Abstract

This paper is concerned with the existence of traveling waves for a delayed SIRS epidemic diffusion model with saturation incidence rate. By using the cross-iteration method and Schauder’s fixed point theorem, we reduce the existence of traveling waves to the existence of a pair of upper-lower solutions. By careful analyzsis, we derive the existence of traveling waves connecting the disease-free steady state and the endemic steady state through the establishment of the suitable upper-lower solutions.

1. Introduction

Since Kermack and Mckendrick [1] proposed an ordinary differential system to study epidemiology in 1927, various models have been used to describe various kinds of epidemics, and the dynamics of these systems have been investigated. Let represent the number of individuals who are susceptible to the disease, let represent the number of infected individuals who are infectious and are able to spread the disease by contact with susceptible individuals, and let represent the number of individuals who have been infected and then removed from the possibility of being infected again. Mena-Lorca and Hethcote [2] considered the following SIRS epidemic model: where the parameters , , , , , are positive constants and is the recruitment rate of the population, is the natural death rate of the population, is the transmission rate, is the rate at which recovered individuals lose immunity and return to the susceptible class, is the recovery rate of the infective individuals, and is the death rate of the infective individuals due to disease. The SIRS model assumes that the recovered individuals have only temporary immunity, which is reasonable in the study of some communicable diseases.

However, due to the diseases latency or immunity, the presence of time delays in such models makes them more realistic. On the other hand, the environment in which an individual lives is actually heterogeneous and the mobility of people within a country or even worldwide is large; introducing the spatial diffusion in these epidemic models is unavoidable. In recent years, the dynamics of the delayed epidemic diffusion model have been widely studied by many researchers (see, e.g., [3–6]), and these studies are mainly focused on the global attractivity, basic reproductive number, and especially the epidemic waves. For example, Gan et al. [7] considered the following delayed SIRS epidemic model with spatial diffusion: and obtained the existence of traveling wave solutions.

In systems (1) and (2), the terms and are called incidence rate and both of them are bilinear. However, as the number of susceptible individuals is large, it is reasonable to consider the saturation incidence rate (see [8]) instead of the bilinear incidence rate. Motivated by the works mentioned above, we will consider the following delayed SIRS epidemic diffusion model with nonlinear saturation rate and study its traveling wave solutions. The main tool is the upper-lower solutions coupled with cross-iteration method established by Ma [9]. We point out that the nonlinear terms in (3) do not satisfy the common various (exponential) monotonicity conditions such as in [10–12]; thus the main difficulty is the construction and verification of the upper-lower solutions.

2. Preliminaries and Lemmas

Throughout this paper, we employ the usual notations for the standard ordering in . That is, for and , we denote if , ; if but ; and if but , . Let denote the Euclidean norm in .

First, we assume that for (3). Denoting , then (3) reduces to the following system:

By making changes of variables , , and dropping the tildes, (4) is converted to the following system:

Consider the equilibrium equation of system (5): Obviously, system (5) often has a trivial equilibrium . From the first and the third equation of (6), we know that , . Substituting the expressions into the second equation of (6), if , we get a positive equilibrium of system (5), where

By calculating, we can obtain that , which is important in the following text. In fact,

Now, we study the existence of traveling wave solutions for system (5) connecting and .

Substituting , , into (5), and denoting still by , we derive the following wave profile system from (5):

Note that imply . Moreover, we have We can select suitable , , such that , , which satisfy

Denote , where .

Denote :

For , by a careful calculation, we have where , , .

For the positive constants , , , we define by

Then operators , , have the following properties.

Lemma 1. For , , , one has(i)(ii)

Proof. According to the definitions of and , we have Let , ; we obtain the properties for and .
For (ii), we have Note that , and is nondecreasing; we have that the first term of the last formula is nonnegative, and the second term is bigger than . Let ; we have . Since then, for any positive constant , we have , .

Remark 2. For , we can further conclude that from Lemma 1(ii).

According to the definition of , system (9) can be written as

Define and operator by It is easy to see that satisfy system (20); thus the fixed point of operator satisfies (9), which is a traveling wave solution of system (5). Therefore, we will use Schauder's fixed point theorem to find the fixed point of , where the continuity of is required. For this purpose, let ; we define a norm for by

Define Then it is obvious that is a Banach space. We also need the following definition of upper and lower solutions for system (9).

Definition 3. A pair of continuous functions and are called an upper solution and a lower solution of (9), respectively, if there exist finite points such that , are twice differentiable and bounded on , , and satisfy for , respectively.
We assume that a pair of upper-lower solutions and are given such that(P1)(P2)(P3)
Define the set Then by the property of , we have the property of .