Abstract and Applied Analysis

Volume 2013 (2013), Article ID 917389, 12 pages

http://dx.doi.org/10.1155/2013/917389

Research Article

Dynamics of a Multigroup SIR Epidemic Model with Nonlinear Incidence and Stochastic Perturbation

Yuguo Lin1,2 and Daqing Jiang¹

¹School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China

²School of Mathematics, Beihua University, Jilin, Jilin 132013, China

Received 18 April 2013; Accepted 20 June 2013

Academic Editor: Andrei Korobeinikov

Copyright © 2013 Yuguo Lin and Daqing Jiang. 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 introduce stochasticity into a multigroup SIR model with nonlinear incidence. We prove that when the intensity of white noise is small, the solution of stochastic system converges weakly to a singular measure (i.e., a distribution) if and there exists an invariant distribution which is ergodic if . This is the same situation as the corresponding deterministic case. When the intensity of white noise is large, white noise controls this system. This means that the disease will extinct exponentially regardless of the magnitude of .

1. Introduction

Considering different contact patterns, distinct number of sexual partners, or different geography, and so forth, individual hosts are often divided into groups in modeling epidemic diseases. Multigroup models have been proposed in the literature to describe the transmission dynamics of infectious diseases in heterogeneous host populations. One of the earliest work on multigroup models is the seminal paper by Lajmanovich and Yorke [1] on a class of SIS multigroup models for the transmission dynamics of Gonorrhea. From then on, much research has been done on various forms of multi-group models, see, for example, [2–6]. It is well known that the global stability of the endemic equilibrium of multigroup models is a very challenging problem. Recently, Guo et al. [7, 8] and Li and Shuai [9] proposed a graph-theoretic approach to the method of global Lyapunov functions and completely solved this problem for some multigroup epidemic models. Subsequently, this approach or ideas of [7–9] were applied to the investigation into the dynamics of several classes of multigroup epidemic models (e.g., [10–13]).

Now we consider a deterministic multigroup SIR (susceptible, infective, and recovered) epidemic model with nonlinear incidence:

Here , , and denote the susceptible, infective, and recovered population at time in the th group, respectively, . Suppose the death rates of , , in the th group are different. The parameters in the model (1) are assumed to be positive and summarized in the following list:

: influx of individuals into the th group; : transmission coefficient between compartments and ; : death rate of compartment in the th group; : death rate of compartment in the th group; : death rate of compartment in the th group; : recovery rate of infectious individuals in the th group.

Considering that the death rates of infective and recovered population are usually no less than the susceptible’s, we assume for all .

Throughout this paper, we consider the following basic assumptions on functions ,  , :

is a -function on , for and ; there exist positive constants such that , ; ; , for any and .

It is easy to check that classes of satisfying include common incidence functions such as [7], [14], [15]. Furthermore, , satisfy .

In Sun [13], the author considered a general multigroup SIR models with nonlinear incidence by using the same method as in [9]. Noting that model (1) is a special case of multigroup SIR models appeared in [13], according to the results of [13], the following results hold for system (1).

Proposition 1. Assume that is irreducible and satisfies .

(1)If , then for system (1), is the unique equilibrium and it is globally asymptotically stable in ;(2)if and is satisfied, then there exists an endemic equilibrium for system (1) and is globally asymptotically stable in ,

where ,  , (the spectral radius of ),     ,  and    .

The aim of this paper is to evaluate the effect of stochastic parameter perturbation on system (1). Considering the existence of environmental noise, we introduce randomness into the model (1) by replacing the parameters , , and by

where , , , are mutual independent standard Brownian motions with ,  ,  , and the intensities of white noises , ,  , respectively. Then the stochastic version corresponding to the deterministic model (1) takes the following form:

System (3) with bilinear incidence has been researched by Ji et al. [16]. They proved that if , the solution of the model is fluctuating around a steady state, which is the disease-free equilibrium of the corresponding deterministic model; while if , there is a stationary distribution, which means the disease will prevail.

In this paper, we will prove that when the intensity of white noise is small, the solution of system (3) converges weakly to a singular measure (i.e., a distribution) if and there exists an invariant distribution which is ergodic if . This is the same situation as in Proposition 1. When the intensity of white noise is large, the disease will extinct exponentially regardless of the magnitude of . Compared to the results of Ji et al. [16] our results can provide a deep insight into the dynamics of corresponding multigroup model.

It is worth mentioning that by now there are many excellent works about stochastic single-group SIR model. Beretta et al. [17] considered a stochastic SIR model with time delays and obtained asymptotic mean square stability conditions for positive equilibrium. In this paper, the authors assumed that stochastic perturbations are of white noise type, which are directly proportional to distances ,  ,   from values of ,  ,  , influence on the ,  ,  , respectively. Tornatore et al. [18] discuss a single-group case of (3). They proved that is a sufficient condition for the asymptotic stability of the disease-free equilibrium. And only by computer simulations they showed that if , the disease-free equilibrium is stable and the disease does not occur; if , the disease-free equilibrium is unstable. Ji et al. [19] considered the same model as in [18]. They deduce the globally asymptotical stability and exponential mean square stability of the disease-free equilibrium under some conditions and investigate the asymptotic behavior of the solution around the endemic equilibrium of the deterministic model.

The rest of this paper is organized as follows. In Section 2, we show there is a unique nonnegative solution of system (3). In Section 3, if combined with small or large enough intensity of white noise, we show that the solution converges weakly to a singular measure. Section 4 focuses on the persistence of the disease. By choosing appropriate Lyapunov function, we show that there is a stationary distribution for system (3) and it is ergodic, if . Finally, for the self-contained, we present an Appendix which contains some results used in the previous sections.

Throughout this paper, let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets). Denote

In general, consider a -dimensional stochastic differential equation

with initial value . denotes -dimensional standard Brownian motion defined on the above probability space. Define the differential operator associated with (5) by

If acts on a function , then

where ,  , . By Itô's formula, if , then

Consider (5), assume and for all . So is a solution of (5), called the trivial solution or equilibrium position.

2. Existence and Uniqueness of the Nonnegative Solution

For a population model, one is interested in whether the solution is nonnegative and global. Hence in this section we show that the solution of system (3) is global and nonnegative. By making the change of variables and Lyapunov analysis method (see [20]), we will show global existence and uniqueness of the positive solution. From now on, we denote the solution of system (3) as .

Theorem 2. If assumption holds and is irreducible, then for any initial value , there is a unique solution of system (3) on , and the solution will remain in with probability 1.

Proof. First consider system

With initial value , , , . Noting that holds, the coefficients of system (9) satisfy the local Lipschitz condition; then there is a unique local solution on , where is the explosion time. Therefore, by Itô’s formula, it is easy to see is the unique positive local solution to system (3) with initial value .

Next, we will prove that this solution is global. The following proof is almost the same as in the proof of Theorem 3.1 in [16]. We do not alter any words except replacing by and therefore we can obtain that . Here we omit the details.

3. Exponential Stability of Infectious Disease

It is clear that is the disease-free equilibrium of system (1) but not (3). For system (1), which has been mentioned in Introduction, is globally stable if , which means the disease will die out after some period of time. Hence, it is interesting to study the disease-free equilibrium for controlling infectious disease. Although there is none of disease-free equilibrium of system (3), in this section, we can still present sufficient conditions for the disease to extinct exponentially.

Theorem 3. Assume hold and is irreducible. If ,  , then the solution of system (3) with initial value has the property . If , the disease will extinct exponentially almost surely. Here

Proof. Let , be the solution of the equation:

By comparison theorem for stochastic equations, we have

It is easy to know that has the following property:

As for the proof of the above two properties, the reader may refer to the proof of Theorems 3.1 and 4.1 in [21] for details.

Since is irreducible, ,   and ,  ,  ,   , then is also nonnegative and irreducible. Hence by Lemma A.1, there is a left eigenvector of corresponding to , which is denoted as and ,  ; that is,

Let , . Define -function as follows:

By Itô’s formula, we compute

where

according to and (12). Let

Then

In view of the definition of and (15), we have

Hence,

which combined with (17) yields

By [22, Lemma 2.6], we get

This together with (14) implies

Thus, the proof of Theorem 3 is completed.

Remark 4. From Theorem 3 we know that if and , are small, the disease will extinct exponentially; that is, as . By using the same arguments as in [20, 21], we get and (), where means the weak convergence and is a distribution in such that and its density is , where is a normal constant and , . That is, the solution of system (3) converges weakly to a singular measure (i.e., a distribution) when and , are small.

Theorem 5. Assume , hold and is irreducible. Then the solution of system (3) with initial value has the property . If , the disease will extinct exponentially almost surely. Here  − .

Proof. Define -function by . By Itô's formula, we compute

where

Then

Taking , in view of (24) and (13), we get

Remark 6. Theorem 5 tells us that the large perturbation forces the infective to expire regardless of the magnitude of .

4. Ergodicity of System (3)

When studying epidemic dynamical systems, we are also interested in when the disease will prevail and persist in a population. For a deterministic model, this is usually solved by showing that the endemic equilibrium is a global attractor or is globally asymptotically stable. But, for the stochastic system (3), there is no endemic equilibrium. In this section, we explore a weak stability. We show that there is a stationary distribution based on the theory of [23] (see Appendix), which reveals the disease will prevail.

From the proof of Theorem 2, we obtain . Let . Then and it is clear that as , where . Hence, by Remark 2 of Theorem 4.1 of Hasminskii ([23], p. 86), we obtain that the solution is a homogeneous Markov process in .

Theorem 7. Assume hold, is irreducible, and . If ,  ,  ,   such that

Then, for any initial value , there is a stationary distribution for system (3) and it has ergodic property, where , ,

,  ,  ,   are defined as in the proof, is the endemic equilibrium of system (1), and ,   denote the cofactor of the th diagonal element of , .

Proof. Since and hold, from Proposition 1 there is an endemic equilibrium of system (1). Then

Define

where ,  ,  ,  ,   are positive constants to be determined later, and ,   according to Lemma A.1. Then is positive definite. In view of (32) and the inequality , by direct calculation, we get   () as follows:

From the calculation of , we directly get

By Lemma A.3, we know

Besides note that for ; then

Successively substituting (39), (40), (37), (38), and (41) into (34) yields

In view of , we obtain

Then

By using inequality for again, it follows that

Successively substituting (40), (37), (45), (41), and (43) into (35), we obtain

Hence

Choose and such that and  −   ,  . Then

Furthermore,

Choose such that  +    and choose    such that  −   ; then

Note that

Then the ellipsoid

lies entirely in . We can take to be a neighborhood of the ellipsoid with , so for ,   is a positive constant), which implies condition in Lemma A.5 is satisfied.

Note that the diffusion matrix of system (3) is

For any bounded domain there is

such that

for all ,  , which implies condition is also satisfied. Therefore, by Lemma A.5, the stochastic system (3) has a stationary distribution and it is ergodic.

Appendix

For the completeness of the paper, in this section, we list some theory used in the upper sections.

(1) Some graph theory (see [9, 24]): a directed graph or a digraph contains a set of vertices and a set of arcs leading from initial vertex to terminal vertex . A digraph is weighted if each arc is assigned a positive weight . Given a weighted digraph with vertices, define the weight matrix whose entry equals the weight of arc if it exists, and otherwise. A weighted digraph is denoted as . A digraph is strongly connected if for any pair of distinct vertices, there exists a directed path from one to the other. A weighted digraph is strongly connected if and only if the weight matrix is irreducible [25].

Lemma A.1. If is nonnegative and irreducible, then the spectral radius of is a simple eigenvalue, and has a positive eigenvector corresponding to .

Remark A.2. The result of this lemma is given in P. 27, [25, Theorem 1.4].

The Laplacian matrix of is defined as

Let denote the cofactor of the th diagonal element of , which has following property.

Lemma A.3 (see [9]). Assume .

(1)If is strongly connected, then for . (2)The following identity holds:

where ,   are arbitrary functions.

Remark A.4. Lemma 2.1 of [8] gives the result (1) of Lemma A.3, and the result (2) is referred to [9, Theorem 2.3].

(2) Some theory about stationary distribution (see [23]): let be a homogeneous Markov process in ( denotes Euclidean -space) described by the stochastic equation

The diffusion matrix is

Assumption B. There exists a bounded domain with regular boundary , having the following properties.

In the domain and some neighbourhood thereof, the smallest eigenvalue of the diffusion matrix is bounded away from zero. If , the mean time at which a path issuing from reaches the set is finite, and for every compact subset .

Lemma A.5 (see [23]). If (B) holds, then the Markov process has a stationary distribution . Let be a function integrable with respect to the measure . Then

for all .

Remark A.6. The proof is given in [23]. Exactly, the existence of stationary distribution with density is referred to Theorem 4.1, P. 119 and Lemma 9.4, P. 138. The weak convergence and the ergodicity are obtained in Theorem 5.1, P. 121 and Theorem 7.1, P. 130.

To validate , it suffices to prove is uniformly elliptical in , where ; that is, there is a positive number such that

(see Chapter 3, P. 103 of [26] and Rayleigh's principle in [27] Chapter 6, P. 349). To verify , it is sufficient to show that there exists some neighborhood and a nonnegative -function such that and for any , is negative (for details refer to [28], P. 1163).

Acknowledgments

The work was supported by the Ministry of Education of China (no. 109051), NSFC of China (no. 10971021 and no. 11201008), and Program for Changjiang Scholars and Innovative Research Team in University.

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