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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 216913, 13 pages
http://dx.doi.org/10.1155/2013/216913
Research Article

Traveling Wave Solutions in a Reaction-Diffusion Epidemic Model

1College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China
2College of Physics and Electronic Information Engineering, Wenzhou University, Wenzhou 325035, China

Received 4 February 2013; Revised 17 March 2013; Accepted 17 March 2013

Academic Editor: Anke Meyer-Baese

Copyright © 2013 Sheng Wang 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 traveling wave solutions in a reaction-diffusion epidemic model. The existence of the wave solutions is derived through monotone iteration of a pair of classical upper and lower solutions. The traveling wave solutions are shown to be unique and strictly monotonic. Furthermore, we determine the critical minimal wave speed.

1. Introduction

Recently, great attention has been paid to the study of the traveling wave solutions in reaction-diffusion models [117]. In the sense of epidemiology, the traveling wave solutions describe the transition from a disease-free equilibrium to an endemic equilibrium; the existence and nonexistence of nontrivial traveling wave solutions indicate whether or not the disease can spread [11]. The results contribute to predicting the developing tendency of infectious diseases, to determining the key factors of the spread of infectious disease, and to seeking the optimum strategies of preventing and controlling the spread of the infectious diseases [1821].

Some methods have been used to derive the existence of traveling wave solutions in reaction-diffusion models, and the monotone iteration method has been proved to be an effective one. Such a method reduces the existence of traveling wave solutions to that of an ordered pair of upper-lower solutions [6, 7, 9, 10, 14, 15].

In [22], Berezovsky and coworkers introduced a simple epidemic model through the incorporation of variable population, disease-induced mortality, and emigration into the classical model of Kermack and McKendrick [23]. The total population is divided into two groups of susceptible and infectious ; that is to say, . The model describing the relations between the state variables is where the reproduction of susceptible follows a logistic equation with the intrinsic growth rate and the carrying capacity , denotes the contact transmission rate (the infection rate constant), is the natural mortality; denotes the disease-induced mortality, and is the per-capita emigration rate of uninfected.

For model (1), the epidemic threshold, the so-called basic reproduction number , is then computed as . The disease will successfully invade when but will die out if . is usually a threshold whether the disease goes to extinction or goes to an endemic. Large values of may indicate the possibility of a major epidemic [19]. In addition, the basic demographic reproductive number is given by . It can be shown that if the population grows, while implies that the population does not survive [22].

For simplicity, rescaling model (1) by letting , , and leads to the following model: where is defined by the ratio of the average life span of susceptibles to that of infectious.

For details, we refer the reader to [20, 22].

In this paper, we are interested in the existence of traveling wave solutions in the following reaction-diffusion epidemic model [20]: where are all positive constants, is the diffusion coefficient, and .

We are looking for the traveling wave solutions of model (3) with the following form: satisfying the following boundary value conditions: where are the equilibrium points of model (3).

This paper is arranged as follows. In Section 2, we construct a pair of ordered upper-lower solutions of model (3) and establish the uniqueness and strict monotonicity of the traveling wave solutions.

2. Existence and Asymptotic Decay Rates

In this section, we will establish the existence of traveling wave solutions of model (3) by constructing a pair of ordered upper-lower solutions. The definition of the upper solution and the lower solution is standard. We assume that the inequality between two vectors throughout this paper is componentwise.

Setting then model (3) can be written as

For model (3), the equilibria are and , where and for model (7), the equilibria are and , where Obviously, For simplicity, we define the following functions and constants: And we will always assume the following hypotheses throughout the rest of this paper: [H1] [H2]

Then we can obtain the following.

Lemma 1. If [H1] holds, then and are endemic points of model (3) and model (7), respectively.

Lemma 2. For model (7), if [H1] holds, then is unstable, and is stable.

For the sake of convenience, let . For simplicity, we still use the variables , , and instead of , , and , respectively, then model (7) could be rewritten as

Following the definition of quasi-monotonicity [17], we can obtain the following results.

Lemma 3. Model (14) is a quasi-monotone decreasing system in .

Proof. Let
From [17], we can know that the functions and are said to possess a quasi-monotone nonincreasing system, if the sign of and are both nonpositive.
Since Then, Let then obviously, is the unique real root of .
Since , consider , then we can get And hence, has two positive roots.
Since , thus .
According to conditions and , we can get Then, . Hence, .
That is to say, model (14) is a quasi-monotone system in .

Since the traveling wave solution of model (14) has the following form substituting (23) into model (14), we can get the following model:

Obviously, we can know the following.

Remark 4. Model (24) is also a quasi-monotone system in .

Now we establish the existence of traveling wave solutions of model (24) through monotone iteration of a pair of smooth upper and lower solutions. Following [17], we give the definitions of the upper and lower solutions of model (24) as follows, respectively.

Definition 5. A smooth function   () is an upper solution of model (24) if its derivatives and are continuous on , and satisfies with the following boundary value conditions

Definition 6. A smooth function   () is a lower solution of model (24) if its derivatives and are continuous on , and satisfies with the following boundary value conditions

The construction of the smooth upper-lower solution pair is based on the solution of the following KPP equation: where and in the open interval with , , and [15]. First, let us recall the following result.

Lemma 7 (see [1, 15]). Corresponding to every , model (29) has a unique (up to a translation of the origin) monotonically increasing traveling wave solution for . The traveling wave solution has the following asymptotic behaviors.(i) For the wave solution with noncritical speed , one has where and are positive constants.(ii) For the wave with critical speed , one has where the constant is negative, is positive, and .

For constructing the upper solution of the model (24), we start with the following model:

Define , , one can verify that all of the following conditions are satisfied:(i);(ii)for all and ;(iii), .

From Lemma 7, we know that, for each , equation (32) has a unique traveling wave solution (up to a translation of the origin), satisfying the given boundary value conditions (26).

Define then we can get the following result.

Lemma 8. For each , (33) is a smooth upper solution of model (24).

Proof. On the boundary, As for the component, we have As for the component, since , then . And(i)if , then ;(ii)if , then .
Thus we can get: Hence, forms a smooth upper solution for model (24).

For constructing the lower solution of the model (24), we start with the following model:

Define , . One can easily verify that all of the following conditions hold:(i);(ii), for all and ;(iii), .

From Lemma 7, we know that, for each fixed , model (37) has a unique traveling wave solution (up to a translation of the origin), satisfying the given boundary value conditions (28).

Define then we have the following result:

Lemma 9. For each fixed , (38) is a lower solution of model (24).

Proof. On the boundary, As for the component, we have As for the component, we have Thus forms a smooth lower solution for model (24).

Next, we show that, by shifting the upper solution far enough to the left, then the upper-lower solution in Lemmas 8 and 9 are ordered.

Lemma 10. Let , and be the upper solution and the lower solution defined in (33) and (38), then there exists a positive number , such that for all .

Proof. Our proof is only for , and the proof for the case of is similar to it.
First, we derive the asymptotic behaviors of the upper solution and the lower solution at infinities.
According to Lemma 7, when , we can obtain:
And let , , when , we can get where, , , , are all positive constants.
Since for any , is also a solution of model (32). Thus, is an upper solution of model (24). So, according to Lemma 7, when , we can get:
Since , we can choose a large enough number , such that hence, there exists a large number , such that
By using a similar argument as above, there exists a large enough number , such that
Second, we show that
We deal with such two possible cases:
Case  1. If then, the proof is completed.
Case  2. If there exists a point , such that satisfying   or .
In this case, we use the Sliding Domain method [15].
Step  1. we shift to the left by increasing the number until finding a new number such that on the smaller interval .
Step  2. we shift back to the right by decreasing to a smaller number such that one of the branches of the upper solution touches its counterpart of the lower solution at some point in the interval . On the endpoints of the interval , we still have .
Let and , where
For , we get that where , . Since the above model is monotone and the cube is convex, thus we can deduce by Maximum Principle that for . So does not exist and we can decrease further to . It is calculated that the point does not exist either. The proof of this lemma is completed.

To ease the burden of notations, we still use to denote the shifted upper solution as given in Lemma 8. Let

With such constructed ordered upper-lower solution pair, we can get the following.

Theorem 11. For , model (24) has a unique (up to a translation of the origin) traveling wave solution. The traveling wave solution is strictly increasing and has the following asymptotic properties:(i) if , when , when , and if , then while : where, , , , , , , and are all positive constants.(ii) if , when , when , and if , then while , where , , , and , , , are all positive constants.

Proof. From Lemma 3 and Remark 4, we know that model (24) is a quasi-monotone nonincreasing system in , and by using the monotone iteration scheme given in [3, 13], we can obtain the existence of the solution to the first two equations in model (24) for every , which satisfies
According to the above inequality, we can get that, on the boundary, the solution tends to as and as .
To derive the asymptotic decay rate of the traveling wave solutions as , we just let and be the traveling wave solution of model (24) generated form the monotone iteration, since the case of (ii) is similar to it.
We differentiate model (24) with respect to , and note that satisfies where
Now, we study the exponential decay rate of the traveling wave solution as . The asymptotic model of model (62) as is where
The second equation of model (64) has two independent solutions with the following form:
Relating the second equation of model (62) with the second equation of model (64), we can deduce that has the following property as : for some constants and . Thus, we can obtain that where So we obtain that Thus, .
Now, we consider the first equation of model (64). We rewrite it as
One can verify that is not a characteristic of
The above equation has two independent solutions of the following form: Thus, when , has the following property: for some constants ; . Since , thus . So, when , we have the following formula:
Then, we study the exponential decay rate of the traveling wave solution as . The asymptotic model of model (62) as is
By setting , , we rewrite model (76) as a first order model of ordinary differential equation in the four components :
In the case of (i) , we can obtain that the solution of model (77) has the following form: where and are the eigenvectors of the constant matrix with as the corresponding eigenvalues, are arbitrary constants. Since thus , so when