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We investigate the complex dynamics of cross-diffusion

In epidemiology, epidemic compartmental models, since the pioneer work of Kermack and McKendrick [

In addition, from a biological perspective, the diffusion of individuals may be connected with other things, such as searching for food, escaping high infection risks. In the first case, individuals tend to diffuse in the direction of lower density of a population, where there are richer resources. In the second, individuals may move along the gradient of infectious individuals to avoid higher infection [

In the past decades, it has been shown that the reaction-diffusion system is capable to generate complex spatiotemporal patterns, and the existence of stationary patterns induced by diffusion has attracted the extensive attention of a great number of biologists and mathematicians, and lots of fascinating and important phenomena have been observed [

The main focus of this paper is to investigate how cross-diffusion affects disease’s dynamics through studying the existence of the constant and nonconstant steady states of a cross-diffusion

The rest of this paper is organized as follows. In Section

Assume that the habitat

It is worthy to note that the diffusion coefficients

For model (

Throughout this paper, the positive solution

In this section, we show the existence of unique positive global solution of model (

First, we convert model (

Set

A simple application of a comparison theorem to model (

Applying the comparison principle we get that

The solution of model (

For the sake of simplicity, we omit the proof, and the interested readers may refer to [

In this section, we consider the stability behavior of nonnegative constant steady states to model (

In this subsection, we will discuss the local stability of the constant steady states

Let

For model (

if

if

(a) The linearization of model (

In the following, we prove that there exists

Consequently, the spectrum of

(b) The stability of the semitrivial constant steady state

This subsection is devoted to the global stability of

First, we have the following lemma regarding the persistence property of the susceptible individuals which will play a critical role in the proof of the global stability of

If

For all

Let

Now, we give the result of the global stability of

For model (

if

if

(a) We adopt the Lyapunov function:

(b) We adopt the Lyapunov function:

In this section, we provide some sufficient conditions for the existence and nonexistence of nonconstant positive solution of model (

In order to obtain the desired bounds, we recall the following maximum principle [

Let

Assume that

If

Assume that

If

Let

For convenience, let us denote the constants

Assume that

By applying Lemma

Let

Now, it suffices to verify the lower bounds of

On the contrary, suppose that the conclusion is not true; then there exist sequences

This subsection is devoted to the consideration of the nonexistence for the nonconstant positive solutions of model (

Assume that

Let

In this section, we discuss the global existence of nonconstant positive classical solutions to model (

Unless otherwise specified, in this section, we always require that

Define a compact operator

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

In fact, after calculation, (

Let

To compute

If

Assume that

Suppose that model (

model (

By virtue of Theorem

Set

It is clear that finding the positive solution of model (

Since

Clearly,

On the contrary, by the choice of

In this paper, we investigate the effect of cross-diffusion on the disease’s dynamics through studying the existence and nonexistence positive constant steady states of a spatial

On the other hand, there have been studies of pattern formation in the spatial epidemic model, starting with the pioneering work of Turing [

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

The authors would like to thank the editors and referees for their helpful comments and suggestions. This research was supported by the National Science Foundation of China (11171357, 61272018, and 61373005) and Zhejiang Provincial Natural Science Foundation (R1110261, and LY12A01014).