Stationary Patterns of a Cross-Diffusion Epidemic Model

Abstract

We investigate the complex dynamics of cross-diffusion 𝑆𝐼 epidemic model. We first give the conditions of the local and global stability of the nonnegative constant steady states, which indicates that the basic reproduction number determines whether there is an endemic outbreak or not. Furthermore, we prove the existence and nonexistence of the positive nonconstant steady states, which guarantees the existence of the stationary patterns.

1. Introduction

In epidemiology, epidemic compartmental models, since the pioneer work of Kermack and McKendrick [1], are widely used for increasing the understanding of infectious disease dynamics and for determining preventive measures to control infection spread qualitatively and quantitatively [2]. More recently, many studies have shown that a spatial epidemic model is an appropriate tool for investigating the fundamental mechanism of complex spatiotemporal epidemic dynamics. In these studies, reaction-diffusion equations have been intensively used to describe spatiotemporal dynamics [3–21]. Spatial epidemiology with diffusion has become a principal scientific discipline aiming at understanding the causes and consequences of spatial heterogeneity in disease transmission [20].

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 [12, 22]. Keeping these in view, cross-diffusion arises, which was proposed first by Kerner [23] and first applied in competitive population system by Shigesada et al. [24]. In particular, Sun et al. [16], by using the standard linear analysis, studied the pattern formation in a cross-diffusion 𝑆𝐼 epidemic model. And in [19], the authors presented Turing pattern selection in a cross-diffusion 𝑆𝐼 epidemic model with zero-flux boundary conditions, gave the conditions of Hopf and Turing bifurcations, and derived the amplitude equations for the excited modes.

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 [25–33]. In particular, in the field of epidemiology, there are many contributions to the existence of steady states in the diffusive epidemic models [34–45]. But in the studies on the steady states of diffusive epidemic models, little attention has been paid to study on the effect of cross-diffusion.

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 𝑆𝐼 epidemic model.

The rest of this paper is organized as follows. In Section 2, we derive a cross-diffusion 𝑆𝐼 epidemic model. In Section 3, we give the global existence and positivity of the solution. In Section 4, we study the local and global stability of the nonnegative steady states of the model. In Section 5, we first give a priori estimates for the positive solutions of the model and then give some results on the existence and nonexistence of positive nonconstant steady states of the model. The paper ends with a brief discussion in Section 6.

2. Model Derivation

Assume that the habitat Ω⊂ℝ^𝑚(𝑚≥1) is a bounded domain with smooth boundary 𝜕Ω (when 𝑚>1), and 𝐧 is the outward unit normal vector on 𝜕Ω. We consider the following cross-diffusion 𝑆𝐼 epidemic model: 𝜕𝑆𝜕𝑡−𝑑_𝑆𝑆Δ𝑆=𝑟𝑆1−𝐾−𝛽𝑆𝐼𝑆+𝐼,𝑥∈Ω,𝑡>0,𝜕𝐼𝜕𝑡−𝐷Δ𝑆−𝑑_𝐼Δ𝐼=𝛽𝑆𝐼𝑆+𝐼−𝜇𝐼,𝑥∈Ω,𝑡>0,𝜕𝑆=𝜕𝐧𝜕𝐼𝑆()𝜕𝐧=0,𝑥∈𝜕Ω,𝑥,0=𝑆₀(𝑥)>0,𝐼(𝑥,0)=𝐼₀(𝑥)≥0,𝑥∈Ω, (1) where 𝑆(𝑥,𝑡) and 𝐼(𝑥,𝑡) denote the density of susceptible and infected individuals at location 𝑥∈Ω and time 𝑡, respectively, 𝑑_𝑆 and 𝑑_𝐼 are the self-diffusion coefficients for the susceptible and infected individuals, and 𝐷 is the cross-diffusion coefficient. 𝑟 stands for the susceptible population intrinsic growth rate, 𝛽 the rate of transmission, 𝜇 the death rate of the infected population 𝐼, and 𝐾 the carrying capacity. The symbol Δ 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.

It is worthy to note that the diffusion coefficients 𝑑_𝑆, 𝑑_𝐼, and 𝐷 are such that 𝑑_𝑆,𝑑_𝐼>0,   𝐷≥0 and 𝑑_𝑆>𝑑_𝐼,  𝐷²<4𝑑_𝑆𝑑_𝐼 which is the parabolic condition.

For model (1), the basic reproduction number is defined as 𝑅₀=𝛽𝜇. (2) The steady states of model (1) satisfy −𝑑_𝑆𝑆Δ𝑆=𝑟𝑆1−𝐾−𝛽𝑆𝐼𝑆+𝐼,𝑥∈Ω,−𝐷Δ𝑆−𝑑_𝐼Δ𝐼=𝛽𝑆𝐼𝑆+𝐼−𝜇𝐼,𝑥∈Ω,𝜕𝑆=𝜕𝐧𝜕𝐼𝜕𝐧=0,𝑥∈𝜕Ω,𝑡>0. (3)

Throughout this paper, the positive solution (𝑆,𝐼) satisfying (1) refers to a classical one with 𝑆>0,  𝐼>0 on Ω. Clearly, model (1) has a semitrivial constant solution (disease-free equilibrium) 𝐸₀=(𝐾,0) and a positive constant solution (endemic equilibrium) 𝐸^∗=𝑆^∗,𝐼^∗=𝐾𝑟𝑟−𝑅₀𝜇,𝐾−1𝑟𝑟−𝑅₀−1𝜇𝑅₀−1 (4) if 1<𝑅₀<1+(𝑟/𝜇).

3. Global Existence and Positivity of the Solution

In this section, we show the existence of unique positive global solution of model (1).

First, we convert model (1) into an abstract first-order system 𝐶(Ω)×𝐶(Ω) of the form 𝑈^′(𝑡)(𝑡)(𝑈(𝑡𝑈(0)=𝐴𝑈+𝐹)),𝑡>0,=𝑈₀∈𝐶Ω×𝐶Ω, (5) where 𝐴𝑈(𝑡)=𝑑_𝑆Δ𝑆,𝐷Δ𝑆+𝑑_𝐼,𝐹(𝑈(𝑡=𝑆Δ𝐼))𝑟𝑆1−𝐾−𝛽𝑆𝐼,𝑆+𝐼𝛽𝑆𝐼.𝑆+𝐼−𝜇𝐼 (6) Since 𝐹 is locally Lipschitz in 𝐶(Ω)×𝐶(Ω), for every initial date 𝑈₀∈𝐶(Ω)×𝐶(Ω), system (5) admits a unique local solution on [0,𝑇_max), where 𝑇_max is the maximal existence time for solution of system (5) [46].

Set 𝑍=𝐼−(𝐷/(𝑑_𝑆−𝑑_𝐼))𝑆. Then model (1) leads to 𝜕𝑆𝜕𝑡−𝑑_𝑆𝑆Δ𝑆=𝑟𝑆1−𝐾−𝛽𝑆𝐼()𝑆+𝐼∶=Λ𝑆,𝑍,𝑥∈Ω,𝑡>0,𝜕𝑍𝜕𝑡−𝑑_𝐼𝐷Δ𝑍=𝑑_𝑆−𝑑_𝐼𝑆𝑟𝑆1−𝐾−𝛽𝑆𝐼+𝑆+𝐼𝛽𝑆𝐼()𝑆+𝐼−𝜇𝐼∶=Υ𝑆,𝑍,𝑥∈Ω,𝑡>0,𝜕𝑆=𝜕𝐧𝜕𝑍𝑆()𝜕𝐧=0,𝑥∈𝜕Ω,𝑡>0,𝑥,0=𝑆₀(𝑥)𝑍()>0,𝑥,0=𝑍₀(𝑥)=𝐼₀(𝑥)−𝐷𝑑_𝑆−𝑑_𝐼𝑆₀(𝑥),𝑥∈Ω. (7)

A simple application of a comparison theorem to model (7) implies (see [47]) that for positive initial date 𝑆₀(𝑥)>0 and 𝑍₀(𝑥)≥0 we have that 𝑆()(𝑥)≥𝐷𝑥,𝑡>0,𝐼𝑑_𝑆−𝑑_𝐼𝑆(𝑥,𝑡),∀𝑥∈Ω,𝑡>0. (8)

Applying the comparison principle we get that 𝑆(𝑥,𝑡)≤max{‖𝑆₀‖_∞,𝐾}. To establish the uniform boundedness of 𝐼(𝑥,𝑡), it is sufficient to show the uniform boundedness of 𝑍(𝑥,𝑡). This task is carried out using a result found in Henry [48], from which it is sufficient to derive a uniform estimate for ‖Υ(𝑆,𝑍)‖_𝑝. Hence, we apply the same method of [49, 50] to study the existence of global solution of model (1). And we have the following result.

Theorem 1. The solution of model (1) is global and uniformly bounded in [0,∞[.

For the sake of simplicity, we omit the proof, and the interested readers may refer to [49, 50] for details.

4. Stability of Nonnegative Constant Steady States

In this section, we consider the stability behavior of nonnegative constant steady states to model (1).

4.1. Local Stability of Nonnegative Steady States

In this subsection, we will discuss the local stability of the constant steady states 𝐸₀=(𝐾,0) and 𝐸^∗=(𝑆^∗,𝐼^∗). For this purpose, we need to introduce some notations.

Let 0=𝜆₀<𝜆₁<𝜆₂<⋯ be the eigenvalues of the operator-Δ on Ω with the homogeneous Neumann boundary conditions. Let 𝐗={(𝑆,𝐼)∈[𝐶²(Ω)]²∣𝜕𝑆/𝜕𝐧=𝜕𝐼/𝜕𝐧=0,𝑥∈𝜕Ω}, {𝜙_𝑖𝑗∣𝑗=1,…,dim𝐸(𝜆_𝑖)} be an orthonormal basis of 𝐸(𝜆_𝑖), and 𝐗_𝑖𝑗={𝐜𝜙_𝑖𝑗∣𝐜∈ℝ²}; then 𝐗=⨁^∞_𝑖=1𝐗_𝑖 where 𝐗_𝑖=⨁^{dim𝐸(𝜆𝑖)}_𝑗=1𝐗_𝑖𝑗.

Theorem 2. For model (1),(a)if 1<𝑅₀<1+(𝑟/𝜇) and 𝑑_𝐼(𝜇𝑅²₀−𝑟𝑅₀−𝜇)+𝐷𝜇(𝑅₀−1)²<𝑑_𝑆𝜇(𝑅₀−1), the positive constant steady state 𝐸^∗=(𝑆^∗,𝐼^∗) is locally asymptotically stable;(b)if 𝑅₀≤1, the semitrivial constant steady state 𝐸₀=(𝐾,0) is locally asymptotically stable.

Proof. (a) The linearization of model (1) at the positive constant steady state 𝐸^∗=(𝑆^∗,𝐼^∗) can be expressed by 𝜕𝜉𝜕𝑡=𝑑_𝑆Δ𝜉+𝜇𝑅²₀−𝑟𝑅₀−𝜇𝑅₀𝜇𝜉−𝑅₀𝜁,𝑥∈Ω,𝑡>0,𝜕𝜁𝜕𝑡=𝐷Δ𝜉+𝑑_𝐼𝜇𝑅Δ𝜁+⁰−1²𝑅₀𝜉−𝜇𝑅₀−1𝑅₀𝜁,𝑥∈Ω,𝑡>0,𝜕𝜉=𝜕𝐧𝜕𝜁𝜕𝐧=0,𝑥∈𝜕Ω. (9) Let 𝑑£=^𝑆Δ+𝜇𝑅²₀−𝑟𝑅₀−𝜇𝑅₀−𝜇𝑅₀𝜇𝑅𝐷Δ+⁰−1²𝑅₀𝑑_𝐼𝜇𝑅Δ−⁰−1𝑅₀. (10) For each 𝑖≥0, 𝐗_𝑖 is invariant under the operator £, and 𝜂 is an eigenvalue of £ if and only if 𝜂 is an eigenvalue of the matrix 𝐴_𝑖=−𝑑_𝑆𝜆_𝑖+𝜇𝑅²₀−𝑟𝑅₀−𝜇𝑅₀−𝜇𝑅₀−𝐷𝜆_𝑖+𝜇𝑅₀−1²𝑅₀−𝑑_𝐼𝜆_𝑖−𝜇𝑅₀−1𝑅₀, (11) for some 𝑖≥0. Thus the stability of the positive constant steady state is reduced to consider the characteristic equation: det𝜂𝐼−𝐴_𝑖=𝜂²−trace𝐴_𝑖𝜂+det𝐴_𝑖∶=𝜑_𝑖𝜂, (12) where trace𝐴_𝑖=−𝜆_𝑖𝑑_𝑆+𝑑_𝐼−𝑟−𝜇𝑅₀𝐴−1<0,det_𝑖=𝑑_𝑆𝑑_𝐼𝜆²_𝑖+1𝑅₀𝑑_𝑆𝜇𝑅₀−1−𝑑_𝐼𝜇𝑅²₀−𝑟𝑅₀𝑅−𝜇−𝐷𝜇₀−1²𝜆_𝑖+𝜇𝑅₀−1𝑟−𝜇𝑅₀−1𝑅₀. (13) It follows from 𝑑_𝐼(𝜇𝑅²₀−𝑟𝑅₀−𝜇)+𝐷𝜇(𝑅₀−1)²<𝑑_𝑆𝜇(𝑅₀−1) that det(𝐴_𝑖)>0. Therefore, the eigenvalues of the matrix 𝐴_𝑖 have negative real parts. It thus follows from the Routh-Hurwitz criterion that, for each 𝑖≥0, the two roots 𝜂_𝑖1 and 𝜂_𝑖2 of 𝜑_𝑖(𝜂)=0 all have negative real parts.
In the following, we prove that there exists 𝜅>0 such that ℜ𝜂_𝑖1,ℜ𝜂_𝑖2≤−𝜅. (14) Let 𝜂=𝜆_𝑖𝜉; then 𝜑_𝑖(𝜂)=𝜆²_𝑖𝜉²−tr(𝐴_𝑖)𝜆_𝑖𝜉+det(𝐴_𝑖)∶=𝜑_𝑖(𝜂). Since 𝜆_𝑖→∞ as 𝑖→∞, it follows that lim_𝑖→∞𝜑_𝑖𝜂𝜆²_𝑖=𝜉²+𝑑_𝑆+𝑑_𝐼𝜉+𝑑_𝑆𝑑_𝐼∶=𝜑(𝜉). (15) Clearly, 𝜑(𝜉)=0 has two negative roots: −𝑑_𝑆,−𝑑_𝐼. Let 𝑑=min{𝑑_𝑆,𝑑_𝐼}. By continuity, we see that there exists 𝑖₀ such that the two roots 𝜉_𝑖1, 𝜉_𝑖2 of 𝜑_𝑖(𝜂)=0 satisfy ℜ{𝜉_𝑖1},ℜ{𝜉_𝑖2}≤−(𝑑/2), ∀𝑖≥𝑖₀. In turn, ℜ{𝜂_𝑖1},ℜ{𝜂_𝑖2}≤−(𝜆_𝑖𝑑/2)≤−(𝑑/2), ∀𝑖≥𝑖₀. Let −𝜅=max_{1≤𝑖≤𝑖0}{ℜ{𝜂_𝑖1},ℜ{𝜂_𝑖2}}. Then 𝜅>0 and (14) holds for 𝜅=min{𝜅,𝑑/2}.
Consequently, the spectrum of £ which consists of eigenvalues lies in {ℜ{𝜂}≤−𝜅}. In the sense of [48], we obtain that the positive constant steady state solution 𝐸^∗=(𝑆^∗,𝐼^∗) of model (1) is uniformly asymptotically stable.
(b) The stability of the semitrivial constant steady state 𝐸₀=(𝐾,0) is reduced to consider the characteristic equation: det𝜂𝐼−𝐴_𝑖=𝜂²−trace𝐴_𝑖𝜂+det𝐴_𝑖, (16) where trace𝐴_𝑖=−𝜆_𝑖𝑑_𝑆+𝑑_𝐼−𝑟+𝜇𝑅₀𝐴−1<0,det_𝑖=𝑑_𝑆𝑑_𝐼𝜆²_𝑖+𝑑_𝑆𝜇1−𝑅₀+𝑑_𝐼𝑟𝜆_𝑖+𝑟𝜇1−𝑅₀>0. (17) The remaining arguments are rather similar as above. The proof is complete.

4.2. Global Stability of the Nonnegative Steady States

This subsection is devoted to the global stability of 𝐸₀=(𝐾,0) and 𝐸^∗=(𝑆^∗,𝐼^∗) for model (1).

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 𝐸^∗=(𝑆^∗,𝐼^∗).

Lemma 3. If 𝑅₀<𝑟/𝜇, 𝑆(𝑥,𝑡) satisfies liminf_𝑡→∞min_Ω𝑆()𝑅𝑥,𝑡≥𝐾1−⁰𝜇𝑟. (18)

Proof. For all 𝑡≥0, 𝑆(𝑥,𝑡) is an upper solution of the following problem: 𝜕𝑧𝜕𝑡−𝑑_𝑆𝛽Δ𝑧=𝑟𝑧1−𝑟−𝑧𝐾,𝑥∈Ω,𝑡>0,𝜕𝑧𝑧()𝜕𝐧=0,𝑥∈𝜕Ω,𝑡>0,𝑥,0=𝑆₀(𝑥)>0,𝑥∈Ω. (19)
Let 𝑆(𝑡) be the unique positive solution of the problem 𝜕𝑤𝛽𝜕𝑡=𝑟𝑤1−𝑟−𝑤𝐾𝑤(0),𝑡>0,=max_Ω𝑆₀(𝑥)>0. (20) Then 𝑆(𝑡) is a lower solution of (20). Since 𝛽<𝑟 we have lim_𝑡→∞𝑆(𝑡)𝛽=𝐾1−𝑟𝑅=𝐾1−⁰𝜇𝑟. (21) It follows by a comparison argument that liminf_𝑡→∞min_Ω𝑆()𝑅𝑥,𝑡≥𝐾1−⁰𝜇𝑟. (22) The proof is complete.

Now, we give the result of the global stability of 𝐸₀ and 𝐸^∗.

Theorem 4. For model (1),(a)if 𝜇<𝑟 and 1<𝑅₀<(𝑟+𝜇)/2𝜇, then positive constant steady state 𝐸^∗=(𝑆^∗,𝐼^∗) of model (1) globally asymptotically stable;(b)if 𝑅₀≤1, the semitrivial constant steady state 𝐸₀=(𝐾,0) of model (1) is globally asymptotically stable.

Proof. (a) We adopt the Lyapunov function: 𝑉(𝑡)=_Ω𝑉₁(𝑆(𝑥,𝑡))+𝜈𝑉₂(𝐼(𝑥,𝑡))𝑑𝑥, (23) where 𝑉₁(𝑆)=∫^𝑆_𝑆∗((𝜉−𝑆^∗)/𝜉)𝑑𝜉, 𝑉₂(𝐼)=𝜈∫^𝐼_𝐼∗((𝜂−𝐼^∗)/𝜂)𝑑𝜂, and 𝜈=𝑆^∗/𝐼^∗=1/(𝑅₀−1). Then 𝑉(𝑡)≥0 and 𝑉(𝑡)=0 if and only if (𝑆,𝐼)=(𝑆^∗,𝐼^∗). Then, 𝑑𝑉=𝑑𝑡_Ω𝑆−𝑆^∗𝑆𝜕𝑆+𝜈𝜕𝑡𝐼−𝐼^∗𝐼𝜕𝐼=𝜕𝑡𝑑𝑥_Ω𝑆−𝑆^∗𝑟−𝑟𝑆𝐾−𝛽𝐼𝑆+𝐼+𝑑_𝑆Δ𝑆𝑆+𝑑𝑥_Ω𝜈𝐼−𝐼^∗𝛽𝑆𝑆+𝐼−𝜇+𝑑_𝐼Δ𝐼𝐼+𝐷Δ𝑆𝐼=𝑑𝑥_Ω𝑃()−𝑆,𝐼𝑑𝑥_Ω𝑑_𝑆𝑆^∗𝑆²|∇𝑆|²+𝑑_𝐼𝜈𝐼^∗𝐼²|∇𝐼|²+𝐷𝜈𝐼^∗𝐼²|∇𝑆||∇𝐼|𝑑𝑥, (24) where 𝑃(𝑆,𝐼)=𝑆−𝑆^∗𝑟−𝑟𝑆𝐾−𝛽𝐼𝑆+𝐼+𝜈𝐼−𝐼^∗𝛽𝑆.𝑆+𝐼−𝜇 (25) It follows from Lemma 3 that, for any 𝜀>0, there exists 𝑡₀>0, such that 𝑆+𝐼≥𝑆≥𝐾(1−(𝜇𝑅₀/𝑟))−𝜀 for all 𝑥∈Ω and 𝑡≥𝑡₀. And by some computational analysis, we have 𝑃(𝑆,𝐼)=𝑆−𝑆^∗2−𝑟𝐾+𝛽𝐼^∗(𝑆^∗+𝐼^∗)−)(𝑆+𝐼𝜈𝑆^∗(𝑆^∗+𝐼^∗)(𝑆+𝐼)𝐼−𝐼^∗2=−𝑆−𝑆^∗2𝑟𝐾−𝜇𝑅₀−1−1𝑆+𝐼𝑅₀𝑅₀−1(𝑆+𝐼)𝐼−𝐼^∗2≤−𝑆−𝑆^∗2𝑟𝐾−𝜇𝑅₀−1𝐾1−𝜇𝑅₀−1/𝑟−𝜀𝑅₀𝑅₀−1(𝑆+𝐼)𝐼−𝐼^∗2. (26) Hence, in view of the conditions of the theorem and the arbitrariness of 𝜀, we have 𝑃(𝑆,𝐼)≤0; that is,𝑑𝑉/𝑑𝑡≤0 for all 𝑥∈Ω and 𝑡≥𝑡₀. So 𝑉(𝑡) decreases monotonically along a solution orbit and 𝐸^∗ is globally asymptotically stable under the assumptions of the theorem.
(b) We adopt the Lyapunov function: 𝑉(𝑡)=_Ω𝐼𝑑𝑥. (27) Then 𝑉(𝑡)≥0 and 𝑉(𝑡)=0 if and only if 𝐼=0. Then, if 𝑅₀≤1, we obtain 𝑑𝑉=𝑑𝑡_Ω𝜕𝐼=𝜕𝑡𝑑𝑥_Ω𝛽𝑆𝐼𝑆+𝐼−𝜇𝐼+𝑑_𝐼≤Δ𝐼+𝐷Δ𝑆𝑑𝑥_Ω𝜇𝐼𝑅₀−1𝑑𝑥≤0. (28) It follows from 𝐼=0 and the second equation of model (1) that 𝑆 is a constant. As a consequence, from the first equation of model (1) and 𝑆≠0, we have 𝑆=𝐾. Hence, 𝐸₀=(𝐾,0) is globally asymptotically stable.

5. 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 (3) by using the Leray-Schauder degree theory [51]. For the purpose, it is necessary to establish a priori positive upper and lower bounds for the positive solution of model (3).

5.1. A Priori Estimates

In order to obtain the desired bounds, we recall the following maximum principle [22] and Harnack inequality [52].

Lemma 5 (maximum principle, see [22]). Let Ω be a bounded Lipschitz domain in ℝ^𝑚 and 𝑔∈𝐶(Ω×ℝ).(a) Assume that 𝑤∈𝐶²(Ω)∩𝐶¹(Ω) and satisfies Δ𝑤(𝑥)+𝑔(𝑥,𝑤(𝑥))≥0inΩ,𝜕𝑤𝜕𝐧≤0on𝜕Ω. (29) If 𝑤(𝑥₀)=max_Ω𝑤(𝑥), then 𝑔(𝑥₀,𝑤(𝑥₀))≥0.(b) Assume that 𝑤∈𝐶²(Ω)∩𝐶¹(Ω) and satisfies Δ𝑤(𝑥)+𝑔(𝑥,𝑤(𝑥))≤0inΩ,𝜕𝑤𝜕𝐧≥0on𝜕Ω. (30) If 𝑤(𝑥₀)=min_Ω𝑤(𝑥), then 𝑔(𝑥₀,𝑤(𝑥₀))≤0.

Lemma 6 (Harnack inequality, see [52]). Let 𝑤∈𝐶²(Ω)∩𝐶¹(Ω) be a positive solution to Δ𝑤(𝑥)+𝑐(𝑥)𝑤(𝑥)=0, where 𝑐∈𝐶(Ω), satisfying the homogeneous Neumann boundary conditions. Then there exists a positive constant 𝐶^∗=𝐶^∗(‖𝑐‖_∞,Ω), such that max_Ω𝑤≤𝐶^∗min_Ω𝑤. (31)

For convenience, let us denote the constants 𝑟,𝐾,𝛽, and 𝜇 collectively by Λ. The positive constants 𝐶,𝐶,𝐶, and so forth will depend only on the domains Ω and Λ.

Theorem 7. Assume that 1<𝑅₀<𝑟/𝜇. Let 𝐷₁ be an arbitrary fixed positive number. Then there exist positive constants 𝐶=𝐶(Λ,Ω,𝐷₁) and 𝐶=𝐶(Λ,Ω,𝐷₁), such that if 𝑑_𝑆,𝑑_𝐼≥𝐷₁, any positive solution (𝑆(𝑥),𝐼(𝑥)) of model (3) satisfies 𝐶≤𝑆(𝑥),𝐼(𝑥)≤𝐶. (32)

Proof. By applying Lemma 5, we have 𝐾(1−(𝜇𝑅₀/𝑟))<𝑆(𝑥)<𝐾 for all 𝑥∈Ω. Let 𝜙=𝐷𝑆+𝑑_𝐼𝐼>0 and set 𝜙(𝑥₀)=max_Ω𝜙. Then, by applying (i) of Lemma 5 again, we have that 𝐼𝑥₀≤𝑅₀−1𝑆𝑥₀<𝑅₀−1𝐾. (33) Thus, max_Ω𝐼≤𝑑_𝐼⁻¹max_Ω𝜙≤𝐾𝑅₀−1+𝐷𝑑_𝐼⁻¹. (34)
Let 𝑐₁(𝑥)=(1/𝑑_𝑆)(𝑟𝑆(1−(𝑆/𝐾))−(𝛽𝑆𝐼/(𝑆+𝐼))). Then, there exists a positive constant 𝐶=𝐶(Λ) such that ‖𝑐₁(𝑥)‖_∞≤𝐶 provided that 𝑑_𝑆>𝐷₁. It follows from Lemma 6 that there exists a positive constant 𝐶^∗=𝐶^∗(Λ,𝐷₁) such that max_Ω𝑆≤𝐶^∗min_Ω𝑆. On the other hand, 𝜙 satisfies −Δ𝜙=𝛽−𝜇𝑆𝐼−𝜇𝐼²(𝑆+𝐼)𝐷𝑆+𝑑_𝐼𝐼𝜙,𝑥∈Ω,𝜕𝜙𝜕𝐧=0,𝑥∈𝜕Ω. (35) Set 𝑐(𝑥)=((𝛽−𝜇)𝑆𝐼−𝜇𝐼²)/(𝑆+𝐼)(𝐷𝑆+𝑑_𝐼𝐼). Then ‖𝑐(𝑥)‖_∞≤𝛽−𝜇𝑑_𝐼+𝜇𝑑_𝐼=𝛽𝑑_𝐼≤𝛽𝐷₁. (36) It follows from Lemma 6 that there exists a positive constant 𝐶^∗=𝐶^∗(Λ,𝐷₁) such that max_Ω𝜙≤𝐶^∗min_Ω𝜙. As a consequence, max_Ω𝐼min_Ω𝐼=max_Ω𝜙−𝐷min_Ω𝑆min_Ω𝜙−𝐷max_Ω𝑆≤𝐶^∗min_Ω𝜙min_Ω𝜑−𝐷max_Ω𝑆≤𝐶. (37)
Now, it suffices to verify the lower bounds of 𝐼(𝑥). We will verify the conclusion by a contradiction argument.
On the contrary, suppose that the conclusion is not true; then there exist sequences {𝑑_𝑆,𝑖}^∞_𝑖=1, {𝑑_𝐼,𝑖}^∞_𝑖=1, and {𝐷_𝑖}^∞_𝑖=1 with 𝑑_𝑆,𝑖≥𝑑,𝑑_𝐼,𝑖>𝐷₁, 𝐷_𝑖≥0, and the positive solution (𝑆_𝑖,𝐼_𝑖) of model (3) corresponding to (𝑑_𝑆,𝑑_𝐼,𝐷)=(𝑑_𝑆,𝑖,𝑑_𝐼,𝑖,𝐷_𝑖), such that min_Ω𝐼_𝑖(𝑥)⟶0as𝑖⟶∞. (38) It follows from Lemma 6 that 𝐼_𝑖(𝑥)⟶0uniformlyonΩas𝑖⟶∞. (39)(𝑆_𝑖,𝐼_𝑖) satisfies −𝑑_𝑆,𝑖Δ𝑆_𝑖=𝑟𝑆_𝑖𝑆1−^𝑖𝐾−𝛽𝑆_𝑖𝐼_𝑖𝑆_𝑖+𝐼_𝑖,𝑥∈Ω,−𝐷_𝑖Δ𝑆_𝑖−𝑑_𝐼,𝑖Δ𝐼_𝑖=𝛽𝑆_𝑖𝐼_𝑖𝑆_𝑖+𝐼_𝑖−𝜇𝐼_𝑖,𝑥∈Ω,𝜕𝑆_𝑖=𝜕𝐧𝜕𝐼_𝑖𝜕𝐧=0,𝑥∈𝜕Ω. (40) Integrating by parts, we obtain that, for 𝑖=1,2,…, _Ω𝑆_𝑖𝑟𝑆1−^𝑖𝐾−𝛽𝐼_𝑖𝑆_𝑖+𝐼_𝑖𝑑𝑥=0,_Ω𝐼_𝑖𝛽𝑆_𝑖𝑆_𝑖+𝐼_𝑖−𝜇𝑑𝑥=0. (41) By the regularity theory for elliptic equations [53], we see that there exist a subsequence of {(𝑆_𝑖,𝐼_𝑖)}^∞_𝑖, which we will still denote by {(𝑆_𝑖,𝐼_𝑖)}^∞_𝑖 and two nonnegative functions 𝑆,𝐼̃∈𝐶²(Ω), such that (𝑆_𝑖,𝐼_𝑖)→(𝑆,𝐼)̃ in [𝐶²(Ω)]² as 𝑖→∞. By (39), we have that 𝐼̃≡0. Letting 𝑖→∞ in (41) we obtain that _Ω𝑆𝑟𝑟−𝐾𝛽𝐼̃𝑆−𝑆+𝐼̃𝑑𝑥=0,_Ω𝐼̃𝛽𝑆𝑆+𝐼̃−𝜇𝑑𝑥=0. (42) Since 𝐼̃=0, the first equation of (42) becomes ∫_Ω𝑆(𝑟−(𝑟/𝐾)𝑆)𝑑𝑥=0. As 𝐾(1−𝛽)<𝑆(𝑥)<𝐾, we derive a contradiction. This completes the proof.

5.2. Nonexistence of Positive Nonconstant Steady States

This subsection is devoted to the consideration of the nonexistence for the nonconstant positive solutions of model (3), and, in the below results, the diffusion coefficients do play a significant role.

Theorem 8. Assume that 𝑅₀>1. Let 𝜀>0 be an arbitrary constant with 4𝑑_𝑆𝜀>𝐷² and 𝑑_𝐼>𝜀. Then model (3) has no positive nonconstant solution provided that (4𝑑_𝑆𝜀−𝐷²)𝜆₁>4(𝑟𝜀+𝛽²) and (𝑑_𝐼−𝜀)𝜆₁>𝜇(𝑅₀−1)+𝜀.

Proof. Let (𝑆(𝑥),𝐼(𝑥)) be any positive solution of model (3) and denote 𝑔=|Ω|⁻¹∫_Ω𝑔𝑑𝑥. Then, multiplying the first equation of model (3) by (𝑆−𝑆), integrating over Ω, we have that 𝑑_𝑆Ω|∇𝑆|²=𝑑𝑥_Ω𝑆−𝑆𝑆𝑟−𝑟𝑆𝐾−𝛽𝐼=𝑆+𝐼𝑑𝑥_Ω𝑆−𝑆²𝑟𝑟−𝐾𝑆+𝑆−𝛽𝐼𝐼𝑆+𝐼)−(𝑆+𝐼𝑑𝑥_Ω𝛽𝑆𝑆𝑆+𝐼(𝑆+𝐼)𝑆−𝑆𝐼−𝐼𝑑𝑥≤𝑟_Ω𝑆−𝑆²+𝛽_Ω𝑆−𝑆𝐼−𝐼𝑑𝑥. (43) In a similar manner, we multiply the second equation in model (3) by (𝐼−𝐼) to have _Ω𝑑_𝐼|∇𝐼|²=+𝐷∇𝑆⋅∇𝐼𝑑𝑥_Ω𝐼−𝐼𝛽𝑆𝐼=𝑆+𝐼−𝜇𝐼𝑃𝑑𝑥_Ω𝐼−𝐼²𝛽−𝜇+𝑆𝑆𝑆+𝐼(𝑆+𝐼)𝑑𝑥+_Ω𝛽𝐼𝐼𝑆+𝐼(𝑆+𝐼)𝑆−𝑆𝐼−𝐼𝑅𝑑𝑥≤𝜇₀−1_Ω𝐼−𝐼²𝑑𝑥+_Ω𝛽𝑆−𝑆𝐼−𝐼𝑑𝑥. (44) It follows from (43), (44), and the 𝜀-Young inequality that _Ω𝑑_𝑆|∇𝑆|²+𝑑_𝐼|∇𝐼|²≤𝑑𝑥_Ω𝛽𝑟+²𝜀𝑆−𝑆²+𝜇𝑅₀−1+𝜀𝐼−𝐼²+𝑑𝑥_Ω𝐷²||4𝜀∇𝑆²+𝜀|∇𝐼|²𝑑𝑥. (45) Thanks to the well-known Poincaré inequality 𝜆_1Ω𝑔−𝑔²𝑑𝑥≤_Ω∇𝑔−𝑔²𝑑𝑥, (46) it follows that _Ω𝑑_𝑆|∇𝑆|²+𝑑_𝐼|∇𝐼|²≤𝑑𝑥_Ω4𝑟𝜀+𝛽²+𝐷²𝜆₁4𝜀𝜆₁|∇𝑆|²+𝜇𝑅𝑑𝑥₀−1+𝜀1+𝜆₁𝜆_1Ω|∇𝐼|²𝑑𝑥. (47) Since 4𝑑_𝑆𝜀𝜆₁>4(𝑟𝜀+𝛽²)+𝐷²𝜆₁ and 𝑑_𝐼𝜆₁>𝜇(𝑅₀−1)+𝜀(1+𝜆₁) from the assumption, one can conclude that 𝑆=𝑆 and 𝐼=𝐼, which asserts our results.

5.3. Existence of Positive Nonconstant Steady States

In this section, we discuss the global existence of nonconstant positive classical solutions to model (3), which guarantees the existence of the stationary patterns [25, 27, 29, 30].

Unless otherwise specified, in this section, we always require that 1<𝑅₀<1+(𝑟/𝜇), which guarantees that model (3) has one positive constant solution 𝐸^∗. From now on, let us denote 𝑊=𝐷𝑆+𝑑_𝐼𝐼,𝑊^∗=𝐷𝑆^∗+𝑑_𝐼𝐼^∗. (48) We also define (),𝐮𝐮=𝑆,𝑊_∗=𝑆^∗,𝑊^∗=𝐾𝑟𝑟−𝜇𝑅₀,𝐾−1𝑟𝑟−𝜇𝑅₀−1𝑑_𝐼𝑅₀−1+𝐷. (49)(50) Let 𝐗={𝐮∈[𝐶²(Ω)]²∣𝜕𝐮/𝜕𝐧=0,𝑥∈𝜕Ω} and 𝐗⁺={𝐮∈𝐗∣𝑆,𝑊>0,𝑥∈Ω}. Then we write model (3) in the form −Δ𝐮=𝐆(𝐮),𝑥∈Ω,𝜕𝐮𝜕𝐧=0,𝑥∈𝜕Ω, (51) where 𝐆(𝐮)=𝑆𝑑_𝑆𝑟𝑟−𝐾𝛽()𝑆−𝑊−𝐷𝑆𝑑_𝐼()−𝐷𝑆+𝑊𝑊−𝐷𝑆𝛽𝑆𝑑_𝐼−𝜇−𝐷𝑆+𝑊𝑑_𝐼. (52)

Define a compact operator ℱ∶𝐗⁺→𝐗⁺ by ℱ(𝐮)∶=(𝐈−Δ)⁻¹{𝐆(𝐮)+𝐮}, (53) where (𝐈−Δ)⁻¹ is the inverse operator of 𝐈−Δ subject to the zero-flux boundary condition. Then 𝐮 is a positive solution of model (51) if and only if 𝐮 satisfies (𝐈−ℱ)𝐮=0,𝑥∈Ω. (54)

To apply the index theory, we investigate the eigenvalue of the problem −𝐈−ℱ_𝐰𝐰^∗Ψ=𝜉Ψ,Ψ≠𝟎, (55) where Ψ=(Ψ₁,Ψ₂)^𝑇 and ℱ_𝐰(𝐰^∗)=(𝐈−Δ)⁻¹(𝐈+𝐀) with 𝐀=𝜇𝑑_𝐼𝑅²₀−𝑟𝑑_𝐼𝑅₀−𝜇𝑑_𝐼+𝜇𝐷𝑑_𝑆𝑑_𝐼𝑅₀−𝜇𝑑_𝑆𝑑_𝐼𝑅₀𝜇𝑑_𝐼𝑅₀−𝑑_𝐼+𝐷𝑅₀−1𝑑_𝐼𝑅₀−𝜇𝑅₀−1𝑑_𝐼𝑅₀. (56) If 0 is not an eigenvalue of (55), by Theorem  2.8.1 in [51], the index of 𝐈−ℱ at 𝐮^∗ is given by index𝐈−ℱ,𝐮^∗=(−1)^𝛾, (57) where 𝛾=∑_𝜆>0𝑛_𝜆 and 𝑛_𝜆 is the algebraic multiplicity of the positive eigenvalue 𝜉 of (55).

In fact, after calculation, (55) can be rewritten as −(𝜉+1)ΔΨ+(𝜉𝐈−𝐀)=0,𝑥∈Ω,𝜕Ψ𝜕𝐧=0,𝑥∈𝜕Ω. (58) Observe that (58) has a nontrivial solution if and only if det(𝜉𝐈+(𝜆_𝑖+1)⁻¹(𝜆_𝑖𝐈−𝐀))=0 for some 𝜉≥0 and 𝑖≥0. That is to say, 𝜉 is an eigenvalue of (55), and so (58), if and only if 𝜉 is an eigenvalue of the matrix (𝜆_𝑖+1)⁻¹(𝜆_𝑖𝐈−𝐀) for any 𝑖≥0. Therefore, 𝐈−ℱ_𝐮(𝐮^∗) is invertible if and only if, for any 𝑖≥0 the matrix 𝜆_𝑖𝐈−𝐀 is invertible.

Let 𝑚(𝜆_𝑖) be the multiplicity of 𝜆_𝑖. For the sake of convenience, we denote 𝐻𝜆_𝑖=det𝜆_𝑖𝐈−𝐀. (59) Then if 𝜆_𝑖𝐈−𝐀 is invertible for any 𝑖≥0, with the same arguments as in [25], we have indexℱ,𝐮^∗=(−1)^𝛾,where𝛾=_{𝜆𝑖≥0,𝐻𝑖<0}𝑚𝜆_𝑖. (60)

To compute index(ℱ,𝐮^∗), we have to consider the sign of 𝐻(𝜆_𝑖). A straightforward computation yields 𝐻𝜆_𝑖=𝜆²_𝑖−trace(𝐀)𝜆_𝑖+det(𝐀), (61) where (𝐀)=trace𝜇𝑑_𝐼𝑅²₀−𝜇𝑑_𝑆+𝑟𝑑_𝐼𝑅₀+𝜇𝑑_𝑆−𝜇𝑑_𝐼+𝜇𝐷𝑑_𝑆𝑑_𝐼𝑅₀,(𝐀)=𝜇𝑅det⁰−1𝑟−𝜇𝑅₀−1𝑑_𝐼𝑅₀. (62)

If trace(𝐀)²−4det(𝐀)≥0, then 𝐻(𝜆)=0 has two positive solutions 𝜆^± given by 𝜆^±=12trace(𝐀)±trace(𝐀)²−4det(𝐀). (63)

Theorem 9. Assume that 𝑡𝑟𝑎𝑐𝑒(𝐀)²−4det(𝐀)≥0. Then if 𝜆⁻∈(𝜆_𝑖,𝜆_𝑖+1) and 𝜆⁺∈(𝜆_𝑗,𝜆_𝑗+1) for some 0≤𝑖<𝑗 and ∑^𝑗_𝑘=𝑖+1𝑚(𝜆_𝑘) is odd, then model (3) has at least one nonconstant solution.

Proof. Suppose that model (3) has no nonconstant positive solution. By Theorem 8, we can fix 𝐷>𝑑_𝑆 such that, (i)model (3) with diffusion coefficients 𝐷,𝑑_𝐼, and 𝐷 has no nonconstant solutions;(ii)𝐻(𝜆_𝑖)>0 for all 𝜆_𝑖≥0.
By virtue of Theorem 7, there exists a positive constant 𝐶=𝐶(Λ,Ω) such that, for 𝑑_𝑆≥𝑑_𝑆, any solution (𝑆(𝑥),𝐼(𝑥)) of model (3) with diffusion coefficients 𝑑_𝑆,𝑑_𝐼, and 𝐷 satisfies 𝐶⁻¹<𝑆,  𝐼<𝐶.
Set ℳ=(𝑆,𝑊)∈𝐶Ω×𝐶Ω∶𝐶⁻¹<𝑆,𝑊<𝐶,𝑥∈Ω, (64) and define Φ∶ℳ×[0,1]⟶𝐶Ω×𝐶Ω (65) by Φ(𝐮,𝜃)=(𝐈−Δ)⁻¹{𝐆(𝐮,𝜃)+𝐮}, (66) where 𝐆()=𝐮,𝜃𝜃𝑑_𝑆+(1−𝜃)𝐷⁻¹𝑆𝑟𝑟−𝐾𝛽()𝑆−𝑊−𝐷𝑆𝑑_𝐼()−𝐷𝑆+𝑊𝑊−𝐷𝑆𝛽𝑆𝑑_𝐼−𝜇−𝐷𝑆+𝑊𝑑_𝐼. (67)
It is clear that finding the positive solution of model (51) becomes equivalent to finding the fixed point of Φ(𝐮,1) in ℳ. Φ(𝐮,𝜃) has no fixed points in 𝜕ℳ for all 0≤𝜃≤1.
Since Φ(𝐮,𝑡) is compact, the Leray-Schauder topological degree deg(𝐈−Φ(𝐮,𝜃),ℳ,0) is well defined. From the invariance of Leray-Schauder degree at the homotopy, we deduce deg(𝐈−Φ(𝐮,1),ℳ,0)=deg(𝐈−Φ(𝐮,0),ℳ,0). (68)
Clearly, 𝐈−Φ(𝐮,1)=𝐈−ℱ. Thus, if model (3) has no other solutions except the constant one 𝐮^∗, we have deg(𝐈−Φ(𝐮,1),ℳ,0)=index𝐈−ℱ,𝐮^∗=(−1)^{∑𝑗𝑘=𝑖+1𝑚(𝜆𝑘)}=−1. (69)
On the contrary, by the choice of 𝐷, we have that 𝐮^∗ is the only solution of Φ(𝐮,0)=0 and therefore deg(𝐈−Φ(𝐮,0),ℳ,0)=index𝐈−ℱ,𝐮^∗=(−1)⁰=1. (70) From (68) to (70), we get a contradiction. Therefore, there exists a nonconstant solution of model (3). The proof is completed.

6. Concluding Remarks

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 𝑆𝐼 epidemic model. The values of this study lie in twofolds. First, we show the local and global stability of the nonnegative steady states, which indicates that the disease reproduction number 𝑅₀ determines whether there is an endemic outbreak or not: the disease free dynamics occurs if 𝑅₀≤1 while the unique endemic steady state is globally stable if 𝜇<𝑟 and 1<𝑅₀<(𝑟+𝜇)/2𝜇. Second, we show that even though the unique positive constant steady state (endemic state) is uniformly asymptotically stable for (1), nonconstant positive steady states can exist due to the emergence of cross-diffusion, which demonstrates that stationary patterns can be found as a result of cross-diffusion.

On the other hand, there have been studies of pattern formation in the spatial epidemic model, starting with the pioneering work of Turing [54]. Turing’s revolutionary idea was that passive diffusion could interact with the chemical reaction in such a way that even if the reaction by itself has no symmetry-breaking capabilities, diffusion can destabilize the symmetry so that the system with diffusion can have them. Spatial epidemiology with self-diffusion has become a principal scientific discipline aiming at understanding the causes and consequences of spatial heterogeneity in disease transmission. And in the present paper, we prove the existence and nonexistence of the positive nonconstant steady states, which guarantees the existence of the stationary Turing patterns. The numerical results about the Turing patterns for model (1) can be found in [19].

Conflict of Interests

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

Acknowledgments

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).