About this Journal Submit a Manuscript Table of Contents
ISRN Applied Mathematics
Volume 2013 (2013), Article ID 606431, 6 pages
http://dx.doi.org/10.1155/2013/606431
Research Article

Stability of SIRS Epidemic Model with Stochastic Perturbation and Distributed Delays

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

Received 18 April 2013; Accepted 26 May 2013

Academic Editors: K. Djidjeli, J. R. Fernandez, and F. Jauberteau

Copyright © 2013 Ramziya Rifhat et al. 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.

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 [17] and the references are cited therein.

In [8], the authors studied the following classical SIRS epidemiological model with distributed time delay: Individual are susceptible, then infectious, recovered with temporary immunity, and then susceptible again when the immunity is lost. Then parameter is the rate of susceptible individuals recruited into the population (either by birth or immigration) and is the natural death rate. The parameter describes the rate at which the infectious population becomes recovered, and denotes the rate at which the recovered population loses immunity. The positive constant is the average number of contacts per infective per day. The nonnegative constant 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 as density function and satisfies. For detailed biological background, we refer to [8, 9].

In [8], the authors derived the disease-free equilibrium and the basic reproductive number of system (1), where , is the transmission coefficient, and is the average residence time in the infectious individuals class. By straightforward computations, it can be seen that, for , system (1) has a unique endemic equilibrium state in the interior of , where

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 are the intensity of white noise, and 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 , and are the intensity of white noise and are standard Wiener process defined on a complete probability space . The detailed definition of Wiener process can be found in [2].

Let , where . Then, system (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 delays

2. Preliminaries

Consider the following stochastic differential equation [2]: where function defined in and is an matrix, with , is-dimensional Wiener process. We assume that is a trivial solution of system (8), that is, , for all. Denote with the space of -adapted random variables , with , and , ( is the mathematical expectation). Let be a functional defined for ,.

Generating operator of (8) is defined [2] by formula where a scalar functional is defined by , , and is the solution of (8) by with initial function . Let us describe one class of functionals for which the operator can be calculated in final form. We reduce the arbitrary functional , , , to the form , and define the function Let be the class of functionals for which functional has two continuous derivations with respect to and one bounded derivative with respect to for almost all . For functionals from the generating operator of (8) is defined and is equal to the following:

Definition 1. The zero solution of (8) is called mean square stable if for any , there exists a such that for any provided that .

Definition 2. The zero solution of (8) is called asymptotically mean square stable if it is mean square stable and .

Definition 3. The zero solution of (8) is called stable in probability if for any and , there exists such that solution of (8) satisfies for any initial function such that . Here 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.

Theorem 5 (see [2]). Let there exist the functional such that for any function such that , where 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 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: Set and . From the condition (15), it follows By this way, we obtain the following: Choose constant such that From (15), it follows that there exists such that , where . 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: where Therefore, there exists a such that , where . 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: Then, we obtain the following: From Theorem 5, there exists a sufficiently small such that . Then Therefore, Hence, for sufficiently small , we obtain . 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 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).

References

  1. E. Beretta, V. Kolmanovskii, and L. Shaikhet, “Stability of epidemic model with time delays influenced by stochastic perturbations,” Mathematics and Computers in Simulation, vol. 45, no. 3-4, pp. 269–277, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. Ĭ. Ī. Gīhman and A. V. Skorohod, The Theory of Stochastic Processes. I, Springer, Berlin, Germany, 1974. View at MathSciNet
  3. M. Carletti, “On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment,” Mathematical Biosciences, vol. 175, no. 2, pp. 117–131, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. V. B. Kolmanovskiĭ and L. E. Shaĭkhet, “On one method of Lyapunov functional construction for stochastic hereditary systems,” Differentsialniye uravneniya, vol. 29, no. 11, pp. 1909–1920, 1993 (Russian). View at MathSciNet
  5. V. Kolmanovskiĭ and L. Shaĭkhet, “New results in stability theory for stochastic functional-differential equations (SFDEs) and their applications,” in Proceedings of Dynamic Systems and Applications, pp. 167–171, Dynamic Publishers, Atlanta, Ga, USA, 1994. View at MathSciNet
  6. V. Kolmanovskii and L. Shaikhet, “General method of Lyapunov functionals construction for stability investigation of stochastic difference equations,” in Dynamical Systems and Applications, vol. 4 of World Scientific Series in Applicable Analysis, pp. 397–439, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  7. V. B. Kolmanovskiĭ and L. E. Shaĭkhet, “Stability of stochastic systems with aftereffect,” Rossiĭskaya Akademiya Nauk. Avtomatika i Telemekhanika, no. 7, pp. 66–85, 1993. View at MathSciNet
  8. C. V. De-Léon and G. Gómez-Alcaraz, “Globel stability condition of delayed SIRS epidemic model for vector diseases,” Revista Electrónica de Contenido Matemáico, vol. 28, no. 2, pp. 1–18, 2010.
  9. J. Zhen, Z. Ma, and M. Han, “Global stability of an SIRS epidemic model with delays,” Acta Mathematica Scientia, vol. 26, no. 2, pp. 291–306, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet