Abstract and Applied AnalysisVolume 2012 (2012), Article ID 132095, 12 pagesdoi:10.1155/2012/132095

Research Article

Global Stability of Two-Group Epidemic Models with Distributed Delays and Random Perturbation

Xiaoming Fan, Zhigang Wang, and Xuelian Xu

School of Mathematics Science, Harbin Normal University, Harbin 150025, China

Received 1 November 2011; Accepted 31 December 2011

Academic Editor: Norimichi Hirano

Copyright © 2012 Xiaoming Fan 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.

We discuss a two-group SEIR epidemic model with distributed delays, incorporating random fluctuation around the endemic equilibrium. Our research shows that the endemic equilibrium of the model with distributed delays and random perturbation is stochastically asymptotically stable in the large. In addition, a sufficient stability condition is obtained by constructing suitable Lyapunov function.

1. Introduction

For the research of control of disease in populations, significant progress has been made in the theory and application of epidemiology modeling by mathematical research [1–8]. One of the main problems for the theory of differential equations and their applications is connected with stability. Most traditional compartmental models in mathematical epidemiology descend from the classical SIR model of Kermack and McKendrick [9], where the population is divided into the classes of susceptible, infected, and recovered individuals. For some diseases, such as influenza and tuberculosis, on adequate contact with an infectious individual, a susceptible becomes exposed for a while, that is, infected but not yet infectious. Thus it is realistic to introduce a latent compartment; the total population can be partitioned into four compartments: susceptible, latent or exposed, infectious, and recovered, with sizes denoted by S, E, I, and R, respectively. The resulting models are of SEI, SEIR, or SEIRS type, respectively. SEIR model has been widely discussed in the literature. Local and global stability analysis of the disease-free and endemic equilibria has been carried out using different assumptions and contact rates in [6, 7]. Greenhalgh [5] considered SEIR models that incorporate density dependence in the death rate. Korobeinikov [6] considers the global properties for SEIR and SEIS by means of the Lyapunov functions. In fact, there are real benefits to be gained in using stochastic models because real life is full of randomness and stochasticity. Recently, some stochastic epidemic models have been studied by many authors, see [3, 10]. Dalal et al. [4] showed that stochastic models had nonnegative solutions and carried out analysis on the asymptotic stability of models. Tornatore et al. [10] studied the stability of disease-free equilibrium of a stochastic SIR model with or without distributed time delay. On the other hand, taking into account environmental variability, white noise stochastic perturbations around the positive endemic equilibrium of epidemic models was considered in [3, 11]. Beretta et al. proved the stability of epidemic model with stochastic time delays influenced by probability under certain conditions [3]. Such type of stochastic perturbations firstly was proposed in [3, 12] and later was successfully used in many other papers for many other different systems (see, for instance, [13–20]). A more general multigroup epidemic model is proposed to describe the disease spread in a heterogeneous host population with general age structure and varying infectivity by Li et al. [1]. They investigated a class of multigroup epidemic models with distributed delays and established the global dynamics determined by the basic reproduction number . More specifically, they proved that, if , then the disease-free equilibrium is globally asymptotically stable; if , then there exists a unique endemic equilibrium, and it is globally asymptotically stable. However, to the best of the authors’ knowledge, no literature exists regarding SEIR model with random perturbation. Thus, the current study hopes to serve such a need and is inspired by the report of [1]. In this paper, based on the SEIR model of [1], we consider the white noise stochastic perturbations around its endemic equilibrium and use the methods, which is similar to [3]. We construct a class of the Lyapunov functions, as it is useful to study the global properties of stochastic models. By means of it, we prove the SEIR model is stochastically asymptotically stable in the large under certain condition.

The paper is organized as follows. In Section 2 we recall the deterministic SEIR model and its main results by Li et al. [1]. We introduce the model with stochastic perturbations around the endemic equilibrium in Section 3. In Section 4 the global stability of the endemic equilibrium is proved by the method of the Lyapunov functions.

2. Preliminaries

We briefly review the following results obtained by Li et al. [1]. Let , and denote the susceptible, infected but noninfectious, infectious, and recovered populations in the th group, respectively. Let denote the population of infectious individuals in the th group with respect to the age of infection at time , and . Let be a continuous kernel function that represents the infectivity at the age of infection . The disease incidence in the -th group, assuming a bilinear incidence form, can be calculated as , where the sum takes into account cross-infections from all groups and represents the transmission coefficient between compartments and . In the special case , the incidence becomes as in [2]. Therefore, the model in [2] can be generalized to the following system of differential equations Here represents influx of individuals into the th group, , and represent death rates of , and populations in the th group, respectively, represents the rate of becoming infectious after a latent period in the -th group, and represents the recovery rate of infectious individuals in the -th group. All parameter values are assumed to be nonnegative and for all . Note that whose solution is Substituting (2.3) into (2.1), we obtain

Since the variables and do not appear in the first two equations of (2.4), Li et al. consider the following reduced system with distributed time delays and general kernel functions [1]:

Here the kernel function is continuous and . System (2.5) can be interpreted as a multigroup model for an infectious disease whose latent period in hosts has a general probability density function , for the -th group. Let . The next-generation matrix for system (2.5) is Define the basic reproduction number as the spectral radius of , In the special case when is an exponential function, reduces to that for the resulting ODE models. Make the following assumption on the kernel function in (2.5): where is a positive number, . Define the following Banach space of fading memory type: with norm . For , let be such that . Let and such that . We consider solutions of system (2.5) with initial conditions

Standard theory of functional differential equations implies for . We consider system (2.5) in the phase space It can be verified that solutions of (2.5) in initial conditions (2.10) remain nonnegative. In particular, for . The following set is positively invariant for system (2.5): All positive semiorbits in are precompact in and thus have nonempty -limit sets. We have the following results [1].

Lemma 2.1. All positive semi-orbits in have non-empty -limit sets. Let It can be shown that is the interior of .

Lemma 2.2. Assume that is irreducible. (1)If , then is the only equilibrium for system (2.5) in .(2)If , then there exist two equilibria for system (2.5) in : the disease-free equilibrium and a unique endemic equilibrium .

Lemma 2.3. Assume that is irreducible. (1)If , then the disease-free equilibrium of system (2.5) is globally asymptotically stable in . If , then is unstable.(2)If , then the endemic equilibrium of system (2.5) is globally asymptotically stable in .

Biologically, Lemma 2.3 implies that if the basic reproduction number , then the disease always dies out from all groups; if , then the disease always persists in all groups at the unique endemic equilibrium level, irrespective of the initial conditions.

3. Stochastic Model Derivation

In this paper, based on system (2.5), we consider the case of in the following system(3.1): It is easy to see that equilibrium for system (3.1) is given by ,

We assume stochastic perturbations are of white noise type, which are directly proportional to distances from values of , influence the respectively. So system (2.4) results in where are independent standard Brownian motions and represent the intensities of , respectively. Obviously, stochastic system (3.3) has the same equilibrium points as system (3.1). In the next section, we will investigate the stability of the equilibrium of system (3.3). Below we will construct a class of different Lyapunov functions to achieve our proof under certain conditions.

4. Stochastic Stability of the Endemic Equilibrium

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 null sets). Let be the Brownian motions defined on this probability space. If , then the stochastic system (3.3) can be centered at its endemic equilibrium , by the change of variables we obtain

It is easy to see that the stability of the equilibrium of the system (3.3) is equivalent to the stability of zero solution of system (4.2). Before proving the main theorem we put forward a lemma in [21]. Consider the -dimensional stochastic differential equation Assume that the assumptions of the existence-and-uniqueness theorem are fulfilled. Hence, for any given initial value , (4.3) has a unique global solution that is denoted by . Assume furthermore that and for all . So (4.3) has the solution corresponding to the initial value . This solution is called the trivial solution or equilibrium position. Denote by the family of all nonnegative functions defined on such that they are continuously twice differentiable in and once in . Define the differential operator associated with (4.3) by If acts on a function , then

Definition 4.1. (1) The trivial solution of (4.3) 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.(2) The trivial solution is said to be stochastically asymptotically stable if it is stochastically stable, and, moreover, for every , there exists a such that whenever .(3)The trivial solution is said to be stochastically asymptotically stable in the large if it is stochastically asymptotically stable and, moreover, for all ,

Lemma 4.2 (see [21]). If there exists a positive-definite decrescent radially unbounded function such that is negative definite, then the trivial solution of (4.3) is stochastically asymptotically stable in the large.

From the above lemma, we can obtain the stochastically asymptotically stability of equilibrium as follows.

Theorem 4.3. Assume that is irreducible and ; then, if the following condition is satisfied the endemic equilibrium of system (3.3) is stochastically asymptotically stable in the large.

Proof. It is easy to see that we only need to prove the zero solution of (4.2) is stochastically asymptotically stable in the large. Let . We define the Lyapunov function as follows: where are real positive constants to be chosen later. So it is obvious that is positive definite and decrescent.Using Itô’s formula, we compute Using (3.2), we obtain In (4.12), we choose Then Moreover, using the Cauchy inequality to and , we can obtain Substituting (4.15) into (4.14) as well as using , yields where From (4.9) it follows that there exists such that Therefore, there exists such that .Let us suppose that . Then Therefore Hence for sufficiently small , is negative definite in a sufficiently small neighborhood of for . According to Lemma 4.2, we therefore conclude that the zero solution of (4.2) is stochastically asymptotically stable in the large. The proof is complete.

Acknowledgment

The first author was partially supported by the NSF of China no. 10926106.

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