Abstract and Applied Analysis

Volume 2013 (2013), Article ID 921401, 7 pages

http://dx.doi.org/10.1155/2013/921401

Research Article

The Existence of Positive Nonconstant Steady States in a Reaction: Diffusion Epidemic Model

Yuan Yuan,¹ Hailing Wang,² and Weiming Wang¹

¹College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China

²Department of Mathematics, Hubei Minzu University, Enshi, Hubei 445000, China

Received 19 November 2013; Accepted 1 December 2013

Academic Editor: Kaifa Wang

Copyright © 2013 Yuan Yuan 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 investigate the disease’s dynamics of a reaction-diffusion epidemic model. We first give a priori estimates of upper and lower bounds for positive solutions to model and then give the conditions of the existence and nonexistence of the positive nonconstant steady states, which guarantees the existence of the stationary patterns.

1. Introduction

Infectious diseases are the second leading cause of death worldwide, after heart disease, and are responsible for more deaths annually than cancer [1]. Since the pioneer work of Kermark and McKendrick [2], mathematical models have been contributing to improve our understanding of infectious disease dynamics and help us develop preventive measures to control infection spread qualitatively and quantitatively.

Many studies indicate that spatial epidemiology with self-diffusion has become a principal scientific discipline aiming at understanding the causes and consequences of spatial heterogeneity in disease transmission [3]. In these studies, reaction-diffusion equations have been intensively used to describe spatiotemporal dynamics. In particular, the spatial spread of infections has been studied by analyzing traveling wave solutions and calculating spread rates [4–10].

Besides, there has been some research on pattern formation in the spatial epidemic model, starting with Turing’s seminal paper [11]. Turing’s revolutionary idea was that the passive diffusion could interact with chemical reaction in such a way that even if the reaction by itself has no symmetry-breaking capabilities, diffusion can destabilize the symmetric solutions with the result that the system with diffusion has them [12]. In these studies [3, 13–20], via standard linear analysis, the authors obtained the conditions of Turing instability, and, via numerical simulation, they showed the pattern formation induced by self-diffusion or cross-diffusion and found that model dynamics exhibits a diffusion controlled formation growth to stripes, spots, and coexistence or chaos pattern replication.

Recently, the researchers are interested in research on the stationary patterns due to the existence and nonexistence nonconstant solutions of the reaction-diffusion model [21–29]. But the research on the existence and nonexistence nonconstant solutions of reaction-diffusion epidemic model, seems rare [3].

In this paper, we will focus on the disease’s dynamics through studying the existence of the constant and nonconstant steady states of a simple reaction-diffusion epidemic model.

The rest of this paper is organized as follows. In Section 2, we derive a reaction-diffusion epidemic model. In Section 3, we give a priori estimates of upper and lower bounds for positive solutions to model. In Section 4, we give the main results on the existence and nonexistence of positive nonconstant steady states of the model. The paper ends with a brief discussion in Section 5.

2. Basic Model

In [30], Berezovsky and coworkers introduced a simple epidemic model through the incorporation of variable population, disease induced mortality, and emigration into the classic model of Kermark and McKendrick [2]. The total population () is divided into two groups susceptible () and infectious (); that is, . The model describing the relations between the state variables is

where the birth process incorporates density dependent effects via a logistic equation with the intrinsic growth rate and the carrying capacity ; , represent population densities of susceptible and infected population, respectively; denotes the transmission rate (the infection rate constant); is the natural mortality; denotes the disease-induced mortality; is the per-capita emigration rate of noninfective.

For model (1), the epidemic threshold of basic reproduction number is then computed as

The basic demographic reproductive number is given by

For simplicity, rescalling the model (1) by letting , , and leads to the following model:

where defined by the ratio of the average life-span of susceptibles to that of infections and .

See [30] for more details.

Assume that the habitat () is a bounded domain with smooth boundary (when ), and is the outward unit normal vector on . We consider the following reaction-diffusion epidemic model:

where the nonnegative constants and are the diffusion coefficients of and , respectively. The symbol is the Laplacian operator. The homogeneous Neumann boundary condition implies that the above model is self-contained and there is no infection across the boundary.

The corresponding kinetic model (5) with has been investigated by Wang et al. [20].

In this paper, we concentrated on the steady states of model (5) which satisfy

Throughout this paper, the positive solution satisfying model (6) refers to a classical one with , on . Clearly, model (6) has a unique positive constant solution (endemic equilibrium) if and , where

3. A Priori Estimates for Positive Solutions to Model (6)

The main purpose of this section is to give a priori upper and lower positive bounds for positive solution of model (6). To this aim, we first cite two known results. The first is due to Lin et al. [31] and the second to Lou and Ni [32]. In the following, let us denote the constants , , and collectively by . The positive constants , , , , and so forth will depend only on the domains and .

Lemma 1 (Harnack inequality [31]). Let be a positive solution to , where , satisfying the homogeneous Neumann boundary conditions. Then there exists a positive constant , such that

Lemma 2 (maximum principle [32]). Let be a bounded Lipschitz domain in and .

(a)Assume that and satisfies

 If , then .(b)Assume that and satisfies

 If , then .

Theorem 3. If , then the positive solution of model (6) satisfies

Proof. Assume that is a positive solution of model (6). We set

By applying Lemma 2, we have

and . This clearly gives

Theorem 4. Assume that and . Let and be fixed positive constants. Then there exists a positive constant such that, if , every positive solution of model (6) satisfies

Proof. Let

In view of Theorem 3, there exists a positive constant such that , provided that . As and satisfy

It follows from Lemma 1 that there exists a positive constant such that

for .

Now, on the contrary, suppose that (15) is not true, then there exist sequences , with and the positive solution of model (6) corresponding to , such that

It follows from Lemma 1 that

satisfies

Integrating by parts, we obtain that, for ,

By the regularity theory for elliptic equations [33], we see that there exist a subsequence of , which we will still denote by , and two nonnegative functions , such that in as . By (20), we have that or .

Letting in (22) we obtain that

Case  1   ( , or , ). Since satisfies the second inequality of (18), on . Therefore, on as . Hence, for sufficiently large which contradicts the second integral identity of (22).

Case  2   ( , ). As above, on . It follows from the first integral identity of (23) that

This fact combines with yielding to , which implies that uniformly on as , since uniformly on . As , this contradicts the second integral identity of (23) and the fact that . This completes the proof.

4. Existence and Nonexistence of Positive Nonconstant Steady States

In this section, we provide some sufficient conditions for the existence and nonexistence of nonconstant positive solution of model (6) by using the Leray-Schauder degree theory [34]. From now on, we denote by

the eigenvalues of the operator on with the zero-flux boundary conditions.

4.1. Nonexistence for Positive Nonconstant Steady States to Model (6)

This section is devoted to the consideration of the nonexistence for the nonconstant positive solutions of model (6), and, in the following results, the diffusion coefficients do play a significant role.

Theorem 5. Assume that . Let be a fixed positive constant with . Then there exists a positive constant such that model (6) has no positive nonconstant solution provided that and .

Proof. Let be any positive solution of model (6) and denote . Then, multiplying the first equation of model (6) by , integrating over , by virtue of Theorem 3, we have that

where depend only on . In a similar manner, we multiply the second equation in model (6) by to have

It follows from (26), (27) and the -Young inequality that

where . It follows from the well-known Poincaré inequality that

Since from the assumption, we can find a sufficiently small such that . Finally, by taking one can conclude that and , which asserts our results.

4.2. Existence for Positive Nonconstant Steady States to Model (6)

In this section, we discuss the global existence of nonconstant positive classical solutions to model (6), which guarantees the existence of the stationary patterns [21, 24, 26, 27].

Unless otherwise specified, in this section, we always require that and , which guarantees that model (6) has one positive constant steady state . From now on, let us denote

Let and . Then we write model (6) in the form

where

Define a compact operator by

where is the inverse operator of subject to the zero-flux boundary condition. Then is a positive solution of model (31) if and only if satisfies

To apply the index theory, we investigate the eigenvalue of the problem

where and with

is an eigenvalue of (35) if and only if is an eigenvalue of the matrix for any . Therefore, is invertible if and only if, for any , the matrix

is invertible. A straightforward computation yields

where . For the sake of convenience, we denote

Then .

If , then has two real roots given by

Set , , and the multiplicity of .

To compute , we can assert the following conclusion by Pang and Wang [22].

Lemma 6 (see [22]). Suppose for all . Then

where

In particular, if for all , then .

From Lemma 6, we see that to calculate the index of , the key step is to determine the range of for which .

Theorem 7. Assume that . If , for some , and is odd, then there exists a positive constant such that model (6) has at least one nonconstant solution if .

Proof. Since , equivalently, , it follows that if is large enough, then and . Furthermore,

Since for some , there exists such that

By Theorem 5, we know that there exists such that model (6) with diffusion coefficients and has no nonconstant solutions. Moreover, we can choose so large that . It follows that there exists such that

We shall prove that, for any , model (6) has at least one nonconstant positive solution. On the contrary, suppose that this assertion is not true for some . In the following, we will derive a contradiction by using a homotopy argument.

By virtue of Theorems 3 and 4, there exists a positive constant such that the positive solution of model (6) satisfies .

Set

and define

by

where

It is clear that finding the positive solution of model (31) is equivalent to finding the fixed point of in . Further, by virtue of the definition of , we have that has no fixed point in for all .

Since is compact, the Leray-Schauder topological degree is well defined. From the invariance of Leray-Schauder degree at the homotopy, we deduce

In view of and , we have . Clearly, . Thus, if model (6) has no other solutions except the constant one , then Lemma 6 shows that

On the contrary, by the choice of and , we have that and is the only fixed point of . It therefore follows from Lemma 6 that

From (50)–(52), we get a contradiction. Therefore, there exists a nonconstant solution of model (6). The proof is completed.

5. Discussion

In this paper, we investigate the disease’s dynamics through studying the existence and nonexistence positive constant steady states of a reaction-diffusion epidemic model. We give a priori estimates for positive solutions to model and show that the nonconstant positive steady states exist due to the emergence of diffusion, which demonstrates that stationary patterns can be found as a result of diffusion. The numerical results about the stationary patterns for model (5) can be found in [20].

On the other hand, there are plenty of papers which focus on the pattern formation of reaction-diffusion population models via standard linear analysis method and numerical simulations. But there is little literature analytically concerning the existence of a stationary patterns via theory and methods of partial differential equations infrequently. The methods and results in the present paper may enrich the research of pattern formation in the spatial epidemic model.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the anonymous referee for very helpful suggestions and comments which led to improvement of our original paper. This research was supported by the National Science Foundation of China (61373005) and Zhejiang Provincial Natural Science Foundation (LY12A01014).

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