0 such that . From Theorem 4, we have that the zero solution of system (6) is asymptotic mean square stable. This completes the proof.
Then, we obtain the following: