Stability of SIRS Epidemic Model with Stochastic Perturbation and Distributed Delays

Abstract

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.

1. 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 [1–7] and the references are cited therein.

In [8], the authors studied the following classical SIRS epidemiological model with distributed time delay: 𝑆̇(𝑡)=−𝛽𝑆(𝑡)^ℎ₀𝑓(𝑠)𝐼(𝑡−𝑠)d𝑠−𝜇₁𝑆(𝑡)(𝑡),𝐼̇(𝑡)(𝑡)+𝑏+𝑘𝑅=𝛽𝑆^ℎ₀𝑓(𝑠)𝐼(𝑡−𝑠)d𝑠−𝜇₂𝐼(𝑡),𝑅̇(𝑡)(𝑡)−𝜇+𝜆=𝜆𝐼₃+𝑘𝑅(𝑡). (1) Individual are susceptible, then infectious, recovered with temporary immunity, and then susceptible again when the immunity is lost. Then parameter 𝑏>0 is the rate of susceptible individuals recruited into the population (either by birth or immigration) and 𝜇₁>0 is the natural death rate. The parameter 𝜆>0 describes the rate at which the infectious population becomes recovered, and 𝑘>0 denotes the rate at which the recovered population loses immunity. The positive constant 𝛽>0 is the average number of contacts per infective per day. The nonnegative constant ℎ is the time delay. 𝑓(𝑠) 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∫^ℎ₀𝑓(𝑠)d𝑠=1. For detailed biological background, we refer to [8, 9].

In [8], the authors derived the disease-free equilibrium 𝑃⁰(𝑆⁰,0,0) and the basic reproductive number 𝑅₀=(𝛽𝑏/𝜇₁(𝜆+𝜇₂)) of system (1), where 𝑆⁰=(𝑏/𝜇₁), 𝛽 is the transmission coefficient, and 1/(𝜆+𝜇₂) is the average residence time in the infectious individuals class. By straightforward computations, it can be seen that, for 𝑅₀>1, system (1) has a unique endemic equilibrium state 𝑃^∗(𝑆^∗,𝐼^∗,𝑅^∗) in the interior of ℜ³₊, where 𝑆^∗=𝑆⁰𝑅₀=𝜆+𝜇₂𝛽,𝐼^∗=𝑘+𝜇₃𝜇₁𝑘+𝜆+𝜇₂𝑏−𝜇₁𝑆^∗,𝑅^∗𝐼=𝜆^∗𝑘+𝜇₃. (2)

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: 𝑆̇(𝑡)=−𝛽𝑆(𝑡)^ℎ₀𝑓(𝑠)𝐼(𝑡−𝑠)d𝑠−𝜇₁𝑆(𝑡)+𝑏+𝜎₁𝑆(𝑡)−𝑆^∗𝑤̇₁(𝑡),𝐼̇(𝑡)=𝛽𝑆(𝑡)^ℎ₀𝑓(𝑠)𝐼(𝑡−𝑠)d𝑠−𝜇₂+𝜆𝐼(𝑡)+𝜎₂𝐼(𝑡)−𝐼^∗𝑤̇₂(𝑡),𝑅̇(𝑡)=𝜆𝐼(𝑡)−𝜇₃𝑅(𝑡)+𝜎₃𝑅(𝑡)−𝑅^∗𝑤̇₃(𝑡), (3) where 𝑆^∗,𝐼^∗, and𝑅^∗ are the positive points of equilibrium for the corresponding deterministic system (1), constants 𝜎_𝑖(𝑖=1,2) are the intensity of white noise, and 𝑤_𝑖(𝑡)(𝑖=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: 𝑆̇(𝑡)=−𝛽𝑆(𝑡)^ℎ₀𝑓(𝑠)𝐼(𝑡−𝑠)d𝑠−𝜇₁𝑆(𝑡)+𝑏+𝑘𝑅(𝑡)+𝜎₁𝑆(𝑡)−𝑆^∗𝑤̇₁(𝑡),𝐼̇(𝑡)=𝛽𝑆(𝑡)^ℎ₀𝑓(𝑠)𝐼(𝑡−𝑠)d𝑠−𝜇₂+𝜆𝐼(𝑡)+𝜎₂𝐼(𝑡)−𝐼^∗𝑤̇₂(𝑡),𝑅̇(𝑡)=𝜆𝐼(𝑡)−𝜇₃+𝑘𝑅(𝑡)+𝜎₃𝑅(𝑡)−𝑅^∗𝑤̇₃(𝑡), (4) where the constants 𝜎₁,𝜎₂, and 𝜎₃ are the intensity of white noise 𝑤̇₁(𝑡),𝑤̇₂(𝑡),𝑤̇₃(𝑡) and 𝑤₁(𝑡),𝑤₂(𝑡),𝑤₃(𝑡) are standard Wiener process defined on a complete probability space (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃). The detailed definition of Wiener process can be found in [2].

Let 𝑢₁=𝑆−𝑆^∗,𝑢₂=𝐼−𝐼^∗,𝑢₃=𝑅−𝑅^∗, where 𝑁^∗=𝑆^∗+𝐼^∗+𝑅^∗. Then, system (4) is rewritten in the following form: 𝑢̇₁=−𝛽𝐼^∗+𝜇₁𝑢₁−𝛽𝑆^∗ℎ₀𝑓(𝑠)𝑢₂(𝑡−𝑠)d𝑠−𝛽𝑢₁×^ℎ₀𝑓(𝑠)𝑢₂(𝑡−𝑠)d𝑠+𝑘𝑢₃+𝜎₁𝑢₁𝑤̇₁,𝑢̇₂=𝛽𝐼^∗𝑢₁−𝛽𝑆^∗𝑢₂+𝛽𝑆^∗ℎ₀𝑓(𝑠)𝑢₂(𝑡−𝑠)d𝑠+𝛽𝑢₁×^ℎ₀𝑓(𝑠)𝑢₂(𝑡−𝑠)d𝑠+𝜎₂𝑢₂𝑤̇₂,𝑢̇₃=𝜆𝑢₂−𝜇₃+𝑘𝑢₃+𝜎₃𝑢₃𝑤̇₃. (5)

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): 𝑧̇₁=−𝛽𝐼^∗+𝜇₁𝑧₁−𝛽𝑆^∗ℎ₀𝑓(𝑠)𝑧₂(𝑡−𝑠)d𝑠+𝑘𝑧₃+𝜎₁𝑧₁𝑤̇₁,𝑧̇₂=𝛽𝐼^∗𝑧₁−𝛽𝑆^∗𝑧₂+𝛽𝑆^∗ℎ₀𝑓(𝑠)𝑧₂(𝑡−𝑠)d𝑠+𝜎₂𝑧₂𝑤̇₂,𝑧̇₃=𝜆𝑧₂−𝜇₃+𝑘𝑧₃+𝜎₃𝑧₃𝑤̇₃, (6) and the auxiliary system without delays 𝑦̇₁=−𝛽𝐼^∗+𝜇₁𝑦₁+𝑘𝑦₃+𝜎₁𝑦₁𝑤̇₁,𝑦̇₂=𝛽𝐼^∗𝑦₁−𝛽𝑆^∗𝑦₂+𝜎₂𝑦₂𝑤̇₂,𝑦̇₃=𝜆𝑦₂−𝜇₃+𝑘𝑦₃+𝜎₃𝑦₃𝑤̇₃. (7)

2. Preliminaries

Consider the following stochastic differential equation [2]: d𝑥(𝑡)=𝑎𝑡,𝑥_𝑡d𝑡+𝑏𝑡,𝑥_𝑡d𝑤(𝑡),𝑥₀=𝜑∈𝐻, (8) where function 𝑎(𝑡,𝑥_𝑡) defined in [𝑡₀,∞)×𝑅^𝑛and 𝑏(𝑡,𝑥_𝑡) is an 𝑛×𝑚 matrix, 𝑥_𝑡(𝑠)=𝑥(𝑡+𝑠) with 𝑠≤0, 𝑤(𝑡) is𝑚-dimensional Wiener process. We assume that 𝑥=0 is a trivial solution of system (8), that is, 𝑎(𝑡,0)=0, 𝑏(𝑡,0)=0 for all𝑡≥𝑡₀. Denote with 𝐻 the space of 𝑓₀-adapted random variables 𝜑, with 𝜑(𝑠)∈ℜ^𝑛, 𝑠≤0 and ‖𝜑‖₀=sup_𝑠≤0|𝜑(𝑠)|, ‖𝜑‖²₁=sup_𝑠≤0𝔼|𝜑(𝑠)|² (𝔼 is the mathematical expectation). Let 𝑉∶[0,∞)×𝐻→𝑅 be a functional defined for ≥0,𝜑∈𝐻.

Generating operator 𝐿 of (8) is defined [2] by formula 𝐿𝑉𝑡,𝜑=lim_Δ→0𝔼_𝑡,𝜑𝑉𝑡+Δ,𝑦_𝑡+Δ−𝑉𝑡,𝜑Δ, (9) where a scalar functional 𝑉(𝑡,𝜑) is defined by 𝑡≥0, 𝜑∈𝐻, and 𝑥(𝑠) is the solution of (8) by 𝑠≥𝑡 with initial function 𝑥_𝑡=𝜑∈𝐻. Let us describe one class of functionals 𝑉(𝑡,𝜑) for which the operator 𝐿 can be calculated in final form. We reduce the arbitrary functional 𝑉(𝑡,𝜑), 𝑡≥0, 𝜑∈𝐻, to the form 𝑉(𝑡,𝜑)=𝑉(𝑡,𝜑(0),𝜑(𝑠)), 𝑠<0 and define the function 𝑉_𝜑(𝑡,𝑥)=𝑉𝑡,𝜑=𝑉(𝑡,𝑥,𝑥(𝑡+𝑠)),𝑠<0,𝜑=𝑥_𝑡,𝑥=𝜑(0)=𝑥(𝑡). (10) Let 𝐷 be the class of functionals 𝑉(𝑡,𝜑) for which functional𝑉_𝜑(𝑡,𝑥) has two continuous derivations with respect to 𝑥 and one bounded derivative with respect to 𝑡 for almost all 𝑡≥0. For functionals from 𝐷 the generating operator 𝐿 of (8) is defined and is equal to the following: 𝐿𝑉𝑡,𝑥_𝑡=𝜕𝑉_𝜑(𝑡,𝑥)𝜕𝑡+𝑎^′𝑡,𝑥_𝑡𝜕𝑉_𝜑(𝑡,𝑥)+1𝜕𝑥2Tr𝑏^′𝑡,𝑥_𝑡𝜕²𝑉_𝜑(𝑡,𝑥)𝜕𝑥²𝑏𝑡,𝑥_𝑡. (11)

Definition 1. The zero solution of (8) is called mean square stable if for any 𝜖>0, there exists a 𝛿>0 such that 𝔼|𝑥(𝑡)|²<𝜖 for any 𝑡≥0 provided that ‖𝜑‖²₁<𝛿.

Definition 2. The zero solution of (8) is called asymptotically mean square stable if it is mean square stable and lim_𝑡→∞𝔼|𝑥(𝑡)|²=0.

Definition 3. The zero solution of (8) is called stable in probability if for any 𝜖₁>0 and 𝜖₂>0, there exists 𝛿>0 such that solution 𝑥(𝑡)=𝑥(𝑡,𝜑) of (8) satisfies ℙ𝑥𝑡,𝜑>𝜖₁<𝜖₂ (12) for any initial function 𝜑∈𝐻 such that ℙ{|𝜑|≤𝛿}=1. Here ℙ{⋅} is the probability of the event enclosed in braces.

Theorem 4 (see [2]). Let there exist the functional 𝑉(𝑡,𝜑)∈𝐷 such that 𝑐₁𝔼|𝑥(𝑡)|²≤𝔼𝑉𝑡,𝑥_𝑡≤𝑐₂𝑥_𝑡²₁,𝔼𝐿𝑉𝑡,𝑥_𝑡≤−𝑐₃𝔼|𝑥(𝑡)|²,𝑐_𝑖>0. (13) Then zero solution of (8) is asymptotically mean square stable.

Theorem 5 (see [2]). Let there exist the functional 𝑉(𝑡,𝜑)∈𝐷 such that 𝑐₁|𝑥(𝑡)|²≤𝑉𝑡,𝑥_𝑡≤𝑐₂𝑥_𝑡²₀,𝐿𝑉𝑡,𝑥_𝑡≤0,𝑐_𝑖>0 (14) for any function 𝜑∈𝐻 such that 𝑃{|𝜑|≤𝛿}=1, where 𝛿>0 is sufficiently small. Then the zero solution of (8) is stable in probability.

3. 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 𝜎²₁<2𝜇₁,𝜎²₂<2𝜇₂,𝑞1+𝑝²𝜎²₃<2𝜇₃; (15) then zero solution of system (7) is asymptotic mean square stable.

Proof. Consider the following Lyapunov function: 𝑉₁=𝑝𝑦²₁+𝑦²₂+𝑝²𝑦²₃+𝑞𝑦₁+𝑦₂+𝑦₃², (16) where 𝑝, 𝑞 are real positive constants to be chosen in the following. From the Itô formula (11), we have the following: 𝐿𝑉₁=2𝑝𝑦₁𝑦̇₁+2𝑝𝑦₂𝑦̇₂+2𝑝²𝑦₃𝑦̇₃+2𝑞𝑦₁+𝑦₂+𝑦₃×𝑦̇₁+𝑦̇₂+𝑦̇₃+𝑝+𝑞𝜎²₁𝑦²₁+21+𝑞𝜎²₂𝑦²₂+2𝑝²+𝑞𝜎²₃𝑦²₃=2𝑝𝑦₁+𝑞𝑦₁+𝑦₂+𝑦₃×−𝛽𝐼^∗+𝜇₁𝑦₁+𝑘𝑦₃+2𝑦₂+𝑞𝑦₁+𝑦₂+𝑦₃×𝛽𝐼^∗𝑦₁−𝛽𝑆^∗𝑦₂+2𝑝²𝑦₃+𝑞𝑦₁+𝑦₂+𝑦₃×𝜆𝑦₂−𝜇₃𝑦₃−𝑘𝑦₃+𝑝+𝑞𝜎²₁𝑦²₁+1+𝑞𝜎²₂𝑦²₂+𝑝²+𝑞𝜎²₃𝑦²₃+2𝑞𝜎₁𝜎₂𝑦₁𝑦₂+2𝑞𝜎₂𝜎₃𝑦₂𝑦₃≤+2𝑞𝜎₁𝜎₃𝑦₁𝑦₃−2𝑝𝑦²₁𝛽𝐼^∗+𝜇₁−2𝑞𝑦₁+𝑦₂+𝑦₃×𝜇₁𝑦₁+𝛽𝑆^∗𝑦₂−𝜆𝑦₂+𝜇₃𝑦₃+2𝛽𝐼^∗𝑦₁𝑦₂−2𝛽𝑆^∗𝑦²₂−2𝑝²𝜇₃𝑦²₃+𝑝+𝑞𝜎²₁𝑦²₁+1+𝑞𝜎²₂𝑦²₂+𝑝²𝜆𝑦²₂𝑝+𝑝𝑦²₃+𝑝²+𝑞𝜎²₃𝑦²₃+2𝑞𝜎₁𝜎₂𝑦₁𝑦₂+2𝑞𝜎₂𝜎₃𝑦₂𝑦₃+2𝑞𝜎₁𝜎₃𝑦₁𝑦₃=−2𝑝𝛽𝐼^∗−2𝑝𝜇₁−2𝑞𝜇₁+𝑝+𝑞𝜎²₁+𝑦²₂−2𝑞𝛽𝑆^∗−2𝛽𝑆^∗+1+𝑞𝜎²₂+𝑦²₃−2𝑞𝜇₃−2𝑝²𝜇₃+𝑝²𝜆𝑝−2𝑝²𝑘+𝑝²𝜎²₃+2𝑦₁𝑦₂−𝑞𝜇₁−𝑞𝛽𝑆^∗+𝑞𝜆+𝛽𝐼^∗+𝑞𝜎₁𝜎₂+2𝑦₂𝑦₃−𝑞𝜇₃−𝑞𝛽𝑆^∗+𝑞𝜆+𝑞𝜎₂𝜎₃+2𝑦₁𝑦₃−𝑞𝜇₃−𝑞𝜇₁+𝑝𝑘+𝑞𝜎₁𝜎₃+𝑞𝜎²₃𝑦²₃≤𝑦²₁𝑝+𝑞𝜎²₁−2𝜇₁+𝑦²₂1+𝑞𝜎²₂−2𝛽𝑆^∗+𝑝𝜆+2𝑞𝜆+𝑝²𝑦²₃𝑝𝜆−2𝜇₃+𝜎²₃+2𝑦₁𝑦₂𝛽𝐼^∗+𝑞𝜎₁𝜎₂−𝜇₁−𝛽𝑆^∗+𝜆+2𝑦₁𝑦₃𝑞−𝜇₃−𝜇₁+𝜎₁𝜎₃+𝑝𝑘+𝑞𝜎²₃𝑦²₃+2𝑦₂𝑦₃𝑞−𝜇₃−𝛽𝑆^∗+𝜆+𝜎₂𝜎₃. (17) Set 𝑞=min𝑝𝑘𝜇₁+𝜇₃−𝜎₁𝜎₃,𝛽𝐼^∗𝜇₁+𝛽𝑆^∗−𝜆−𝜎₁𝜎₂, (18) and 𝜎²₃<(1+(𝑞/𝑝²))𝜎²₃<2𝜇₃. From the condition (15), it follows 𝜎₁𝜎₂≤𝜎²₁+𝜎²₂2<𝜇₁+𝛽𝑆^∗𝜎−𝜆,₂𝜎₃≤𝜎²₂+𝜎²₃2<𝜇₃+𝛽𝑆^∗𝜎−𝜆,₁𝜎₃≤𝜎²₁+𝜎²₃2<𝜇₁+𝜇₃. (19) By this way, we obtain the following: 𝐿𝑉₁≤−𝑦²₁𝑝+𝑞2𝜇₁−𝜎²₁−𝑦²₂1+𝑞2𝛽𝑆^∗−𝜎²₂−𝑝𝜆−2𝑞𝜆−𝑝²𝑦²₃2𝜇₃−𝑞1+𝑝²𝜎²₃−𝑝𝜆. (20) Choose constant 𝑝 such that 1𝑝=𝜆min2𝜇₃−𝑞1+𝑝²𝜎²₃,1+𝑞2𝛽𝑆^∗−𝜎²₂−2𝑞𝜆. (21) From (15), it follows that there exists 𝑐>0 such that 𝐿𝑉₁≤−𝑐|𝑦|², where 𝑦=(𝑦₁,𝑦₂,𝑦₃). 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 𝜎²₁<2𝜇₁,𝜎²₂<2𝑞𝜇₂,𝑞1+𝑞1+𝑝²𝜎²₃<2𝜇₃, (22) where 𝑞 is defined in (18); then the zero solution of the system (6) is asymptotic mean square stable.

Proof. Consider the following Lyapunov function; 𝑉₂=𝑝𝑧²₁+𝑧²₂+𝑝²𝑧²₃+𝑞𝑧₁+𝑧₂+𝑧₃². (23) Let 𝐿 be the generating operator [1] of the system (6). We compute the following: 𝐿𝑉₂=2𝑝𝑧₁+𝑞𝑧₁+𝑧₂+𝑧₃×−𝛽𝐼^∗+𝜇₁𝑧₁+𝑘𝑧₃−𝛽𝑆^∗ℎ_𝑜𝑓(𝑠)𝑧₂()𝑧𝑡−𝑠d𝑠+2₂+𝑞𝑧₁+𝑧₂+𝑧₃×𝛽𝐼^∗𝑧₁−𝛽𝑆^∗𝑧₂+𝛽𝑆^∗ℎ_𝑜𝑓(𝑠)𝑧₂()𝑝𝑡−𝑠d𝑠+2²𝑧₃+𝑞𝑧₁+𝑧₂+𝑧₃𝜆𝑧₂−𝜇₃𝑧₃−𝑘𝑧₃+𝑝+𝑞𝜎²₁𝑧²₁+1+𝑞𝜎²₂𝑧²₂+𝑝²+𝑞𝜎²₃𝑧²₃+2𝑞𝜎₁𝜎₂𝑧₁𝑧₂+2𝑞𝜎₂𝜎₃𝑧₂𝑧₃+2𝑞𝜎₁𝜎₃𝑧₁𝑧₃≤−𝑧²₁𝑝+𝑞2𝜇₁−𝜎²₁−𝑧²₂1+𝑞2𝛽𝑆^∗−𝜎²₂−𝑝𝜆−2𝑞𝜆−𝑝²𝑧²₃2𝜇₃−𝜎²₃−𝑝𝜆+𝑞𝜎²₃𝑧²₃+2𝑧₁𝑧₂×𝛽𝐼^∗+𝑞𝜎₁𝜎₂−𝜇₁−𝛽𝑆^∗+𝜆+2𝑧₁𝑧₃𝑞−𝜇₃−𝜇₁+𝜎₁𝜎₃+𝑝𝑘+2𝑧₂𝑧₃𝑞−𝜇₃−𝛽𝑆^∗+𝜆+𝜎₂𝜎₃+2𝛽𝑆^∗𝑧₂−𝑝𝑧₁^ℎ₀𝑓(𝑠)𝑧₂(𝑡−𝑠)d𝑠. (24) Using (18), we obtain the following: 𝐿𝑉₂≤−𝑧²₁𝑞2𝜇₁−𝜎²₁−𝑧²₂1+𝑞2𝛽𝑆^∗−𝜎²₂−𝑝𝜆−2𝑞𝜆−𝑝²𝑧²₃2𝜇₃−𝜎²₃−𝑝𝜆+𝑞𝜎²₃𝑧²₃+𝑝𝛽𝑆^∗𝑧²₁+^ℎ₀𝑓(𝑠)𝑧²₂(𝑡−𝑠)d𝑠+𝛽𝑆^∗𝑧²₂+^ℎ₀𝑓(𝑠)𝑧²₂(𝑡−𝑠)d𝑠=−𝑧²₁𝑞2𝜇₁−𝜎²₁−𝑝𝛽𝑆^∗−𝑧²₂1+𝑞2𝛽𝑆^∗−𝜎²₂−𝑝𝜆−2𝑞𝜆−𝛽𝑆^∗−𝑝²𝑧²₃2𝜇₃−𝜎²₃−𝑝𝜆+𝑞𝜎²₃𝑧²₃+𝛽𝑆^∗×1+𝑝_ℎ0𝑓(𝑠)𝑧²₂(𝑡−𝑠)d𝑠, (25) and we define a Lyapunov function as follows: 𝑉₃=𝛽𝑆^∗1+𝑝^ℎ₀𝑓(𝑠)^𝑡_𝑡−𝑠𝑧²₂(𝜏)d𝜏d𝑠. (26) With the application of the multidimensional Itô’s formula [1], we obtain the following: 𝐿𝑉₃=𝛽𝑆^∗1+𝑝𝑧²₂(𝑡)−𝛽𝑆^∗1+𝑝^ℎ₀𝑓(𝑠)𝑧²₂(𝑡−𝑠)d𝑠. (27) Finally, we define a Lyapunov function 𝑉₄=𝑉₂+𝑉₃. (28)
By (22), (25), and (27), we have the following: 𝐿𝑉₄≤−𝑧²₁𝑞2𝜇₁−𝜎²₁−𝑝𝛽𝑆^∗−𝑧²₂2𝑞𝛽𝑆^∗−𝜆−1+𝑞𝜎²₂−𝑝𝜆+𝛽𝑆^∗−𝑝²𝑧²₃2𝜇₃−𝑞1+𝑝²𝜎²₃−𝑝𝜆, (29) where 𝑞𝑝=min2𝜇₁−𝜎²₁𝛽𝑆^∗,2𝑞𝛽𝑆^∗−𝜆−1+𝑞𝜎²₂𝜆+𝛽𝑆^∗,2𝜇₃−1+𝑞/𝑝²𝜎²₃𝜆. (30) Therefore, there exists a 𝑐>0 such that 𝐿𝑉₄≤−𝑐|𝑧|², where 𝑧=(𝑧₁,𝑧₂,𝑧₃). From Theorem 4, we have that the zero solution of system (6) is asymptotic mean square stable. This completes the proof.

Theorem 8. Suppose that the conditions of Theorem 7 hold; then the zero solution of the system (5) is stable in probability.

Proof. Consider the following Lyapunov function: 𝑉=𝑝𝑢²₁+𝑢²₂+𝑝²𝑢²₃+𝑞𝑢₁+𝑢₂+𝑢₃²+𝛽𝑆^∗1+𝑝^ℎ₀𝑓(𝑠)^𝑡_𝑡−𝑠𝑢²₂(𝜏)d𝜏d𝑠. (31) Then, we obtain the following: 𝐿𝑉=2𝑝𝑢₁+𝑞𝑢₁+u₂+𝑢₃×−𝛽𝐼^∗+𝜇₁𝑢₁+𝑘𝑢₃−𝛽𝑆^∗ℎ₀𝑓(𝑠)𝑢₂(𝑡−𝑠)d𝑠−𝛽𝑢₁^ℎ₀𝑓(𝑠)𝑢₂(𝑡−𝑠)d𝑠+2𝑢₂+𝑞𝑢₁+𝑢₂+𝑢₃×𝛽𝐼^∗𝑢₁−𝛽𝑆^∗𝑢₂+𝛽𝑆^∗ℎ₀𝑓(𝑠)𝑢₂(𝑡−𝑠)d𝑠+𝛽𝑢₁^ℎ₀𝑓(𝑠)𝑢₂()𝑝𝑡−𝑠d𝑠+2²𝑢₃+𝑞𝑢₁+𝑢₂+𝑢₃×𝜆𝑢₂−𝜇₃𝑢₃−𝑘𝑢₃+𝑝+𝑞𝜎²₁𝑢²₁+1+𝑞𝜎²₂𝑢²₂+𝑝²+𝑞𝜎²₃𝑢²₃+2𝑞𝜎₁𝜎₂𝑢₁𝑢₂+2𝑞𝜎₂𝜎₃𝑢₂𝑢₃+2𝑞𝜎₁𝜎₃𝑢₁𝑢₃+1+𝑝𝛽𝑆^∗𝑢²₂−1+𝑝𝛽𝑆^∗ℎ₀𝑓(𝑠)𝑢²₂(𝑡−𝑠)d𝑠≤−𝑢²₁𝑞2𝜇₁−𝜎²₁−𝑝𝛽𝑆^∗−𝑢²₂2𝑞𝛽𝑆^∗−𝜆−1+𝑞𝜎²₂−𝑝𝜆+𝛽𝑆^∗−𝑝²𝑢²₃2𝜇₃−𝜎²₃−𝑝𝜆+2𝛽𝑢₁𝑢₂−𝑝𝑢₁×^ℎ₀𝑓(𝑠)𝑢₂(𝑡−𝑠)d𝑠. (32) From Theorem 5, there exists a sufficiently small 𝛿>0 such that ℙ{|𝑢₂(𝑠)|<𝛿}=1. Then 2𝛽𝑢₁𝑢₂−𝑝𝑢₁^ℎ₀𝑓(𝑠)𝑢₂()≤𝑡−𝑠d𝑠2𝛽𝑢₁𝑢₂^ℎ₀𝑓(𝑠)𝑢₂()+𝑡−𝑠d𝑠2𝛽𝑝𝑢²₁^ℎ₀𝑓(𝑠)𝑢₂(𝑡−𝑠)d𝑠≤2𝛽𝑢₁𝑢₂𝛿+2𝛽𝑝𝑢²₁𝛿=𝛽𝛿2𝑢₁𝑢₂+2𝑝𝑢²₁≤𝛽𝛿𝑢²₁1+2𝑞+𝑢²₂. (33) Therefore, 𝐿𝑉≤−𝑢²₁𝑞2𝜇₁−𝜎²₁−𝑝𝛽𝑆^∗−𝛽𝛿1+2𝑞−𝑢²₂2𝑞𝛽𝑆^∗−𝜆−1+𝑞𝜎²₂−𝑝𝜆+𝛽𝑆^∗−𝛽𝛿−𝑝²𝑢²₃2𝜇₃−𝜎²₃−𝑝𝜆. (34) Hence, for sufficiently small 𝛿>0, we obtain 𝐿𝑉≤0. From Theorem 5, we have that the zero solution of system (5) is asymptotic mean square stable. This completes the proof.

4. Conclusion

In this paper, we considered the SIRS epidemic model with stochastic perturbation and distributed delays and some sufficient conditions on the stability of the zero solution are established. Further, we obtain that the white noise 𝜎_𝑖(𝑖=1,2,3) is favorable for the stability of zero solution and the distributed time delays have no impact on the stability of zero solution. The results obtained in this paper indicate that suitable stochastic perturbation can maintain the stability of zero solution.

Acknowledgment

This work was supported by The National Natural Science Foundation of China (11261056).