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

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 (𝑁) is divided into two groups susceptible (𝑆) and infectious (𝐼); that is, 𝑁=𝑆+𝐼. The model describing the relations between the state variables is 𝑑𝑆𝑁𝑑𝑡=𝑟𝑁1−𝐾−𝛽𝑆𝐼𝑁−𝜇+𝜃𝑆,𝑑𝐼𝑑𝑡=𝛽𝑆𝐼𝑁−𝜇+𝑑𝐼, (1) where the birth process incorporates density dependent effects via a logistic equation with the intrinsic growth rate 𝑟 and the carrying capacity 𝐾; 𝑆(𝑡), 𝐼(𝑡) represent population densities of susceptible and infected population, respectively; 𝛽 denotes the transmission rate (the infection rate constant); 𝜇 is the natural mortality; 𝑑 denotes the disease-induced mortality; 𝜃 is the per-capita emigration rate of noninfective.

For model (1), the epidemic threshold of basic reproduction number 𝑅₀ is then computed as 𝑅₀=𝛽.𝜇+𝑑 (2)

The basic demographic reproductive number 𝑅_𝑑 is given by 𝑅_𝑑=𝑟.𝜇+𝜃 (3)

For simplicity, rescalling the model (1) by letting 𝑆→𝑆/𝐾, 𝐼→𝐼/𝐾, and 𝑡→𝑡/(𝜇+𝑑) leads to the following model: 𝑑𝑆𝑑𝑡=𝜈𝑅_𝑑(𝑆+𝐼)(1−(𝑆+𝐼))−𝑅₀𝑆𝐼𝑆+𝐼−𝜈𝑆,𝑑𝐼𝑑𝑡=𝑅₀𝑆𝐼𝑆+𝐼−𝐼, (4) where 𝜈=(𝜇+𝜃)/(𝜇+𝑑) defined by the ratio of the average life-span of susceptibles to that of infections and 𝑆+𝐼≤1.

See [30] for more details.

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 reaction-diffusion 𝑆𝐼 epidemic model: 𝜕𝑆𝜕𝑡−𝑑_𝑆Δ𝑆=𝜈𝑅_𝑑(𝑆+𝐼)(1−(𝑆+𝐼))−𝑅₀𝑆𝐼𝑆+𝐼−𝜈𝑆,𝑥∈Ω,𝑡>0,𝜕𝐼𝜕𝑡−𝑑_𝐼Δ𝐼=𝑅₀𝑆𝐼𝑆()𝑆+𝐼−𝐼,𝑥∈Ω,𝑡>0,𝑥,0=𝑆₀(𝑥)>0,𝐼(𝑥,0)=𝐼₀(𝑥)≥0,𝑥∈Ω,𝜕𝑆=𝜕𝐧𝜕𝐼𝜕𝐧=0,𝑥∈𝜕Ω,𝑡>0, (5) where the nonnegative constants 𝑑_𝑆 and 𝑑_𝐼 are the diffusion coefficients of 𝑆 and 𝐼, respectively. 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.

The corresponding kinetic model (5) with 𝑚=2 has been investigated by Wang et al. [20].

In this paper, we concentrated on the steady states of model (5) which satisfy −𝑑_𝑆Δ𝑆=𝜈𝑅_𝑑(𝑆+𝐼)(1−(𝑆+𝐼))−𝑅₀𝑆𝐼𝑆+𝐼−𝜈𝑆,𝑥∈Ω,−𝑑_𝐼Δ𝐼=𝑅₀𝑆𝐼𝑆+𝐼−𝐼,𝑥∈Ω,𝜕𝑆=𝜕𝐧𝜕𝐼𝜕𝐧=0,𝑥∈𝜕Ω. (6)

Throughout this paper, the positive solution (𝑆,𝐼) satisfying model (6) refers to a classical one with 𝑆>0, 𝐼>0 on Ω. Clearly, model (6) has a unique positive constant solution (endemic equilibrium) 𝐸^∗=(𝑆^∗,𝐼^∗) if 𝑅_𝑑>(𝜈+𝑅₀−1)/𝑅₀𝜈 and 𝑅₀>1, where 𝑆^∗=𝜈𝑅₀𝑅_𝑑−𝑅₀+1−𝜈𝜈𝑅²₀𝑅_𝑑,𝐼^∗=𝑅₀−1𝑆^∗. (7)

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 𝜈, 𝑅_𝑑, and 𝑅₀ collectively by Λ. The positive constants 𝐶, 𝐶, 𝐶, 𝐶^∗, and so forth will depend only on the domains Ω and Λ.

Lemma 1 (Harnack inequality [31]). 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. (8)

Lemma 2 (maximum principle [32]). 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))≥0,𝑥∈Ω,𝜕𝑤𝜕𝐧≤0,𝑥∈𝜕Ω. (9) 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))≤0,𝑥∈Ω,𝜕𝑤𝜕𝐧≥0,𝑥∈𝜕Ω. (10) If 𝑤(𝑥₁)=min_Ω𝑤(𝑥), then 𝑔(𝑥₁,𝑤(𝑥₁))≤0.

Theorem 3. If 𝑅₀>1, then the positive solution (𝑆(𝑥),𝐼(𝑥)) of model (6) satisfies max_Ω𝑆(𝑥)<14𝑅_𝑑,max_Ω𝐼(𝑥)<14𝑅_𝑑𝑅₀−1. (11)

Proof. Assume that (𝑆(𝑥),𝐼(𝑥)) is a positive solution of model (6). We set 𝑆𝑥₁=max_Ω𝑆(𝑥),𝐼𝑥₂=max_Ω𝐼(𝑥). (12) By applying Lemma 2, we have 14𝑅_𝑑−𝜈𝑆𝑥₁≥𝜈𝑅_𝑑𝑆𝑥₁+𝐼𝑥₁1−𝑆𝑥₁+𝐼𝑥₁−𝜈𝑆𝑥₁≥𝑅₀𝑆𝑥₁𝐼𝑥₁𝑆𝑥₁+𝐼𝑥₁>0, (13) and 𝑅₀𝑆(𝑥₂)𝐼(𝑥₂)/(𝑆(𝑥₂)+𝐼(𝑥₂))≥𝐼(𝑥₂). This clearly gives 𝑆𝑥₁<14𝑅_𝑑,𝐼𝑥₂≤𝑅₀−1𝑆𝑥₂<14𝑅_𝑑𝑅₀−1. (14)

Theorem 4. Assume that 𝑅_𝑑>1 and 𝑅₀>1. Let 𝑑 and 𝐷 be fixed positive constants. Then there exists a positive constant 𝐶=𝐶(Λ,𝑑) such that, if 𝑑_𝑆,𝑑_𝐼>𝑑, every positive solution (𝑆(𝑥),𝐼(𝑥)) of model (6) satisfies min_Ω𝑆(𝑥)>𝐶,min_Ω𝐼(𝑥)>𝐶. (15)

Proof. Let 𝑐₁(𝑥)=1𝑑_𝑆𝜈𝑅_𝑑𝐼1+𝑆((−𝑅1−𝑆+𝐼))⁰𝐼,𝑐𝑆+𝐼−𝜈₂(𝑥)=1𝑑_𝐼𝑅₀𝑆.𝑆+𝐼−1 (16) In view of Theorem 3, there exists a positive constant 𝐶=𝐶(Λ) such that ‖𝑐₁(𝑥)‖_∞≤𝐶, ‖𝑐₂(𝑥)‖_∞≤𝐶 provided that 𝑑_𝑆,𝑑_𝐼>𝑑. As 𝑆 and 𝐼 satisfy Δ𝑆(𝑥)+𝑐₁(𝑥)(𝑥)𝑆=0,𝑥∈Ω,Δ𝐼+𝑐₂(𝑥)𝐼=0,𝑥∈Ω,𝜕𝑆=𝜕𝐧𝜕𝐼𝜕𝐧=0,𝑥∈𝜕Ω. (17) It follows from Lemma 1 that there exists a positive constant 𝐶^∗=𝐶^∗(Λ,𝑑) such that max_Ω𝑆≤𝐶^∗min_Ω𝑆,max_Ω𝐼≤𝐶^∗min_Ω𝐼 (18) for 𝑑_𝑆,𝑑_𝐼≥𝑑.
Now, on the contrary, suppose that (15) is not true, then there exist sequences {𝑑_𝑆,𝑖}^∞_𝑖=1, {𝑑_𝐼,𝑖}^∞_𝑖=1 with (𝑑_𝑆,𝑖,𝑑_𝐼,𝑖)∈[𝑑,∞)×[𝑑,∞) and the positive solution (𝑆_𝑖,𝐼_𝑖) of model (6) corresponding to (𝑑_𝑆,𝑑_𝐼)=(𝑑_𝑆,𝑖,𝑑_𝐼,𝑖), such that min_Ω𝑆_𝑖(𝑥)⟶0ormin_Ω𝐼_𝑖(𝑥)⟶0as𝑖⟶∞. (19) It follows from Lemma 1 that 𝑆_𝑖(𝑥)⟶0or𝐼_𝑖(𝑥)⟶0uniformlyonΩas𝑖⟶∞. (20)(𝑆_𝑖,𝐼_𝑖) satisfies −𝑑_𝑆,𝑖Δ𝑆_𝑖=𝜈𝑅_𝑑𝑆_𝑖+𝐼_𝑖1−𝑆_𝑖+𝐼_𝑖−𝑅₀𝑆_𝑖𝐼_𝑖𝑆_𝑖+𝐼_𝑖−𝜈𝑆_𝑖,𝑥∈Ω,−𝑑_𝐼,𝑖Δ𝐼_𝑖=𝑅₀𝑆_𝑖𝐼_𝑖𝑆_𝑖+𝐼_𝑖−𝐼_𝑖,𝑥∈Ω,𝜕𝑆_𝑖=𝜕𝐧𝜕𝐼_𝑖𝜕𝐧=0,𝑥∈𝜕Ω. (21) Integrating by parts, we obtain that, for 𝑖=1,2,…, _Ω𝜈𝑅_𝑑𝑆_𝑖+𝐼_𝑖1−𝑆_𝑖+𝐼_𝑖−𝑅₀𝑆_𝑖𝐼_𝑖𝑆_𝑖+𝐼_𝑖−𝜈𝑆_𝑖𝑑𝑥=0,_Ω𝐼_𝑖𝑅₀𝑆_𝑖𝑆_𝑖+𝐼_𝑖−1𝑑𝑥=0. (22) By the regularity theory for elliptic equations [33], 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 (20), we have that 𝑆≡0 or 𝐼̃≡0.
Letting 𝑖→∞ in (22) we obtain that _Ω𝜈𝑅_𝑑𝐼̃𝐼̃−𝑅𝑆+1−𝑆+⁰𝑆𝐼̃𝐼̃𝑆𝑆+−𝜈𝑑𝑥=0,_Ω𝐼̃𝑅₀𝑆𝑆+𝐼̃−1𝑑𝑥=0. (23)
Case  1   ( 𝑆≡0, 𝐼̃≢0 or 𝑆≡0, 𝐼̃≡0). Since 𝐼_𝑖 satisfies the second inequality of (18), 𝐼_𝑖>0 on Ω. Therefore, 𝑅₀𝑆_𝑖/(𝑆_𝑖+𝐼_𝑖)−1→−1<0 on Ω as 𝑖→∞. Hence, ∫_Ω𝐼_𝑖(𝑅₀𝑆_𝑖/(𝑆_𝑖+𝐼_𝑖)−1)𝑑𝑥<0 for sufficiently large 𝑖 which contradicts the second integral identity of (22).
Case  2   ( 𝐼̃≡0, 𝑆≢0). As above, 𝑆>0 on Ω. It follows from the first integral identity of (23) that _Ω𝑆𝜈𝑅_𝑑1−𝑆−𝜈𝑑𝑥=0. (24) This fact combines with 0<𝑆≤(1/4)𝑅_𝑑 yielding to 𝑆=1−1/𝑅_𝑑, which implies that 𝑅₀𝑆_𝑖/(𝑆_𝑖+𝐼_𝑖)→𝑅₀ uniformly on Ω as 𝑖→∞, since 𝐼_𝑖→0 uniformly on Ω. As 𝑅₀>1, this contradicts the second integral identity of (23) and the fact that 𝐼_𝑖>0. 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 0=𝜇₀<𝜇₁<𝜇₂<𝜇₃<⋯ (25) the eigenvalues of the operator −Δ on Ω 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 𝑅₀>1. Let 𝐷₂ be a fixed positive constant with 𝐷₂>(𝑅₀−1)/𝜇₁. Then there exists a positive constant 𝐷₁(Λ,𝐷₂) such that model (6) has no positive nonconstant solution provided that 𝑑_𝑆≥𝐷₁ and 𝑑_𝐼≥𝐷₂.

Proof. Let (𝑆(𝑥),𝐼(𝑥)) be any positive solution of model (6) and denote 𝑔=|Ω|⁻¹∫_Ω𝑔𝑑𝑥. Then, multiplying the first equation of model (6) by (𝑆−𝑆), integrating over Ω, by virtue of Theorem 3, we have that 𝑑_𝑆Ω|∇𝑆|²=𝑑𝑥_Ω𝑆−𝑆×𝜈𝑅_𝑑((−𝑅𝑆+𝐼)(1−𝑆+𝐼))⁰𝑆𝐼=𝑆+𝐼−𝜈𝑆𝑑𝑥_Ω𝑆−𝑆²×𝜈𝑅_𝑑−1−𝜈𝑅_𝑑𝑆+𝑆+2𝜈𝑅_𝑑𝑅𝐼−⁰𝐼𝐼𝑆+𝐼()+𝑆+𝐼𝑑𝑥_Ω𝜈𝑅_𝑑1−𝐼+𝐼+2𝑆−𝑅₀𝑆𝑆𝑆+𝐼()×𝑆+𝐼𝑆−𝑆𝐼−𝐼𝑑𝑥,≤𝐶_1Ω𝑆−𝑆²+𝐶_2Ω𝑆−𝑆𝐼−𝐼𝑑𝑥, (26) where 𝐶₁,𝐶₂ depend only on Λ. In a similar manner, we multiply the second equation in model (6) by (𝐼−𝐼) to have 𝑑_𝐼Ω|∇𝐼|²=𝑑𝑥_Ω𝐼−𝐼𝑅₀𝑆𝐼=𝑆+𝐼−𝐼𝑑𝑥_Ω𝐼−𝐼²𝑅−1+⁰𝑆𝑆𝑆+𝐼()+𝑆+𝐼𝑑𝑥_Ω𝑅₀𝐼𝐼𝑆+𝐼(𝑆+𝐼)𝑆−𝑆𝐼−𝐼≤𝑅𝑑𝑥₀−1_Ω𝐼−𝐼²𝑑𝑥+𝑅_0Ω𝑆−𝑆𝐼−𝐼𝑑𝑥. (27) It follows from (26), (27) and the 𝜀-Young inequality that _Ω𝑑_𝑆|∇𝑆|²+𝑑_𝐼|∇𝐼|²≤𝑑𝑥_Ω𝐶₁+𝐶2𝜀𝑆−𝑆²+𝑅₀−1+𝜀𝐶2𝐼−𝐼²𝑑𝑥, (28) where 𝐶=𝐶₂+𝑅₀. It follows from the well-known Poincaré inequality that _Ω𝑑_𝑆|∇𝑆|²+𝑑_𝐼|∇𝐼|²≤1𝑑𝑥𝜇₁𝐶₁+𝐶2𝜀_Ω|∇𝑆|²+1𝑑𝑥𝜇₁𝑅₀−1+𝜀𝐶2_Ω|∇𝐼|²𝑑𝑥. (29) Since 𝑑_𝐼𝜇₁>𝑅₀−1 from the assumption, we can find a sufficiently small 𝜀₀ such that 𝑑_𝐼𝜇₁≥𝑅₀−1+𝜀𝐶/2. Finally, by taking 𝐷₁∶=(1/𝜇₁)(𝐶₁+𝐶/2𝜀₀) one can conclude that 𝑆=𝑆 and 𝐼=𝐼, 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 𝑅_𝑑>(𝜈+𝑅₀−1)/𝑅₀𝜈 and 𝑅₀>1, which guarantees that model (6) has one positive constant steady state 𝐸^∗. From now on, let us denote (),𝐮𝐮=𝑆,𝐼_∗=𝑆^∗,𝐼^∗=𝜈𝑅₀𝑅_𝑑−𝑅₀+1−𝜈𝜈𝑅²₀𝑅_𝑑,𝑅₀−1𝑆^∗. (30)

Let 𝐗={𝐮∈[𝐶²(Ω)]²∣𝜕𝐮/𝜕𝐧=0,𝑥∈𝜕Ω} and 𝐗⁺={𝐮∈𝐗∣𝑆,𝐼>0,𝑥∈Ω}. Then we write model (6) in the form −Δ𝐮=𝐆(𝐮),𝑥∈Ω,𝜕𝐮𝜕𝐧=0,𝑥∈𝜕Ω, (31) where 𝐆(𝐮)=1𝑑_𝑆𝜈𝑅_𝑑((−𝑅𝑆+𝐼)(1−𝑆+𝐼))⁰𝑆𝐼1𝑆+𝐼−𝜈𝑆𝑑_𝐼𝑅₀𝑆𝐼𝑆+𝐼−𝐼. (32)

Define a compact operator ℱ∶𝐗⁺→𝐗⁺ by ℱ(𝐮)∶=(𝐈−Δ)⁻¹{𝐆(𝐮)+𝐮}, (33) 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 (31) if and only if 𝐮 satisfies (𝐈−ℱ)𝐮=0,𝑥∈Ω. (34)

To apply the index theory, we investigate the eigenvalue of the problem −𝐈−ℱ_𝐮𝐮^∗Ψ=𝜆Ψ,Ψ≠𝟎, (35) where Ψ=(Ψ₁,Ψ₂)^𝑇 and ℱ_𝐮(𝐮^∗)=(𝐈−Δ)⁻¹(𝐈+𝐀) with𝐀=4𝑅₀+2𝜈−𝑅²₀−𝜈𝑅₀𝑅_𝑑−𝜈𝑅₀−3𝑑_𝑆𝑅₀−𝜈𝑅₀𝑅_𝑑+3−2𝑅₀−2𝜈𝑑_𝑆𝑅₀𝑅₀−1²𝑑_𝐼𝑅₀−𝑅₀−1𝑑_𝐼𝑅₀𝑑∶=^𝑆−1𝑎₁−𝑑_𝑆⁻¹𝑎₂𝑑_𝐼⁻¹𝑎₃−𝑑_𝐼⁻¹𝑎₄. (36)𝜆 is an eigenvalue of (35) if and only if 𝜆 is an eigenvalue of the matrix (𝜇_𝑖+1)⁻¹(𝜇_𝑖𝐈−𝐀) for any 𝑖≥0. Therefore, 𝐈−ℱ_𝐮(𝐮^∗) is invertible if and only if, for any 𝑖≥0, the matrix 𝑀_𝑖∶=𝜇_𝑖𝜇𝐈−𝐀=^𝑖−𝑑_𝑆⁻¹𝑎₁𝑑_𝑆⁻¹𝑎₂−𝑑_𝐼⁻¹𝑎₃𝜇_𝑖+𝑑_𝐼⁻¹𝑎₄ (37) is invertible. A straightforward computation yields det𝑀_𝑖=𝑑_𝑆⁻¹𝑑_𝐼⁻¹×𝑑_𝑆𝑑_𝐼𝜇²_𝑖+𝑑_𝑆𝑎₄−𝑑_𝐼𝑎₁𝜇_𝑖+𝜌, (38) where 𝜌=(1/𝑅₀)(𝜈𝑅₀𝑅_𝑑−𝑅₀+1−𝜈)(𝑅₀−1)>0. For the sake of convenience, we denote 𝐻𝑑_𝑆,𝑑_𝐼,𝜇=𝑑_𝑆𝑑₂𝜇²_𝑖+𝑑_𝑆𝑎₄−𝑑_𝐼𝑎₁𝜇_𝑖+𝜌. (39) Then 𝐻(𝑑_𝑆,𝑑_𝐼,𝜇)=𝑑_𝑆𝑑₂det(𝑀_𝑖).

If (𝑑_𝑆𝑎₄−𝑑_𝐼𝑎₁)²>4𝑑_𝑆𝑑₂𝜌, then 𝐻(𝑑_𝑆,𝑑_𝐼,𝜇)=0 has two real roots 𝜇^± given by 𝜇⁺𝑑_𝑆,𝑑_𝐼=12𝑑_𝑆𝑑₂𝑑_𝐼𝑎₁−𝑑_𝑆𝑎₄+𝑑_𝑆𝑎₄−𝑑_𝐼𝑎₁²−4𝑑_𝑆𝑑₂𝜌,𝜇⁻𝑑_𝑆,𝑑_𝐼=12𝑑_𝑆𝑑₂𝑑_𝐼𝑎₁−𝑑_𝑆𝑎₄−𝑑_𝑆𝑎₄−𝑑_𝐼𝑎₁²−4𝑑_𝑆𝑑₂𝜌. (40)

Set 𝐵∶=𝐵(𝑑_𝑆,𝑑_𝐼)={𝜇∶𝜇≥0,𝜇⁻(𝑑_𝑆,𝑑_𝐼)<𝜇<𝜇⁺(𝑑_𝑆,𝑑_𝐼)}, 𝑆_𝑝={𝜇₀,𝜇₁,𝜇₂,…}, and 𝑚(𝜇_𝑖) the multiplicity of 𝜇_𝑖.

To compute index(𝐈−ℱ,𝐮^∗), we can assert the following conclusion by Pang and Wang [22].

Lemma 6 (see [22]). Suppose 𝐻(𝑑_𝑆,𝑑_𝐼,𝜇_𝑖)≠0 for all 𝜇_𝑖∈𝑆_𝑝. Then 𝑖𝑛𝑑𝑒𝑥𝐈−ℱ,𝐮^∗=(−1)^𝜎, (41) where ∑𝜎=^{𝜇∈𝐵∩𝑆𝑗}𝑚𝑢_𝑖,𝑖𝑓𝐵∩𝑆_𝑝≠∅,0,𝑖𝑓𝐵∩𝑆_𝑝=∅. (42) In particular, if 𝐻(𝑑_𝑆,𝑑_𝐼,𝜇)>0 for all 𝜇≥0, then 𝜎=0.

From Lemma 6, we see that to calculate the index of index(𝐈−ℱ,𝐮^∗), the key step is to determine the range of 𝜇 for which 𝐻(𝑑_𝑆,𝑑_𝐼,𝜇)<0.

Theorem 7. Assume that 𝑅_𝑑>max{1,(𝜈+𝑅₀−1)/𝑅₀𝜈}. If 4𝑅₀+2𝜈−𝑅²₀−𝜈𝑅₀𝑅_𝑑−𝜈𝑅₀−3>0, (4𝑅₀+2𝜈−𝑅²₀−𝜈𝑅₀𝑅_𝑑−𝜈𝑅₀−3)/𝑑_𝑆𝑅₀∈(𝜇_𝑗,𝜇_𝑗+1) for some 𝑗≥1, and ∑^𝑗_𝑖=1𝑚(𝜇_𝑖) is odd, then there exists a positive constant 𝑑^∗ such that model (6) has at least one nonconstant solution if 𝑑_𝐼>𝑑^∗.

Proof. Since 4𝑅₀+2𝜈−𝑅²₀−𝜈𝑅₀𝑅_𝑑−𝜈𝑅₀−3>0, equivalently, 𝑎₁>0, it follows that if 𝑑_𝐼 is large enough, then (𝑑_𝑆𝑎₄−𝑑_𝐼𝑎₁)²>4𝑑_𝑆𝑑_𝐼𝜌 and 0<𝜇⁻(𝑑_𝑆,𝑑_𝐼)<𝜇⁺(𝑑_𝑆,𝑑_𝐼). Furthermore, 𝜇⁻𝑑_𝑆,𝑑_𝐼⟶0,𝜇⁺𝑑_𝑆,𝑑_𝐼⟶𝑎₁𝑑_𝑆,as𝑑_𝐼⟶∞. (43) Since 𝑎₁/𝑑_𝑆∈(𝜇_𝑗,𝜇_𝑗+1) for some 𝑗≥1, there exists 𝑑₀≫1 such that 𝜇⁺𝑑_𝑆,𝑑_𝐼∈𝜇_𝑗,𝜇_𝑗+1,0<𝜇⁻𝑑_𝑆,𝑑_𝐼<𝜇₁,∀𝑑_𝐼≥𝑑₀. (44)
By Theorem 5, we know that there exists 𝑑>𝑑₀ such that model (6) with diffusion coefficients 𝑑_𝑆=𝑑 and 𝑑_𝐼≥𝑑 has no nonconstant solutions. Moreover, we can choose 𝑑 so large that 𝑎₁/𝑑<𝜇₁. It follows that there exists 𝑑^∗>𝑑 such that 0<𝜇⁻𝑑_𝑆,𝑑_𝐼<𝜇⁺𝑑_𝑆,𝑑_𝐼<𝜇₁,∀𝑑_𝐼≥𝑑^∗. (45) We shall prove that, for any 𝑑_𝐼≥𝑑^∗, model (6) has at least one nonconstant positive solution. On the contrary, suppose that this assertion is not true for some 𝑑^∗_𝐼>𝑑^∗. 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 𝐶=𝐶(Λ,𝑑,𝑑_𝑆,𝑑^∗,𝑑_𝐼^∗) such that the positive solution (𝑆(𝑥),𝐼(𝑥)) of model (6) satisfies 𝐶⁻¹<𝑆,𝐼<𝐶.
Set ℳ=(𝑆,𝐼)∈𝐶Ω×𝐶Ω∶𝐶⁻¹<𝑆,𝐼<𝐶,𝑥∈Ω, (46) and define Φ∶ℳ×[0,1]⟶𝐶Ω×𝐶Ω (47) by Φ(𝐮,𝜃)=(𝐈−Δ)⁻¹{𝐆(𝐮,𝜃)+𝐮}, (48) where 𝐆()=𝐮,𝜃𝜃𝑑_𝑆+(1−𝜃)𝑑⁻¹𝜈𝑅_𝑑((−𝑅𝑆+𝐼)(1−𝑆+𝐼))⁰𝑆𝐼𝑆+𝐼−𝜈𝑆𝜃𝑑_𝐼+(1−𝜃)𝑑^∗−1𝑅₀𝑆𝐼.𝑆+𝐼−𝐼 (49)
It is clear that finding the positive solution of model (31) is equivalent to finding the fixed point of Φ(𝐮,1) in ℳ. Further, by virtue of the definition of ℳ, we have that Φ(𝐮,𝜃)=0 has no fixed point 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). (50)
In view of 𝜇⁻∈(𝜇_𝑖,𝜇_𝑖+1) and 𝜇⁺∈(𝜇_𝑗,𝜇_𝑗+1), we have 𝐵(𝑑_𝑆,𝑑_𝐼)∩𝑆_𝑗={𝜇_𝑖+1,𝜇_𝑖+2,…,𝜇_𝑝}. Clearly, 𝐈−Φ(𝐮,1)=𝐈−ℱ. Thus, if model (6) has no other solutions except the constant one 𝐮^∗, then Lemma 6 shows that deg(𝐈−Φ(𝐮,1),ℳ,0)=index𝐈−ℱ,𝐮^∗=(−1)^{∑𝑗𝑖=1𝑚(𝑢𝑖)}=−1. (51)
On the contrary, by the choice of 𝑑 and 𝑑^∗, we have that 𝐵(𝑑₁,𝑑₂)∩𝑆_𝑝=∅ and 𝐮^∗ is the only fixed point of Φ(𝐮,0). It therefore follows from Lemma 6 that deg(𝐈−Φ(𝐮,0),ℳ,0)=index𝐈−ℱ,𝐮^∗=−1⁰=1. (52) 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).