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

Authors: Yuan Yuan; Hailing Wang; Weiming Wang
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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 Section2, 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
(1)dSdt=rN(1-NK)-βSIN-(μ+θ)S,hhhhdIdt=βSIN-(μ+d)I,
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
(2)R0=βμ+d.The basic demographic reproductive numberRd is given by
(3)Rd=rμ+θ.For simplicity, rescalling the model (1) by letting S→S/K, I→I/K, and t→t/(μ+d) leads to the following model:
(4)dSdt=νRd(S+I)(1-(S+I))-R0SIS+I-νS,hhhhhhhhhdIdt=R0SIS+I-I,
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 habitatΩ⊂ℝm (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:
(5)∂S∂t-dSΔS=νRd(S+I)(1-(S+I))∂S∂t-dSΔS=-R0SIS+I-νS,x∈Ω,t>0,∂I∂t-dIΔI=R0SIS+I-I,x∈Ω,t>0,S(x,0)=S0(x)>0,I(x,0)=I0(x)≥0,x∈Ω,∂S∂n=∂I∂n=0,x∈∂Ω,t>0,
where the nonnegative constants dS and dI 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
(6)-dSΔS=νRd(S+I)(1-(S+I))-R0SIS+I-νS,x∈Ω,fffffffffffffff-dIΔI=R0SIS+I-I,x∈Ω,ffffffffffffffffffff∂S∂n=∂I∂n=0,x∈∂Ω.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 Rd>(ν+R0-1)/R0ν and R0>1, where
(7)S*=νR0Rd-R0+1-ννR02Rd,I*=(R0-1)S*.


## 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, Rd, 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]).
Letw∈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
(8)maxΩ-w≤C*minΩ-w.Lemma 2 (maximum principle [32]).
LetΩ be a bounded Lipschitz domain in ℝm and g∈C(Ω-×ℝ).(a)
Assume thatw∈C2(Ω)∩C1(Ω-) and satisfies
(9)Δw(x)+g(x,w(x))≥0,x∈Ω,hhhhhh∂w∂n≤0,x∈∂Ω.Ifw(x0)=maxΩ-w(x), then g(x0,w(x0))≥0.(b)
Assume thatw∈C2(Ω)∩C1(Ω-) and satisfies
(10)Δw(x)+g(x,w(x))≤0,x∈Ω,hhhhhh∂w∂n≥0,x∈∂Ω.Ifw(x1)=minΩ-w(x), then g(x1,w(x1))≤0.Theorem 3.
IfR0>1, then the positive solution (S(x),I(x)) of model (6) satisfies
(11)maxΩ-S(x)<14Rd,maxΩ-I(x)<14Rd(R0-1).Proof.
Assume that(S(x),I(x)) is a positive solution of model (6). We set
(12)S(x1)=maxΩ-S(x),I(x2)=maxΩ-I(x).
By applying Lemma 2, we have
(13)14Rd-νS(x1)≥νRd(S(x1)+I(x1))(1-(S(x1)+I(x1)))-νS(x1)≥R0S(x1)I(x1)S(x1)+I(x1)>0,
and R0S(x2)I(x2)/(S(x2)+I(x2))≥I(x2). This clearly gives
(14)S(x1)<14Rd,I(x2)≤(R0-1)S(x2)<14Rd(R0-1).Theorem 4.
Assume thatRd>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 dS,dI>d, every positive solution (S(x),I(x)) of model (6) satisfies
(15)minΩ-S(x)>C_,minΩ-I(x)>C_.Proof.
Let(16)c1(x)=1dS(νRd(1+IS)(1-(S+I))-R0IS+I-ν),hhhhhhhhhhhc2(x)=1dI(R0SS+I-1).
In view of Theorem 3, there exists a positive constant C=C(Λ) such that ∥c1(x)∥∞≤C, ∥c2(x)∥∞≤C provided that dS,dI>d. As S and I satisfy
(17)ΔS(x)+c1(x)S=0,x∈Ω,ΔI(x)+c2(x)I=0,x∈Ω,∂S∂n=∂I∂n=0,x∈∂Ω.
It follows from Lemma 1 that there exists a positive constant C*=C*(Λ,d) such that
(18)maxΩ-S≤C*minΩ-S,maxΩ-I≤C*minΩ-I
for dS,dI≥d.
Now, on the contrary, suppose that (15) is not true, then there exist sequences {dS,i}i=1∞, {dI,i}i=1∞ with (dS,i,dI,i)∈[d,∞)×[d,∞) and the positive solution (Si,Ii) of model (6) corresponding to (dS,dI)=(dS,i,dI,i), such that
(19)minΩ¯Si(x)⟶0orminΩ¯Ii(x)⟶0asi⟶∞.
It follows from Lemma 1 that
(20)Si(x)⟶0orIi(x)⟶0uniformlyonΩ-asi⟶∞.(Si,Ii) satisfies
(21)-dS,iΔSi=νRd(Si+Ii)(1-(Si+Ii))-dS,iΔSi=.-R0SiIiSi+Ii-νSi,x∈Ω,-dI,iΔIi=R0SiIiSi+Ii-Ii,x∈Ω,∂Si∂n=∂Ii∂n=0,x∈∂Ω.
Integrating by parts, we obtain that, for i=1,2,…,
(22)∫Ω‍(νRd(Si+Ii)(1-(Si+Ii))-R0SiIiSi+Ii-νSi)dx=0,hhhhhhhhhhhh∫Ω‍Ii(R0SiSi+Ii-1)dx=0.
By the regularity theory for elliptic equations [33], we see that there exist a subsequence of {(Si,Ii)}i∞, which we will still denote by {(Si,Ii)}i∞, and two nonnegative functions S~,I~∈C2(Ω), such that (Si,Ii)→(S~,I~) in [C2(Ω)]2 as i→∞. By (20), we have that S~≡0 or I~≡0.
Lettingi→∞ in (22) we obtain that
(23)∫Ω‍(νRd(S~+I~)(1-(S~+I~))-R0S~I~S~+I~-νS~)dx=0,hhhhhhhhhh∫Ω‍I~(R0S~S~+I~-1)dx=0.
Case  1   (
S
~
≡
0
,
I
~
≢
0
or
S
~
≡
0
,
I
~
≡
0
). Since Ii satisfies the second inequality of (18), Ii>0 on Ω-. Therefore, R0Si/(Si+Ii)-1→-1<0 on Ω- as i→∞. Hence, ∫Ω‍Ii(R0Si/(Si+Ii)-1)dx<0 for sufficiently large i which contradicts the second integral identity of (22).
Case  2   (
I
~
≡
0
,
S
~
≢
0
). As above, S~>0 on Ω-. It follows from the first integral identity of (23) that
(24)∫Ω‍S~(νRd(1-S~)-ν)dx=0.
This fact combines with 0<S~≤(1/4)Rd yielding to S~=1-1/Rd, which implies that R0Si/(Si+Ii)→R0 uniformly on Ω- as i→∞, since Ii→0 uniformly on Ω-. As R0>1, this contradicts the second integral identity of (23) and the fact that Ii>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
(25)0=μ0<μ1<μ2<μ3<⋯
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 thatR0>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 dS≥D1 and dI≥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
(26)dS∫Ω|∇S|2dx‍=∫Ω(S-S-)‍ggggg×(νRd(S+I)(1-(S+I))-R0SIS+I-νS)dx=∫Ω(S-S-)2‍hhhhh×(ν(Rd-1)-νRd(S+S-)+2νRdI-R0II-(S-+I-)(S+I))dx+∫Ω‍(νRd(1-(I+I-)+2S-)-R0SS-(S-+I-)(S+I))gggggggg×(S-S-)(I-I-)dx,≤C1∫Ω‍(S-S-)2+C2∫Ω|S-S-||I-I-|dx,‍
where C1,C2 depend only on Λ. In a similar manner, we multiply the second equation in model (6) by (I-I-) to have
(27)dI∫Ω‍|∇I|2dx=∫Ω‍(I-I-)(R0SIS+I-I)dx=∫Ω‍(I-I-)2(-1+R0S-S(S-+I-)(S+I))dx+∫Ω‍R0I-I(S-+I-)(S+I)(S-S-)(I-I-)dx≤(R0-1)∫Ω‍(I-I-)2dx+R0∫Ω‍|S-S-||I-I-|dx.
It follows from (26), (27) and the ε-Young inequality that
(28)∫Ω‍(dS|∇S|2+dI|∇I|2)dx≤∫Ω‍((C1+C2ε)(S-S-)2+(R0-1+εC2)(I-I-)2)dx,
where C=C2+R0. It follows from the well-known Poincaré inequality that
(29)∫Ω‍(dS|∇S|2+dI|∇I|2)dx≤1μ1(C1+C2ε)∫Ω‍|∇S|2dx+1μ1(R0-1+εC2)∫Ω‍|∇I|2dx.
Since dIμ1>R0-1 from the assumption, we can find a sufficiently small ε0 such that dIμ1≥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 thatRd>(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
(30)hhhhhhhhhhhhhhu=(S,I),u*=(S*,I*)=(νR0Rd-R0+1-ννR02Rd,(R0-1)S*).LetX={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
(31)-Δu=G(u),x∈Ω,hh∂u∂n=0,x∈∂Ω,
where
(32)G(u)=(1dS(νRd(S+I)(1-(S+I))-R0SIS+I-νS)1dI(R0SIS+I-I)).Define a compact operatorℱ:X+→X+ by
(33)ℱ(u):=(I-Δ)-1{G(u)+u},
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
(34)(I-ℱ)u=0,x∈Ω.To apply the index theory, we investigate the eigenvalue of the problem(35)-(I-ℱu(u*))Ψ=λΨ,Ψ≠0,
where Ψ=(Ψ1,Ψ2)T and ℱu(u*)=(I-Δ)-1(I+A) with(36)A=(4R0+2ν-R02-νR0Rd-νR0-3dSR0-νR0Rd+3-2R0-2νdSR0(R0-1)2dIR0-R0-1dIR0):=(dS-1a1-dS-1a2dI-1a3-dI-1a4).λ is an eigenvalue of (35) if and only if λ is an eigenvalue of the matrix (μi+1)-1(μiI-A) for any i≥0. Therefore, I-ℱu(u*) is invertible if and only if, for any i≥0, the matrix
(37)Mi:=μiI-A=(μi-dS-1a1dS-1a2-dI-1a3μi+dI-1a4)
is invertible. A straightforward computation yields
(38)det(Mi)=dS-1dI-1×(dSdIμi2+(dSa4-dIa1)μi+ρ),
where ρ=(1/R0)(νR0Rd-R0+1-ν)(R0-1)>0. For the sake of convenience, we denote
(39)H(dS,dI,μ)=dSd2μi2+(dSa4-dIa1)μi+ρ.
Then H(dS,dI,μ)=dSd2det(Mi).If(dSa4-dIa1)2>4dSd2ρ, then H(dS,dI,μ)=0 has two real roots μ± given by
(40)μ+(dS,dI)=12dSd2(dIa1-dSa4+(dSa4-dIa1)2-4dSd2ρ),μ-(dS,dI)=12dSd2(dIa1-dSa4-(dSa4-dIa1)2-4dSd2ρ).SetB:=B(dS,dI)={μ:μ≥0,μ-(dS,dI)<μ<μ+(dS,dI)}, Sp={μ0,μ1,μ2,…}, and m(μi) the multiplicity of μi.To computeindex(I-ℱ,u*), we can assert the following conclusion by Pang and Wang [22].Lemma 6 (see [22]).
SupposeH(dS,dI,μi)≠0 for all μi∈Sp. Then
(41)index(I-ℱ,u*)=(-1)σ,
where
(42)σ={∑μ∈B∩Sj‍m(ui),ifB∩Sp≠∅,0,ifB∩Sp=∅.
In particular, if H(dS,dI,μ)>0 for all μ≥0, then σ=0.From Lemma6, 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(dS,dI,μ)<0.Theorem 7.
Assume thatRd>max{1,(ν+R0-1)/R0ν}. If 4R0+2ν-R02-νR0Rd-νR0-3>0, (4R0+2ν-R02-νR0Rd-νR0-3)/dSR0∈(μj,μj+1) for some j≥1, and ∑i=1j‍m(μi) is odd, then there exists a positive constant d* such that model (6) has at least one nonconstant solution if dI>d*.Proof.
Since4R0+2ν-R02-νR0Rd-νR0-3>0, equivalently, a1>0, it follows that if dI is large enough, then (dSa4-dIa1)2>4dSdIρ and 0<μ-(dS,dI)<μ+(dS,dI). Furthermore,
(43)μ-(dS,dI)⟶0,μ+(dS,dI)⟶a1dS,asdI⟶∞.
Since a1/dS∈(μj,μj+1) for some j≥1, there exists d0≫1 such that
(44)μ+(dS,dI)∈(μj,μj+1),0<μ-(dS,dI)<μ1,hhhhhhhhhhhhhhhhhhhhhhhhhhhh∀dI≥d0.
By Theorem5, we know that there exists d>d0 such that model (6) with diffusion coefficients dS=d and dI≥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
(45)0<μ-(dS,dI)<μ+(dS,dI)<μ1,∀dI≥d*.
We shall prove that, for any dI≥d*, model (6) has at least one nonconstant positive solution. On the contrary, suppose that this assertion is not true for some dI*>d*. In the following, we will derive a contradiction by using a homotopy argument.
By virtue of Theorems3 and 4, there exists a positive constant C=C(Λ,d,dS,d*,dI*) such that the positive solution (S(x),I(x)) of model (6) satisfies C-1<S,I<C.
Set(46)ℳ={(S,I)∈C(Ω¯)×C(Ω¯):C-1<S,I<C,x∈Ω¯},
and define
(47)Φ:ℳ×[0,1]⟶C(Ω¯)×C(Ω¯)
by
(48)Φ(u,θ)=(I-Δ)-1{G(u,θ)+u},
where
(49)G(u,θ)=((θdS+(1-θ)d)-1(νRd(S+I)(1-(S+I))-R0SIS+I-νS)(θdI+(1-θ)d*)-1(R0SIS+I-I)).
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.
SinceΦ(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
(50)deg(I-Φ(u,1),ℳ,0)=deg(I-Φ(u,0),ℳ,0).
In view ofμ-∈(μi,μi+1) and μ+∈(μj,μj+1), we have B(dS,dI)∩Sj={μi+1,μi+2,…,μp}. 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
(51)deg(I-Φ(u,1),ℳ,0)=index(I-ℱ,u*)=(-1)∑i=1jm(ui)=-1.
On the contrary, by the choice ofd and d*, we have that B(d-1,d-2)∩Sp=∅ and u* is the only fixed point of Φ(u,0). It therefore follows from Lemma 6 that
(52)deg(I-Φ(u,0),ℳ,0)=index(I-ℱ,u*)=(-1)0=1.
From (50)–(52), we get a contradiction. Therefore, there exists a nonconstant solution of model (6). The proof is completed.


## 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 thatR0>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 dS≥D1 and dI≥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
(26)dS∫Ω|∇S|2dx‍=∫Ω(S-S-)‍ggggg×(νRd(S+I)(1-(S+I))-R0SIS+I-νS)dx=∫Ω(S-S-)2‍hhhhh×(ν(Rd-1)-νRd(S+S-)+2νRdI-R0II-(S-+I-)(S+I))dx+∫Ω‍(νRd(1-(I+I-)+2S-)-R0SS-(S-+I-)(S+I))gggggggg×(S-S-)(I-I-)dx,≤C1∫Ω‍(S-S-)2+C2∫Ω|S-S-||I-I-|dx,‍
where C1,C2 depend only on Λ. In a similar manner, we multiply the second equation in model (6) by (I-I-) to have
(27)dI∫Ω‍|∇I|2dx=∫Ω‍(I-I-)(R0SIS+I-I)dx=∫Ω‍(I-I-)2(-1+R0S-S(S-+I-)(S+I))dx+∫Ω‍R0I-I(S-+I-)(S+I)(S-S-)(I-I-)dx≤(R0-1)∫Ω‍(I-I-)2dx+R0∫Ω‍|S-S-||I-I-|dx.
It follows from (26), (27) and the ε-Young inequality that
(28)∫Ω‍(dS|∇S|2+dI|∇I|2)dx≤∫Ω‍((C1+C2ε)(S-S-)2+(R0-1+εC2)(I-I-)2)dx,
where C=C2+R0. It follows from the well-known Poincaré inequality that
(29)∫Ω‍(dS|∇S|2+dI|∇I|2)dx≤1μ1(C1+C2ε)∫Ω‍|∇S|2dx+1μ1(R0-1+εC2)∫Ω‍|∇I|2dx.
Since dIμ1>R0-1 from the assumption, we can find a sufficiently small ε0 such that dIμ1≥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 thatRd>(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
(30)hhhhhhhhhhhhhhu=(S,I),u*=(S*,I*)=(νR0Rd-R0+1-ννR02Rd,(R0-1)S*).LetX={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
(31)-Δu=G(u),x∈Ω,hh∂u∂n=0,x∈∂Ω,
where
(32)G(u)=(1dS(νRd(S+I)(1-(S+I))-R0SIS+I-νS)1dI(R0SIS+I-I)).Define a compact operatorℱ:X+→X+ by
(33)ℱ(u):=(I-Δ)-1{G(u)+u},
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
(34)(I-ℱ)u=0,x∈Ω.To apply the index theory, we investigate the eigenvalue of the problem(35)-(I-ℱu(u*))Ψ=λΨ,Ψ≠0,
where Ψ=(Ψ1,Ψ2)T and ℱu(u*)=(I-Δ)-1(I+A) with(36)A=(4R0+2ν-R02-νR0Rd-νR0-3dSR0-νR0Rd+3-2R0-2νdSR0(R0-1)2dIR0-R0-1dIR0):=(dS-1a1-dS-1a2dI-1a3-dI-1a4).λ is an eigenvalue of (35) if and only if λ is an eigenvalue of the matrix (μi+1)-1(μiI-A) for any i≥0. Therefore, I-ℱu(u*) is invertible if and only if, for any i≥0, the matrix
(37)Mi:=μiI-A=(μi-dS-1a1dS-1a2-dI-1a3μi+dI-1a4)
is invertible. A straightforward computation yields
(38)det(Mi)=dS-1dI-1×(dSdIμi2+(dSa4-dIa1)μi+ρ),
where ρ=(1/R0)(νR0Rd-R0+1-ν)(R0-1)>0. For the sake of convenience, we denote
(39)H(dS,dI,μ)=dSd2μi2+(dSa4-dIa1)μi+ρ.
Then H(dS,dI,μ)=dSd2det(Mi).If(dSa4-dIa1)2>4dSd2ρ, then H(dS,dI,μ)=0 has two real roots μ± given by
(40)μ+(dS,dI)=12dSd2(dIa1-dSa4+(dSa4-dIa1)2-4dSd2ρ),μ-(dS,dI)=12dSd2(dIa1-dSa4-(dSa4-dIa1)2-4dSd2ρ).SetB:=B(dS,dI)={μ:μ≥0,μ-(dS,dI)<μ<μ+(dS,dI)}, Sp={μ0,μ1,μ2,…}, and m(μi) the multiplicity of μi.To computeindex(I-ℱ,u*), we can assert the following conclusion by Pang and Wang [22].Lemma 6 (see [22]).
SupposeH(dS,dI,μi)≠0 for all μi∈Sp. Then
(41)index(I-ℱ,u*)=(-1)σ,
where
(42)σ={∑μ∈B∩Sj‍m(ui),ifB∩Sp≠∅,0,ifB∩Sp=∅.
In particular, if H(dS,dI,μ)>0 for all μ≥0, then σ=0.From Lemma6, 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(dS,dI,μ)<0.Theorem 7.
Assume thatRd>max{1,(ν+R0-1)/R0ν}. If 4R0+2ν-R02-νR0Rd-νR0-3>0, (4R0+2ν-R02-νR0Rd-νR0-3)/dSR0∈(μj,μj+1) for some j≥1, and ∑i=1j‍m(μi) is odd, then there exists a positive constant d* such that model (6) has at least one nonconstant solution if dI>d*.Proof.
Since4R0+2ν-R02-νR0Rd-νR0-3>0, equivalently, a1>0, it follows that if dI is large enough, then (dSa4-dIa1)2>4dSdIρ and 0<μ-(dS,dI)<μ+(dS,dI). Furthermore,
(43)μ-(dS,dI)⟶0,μ+(dS,dI)⟶a1dS,asdI⟶∞.
Since a1/dS∈(μj,μj+1) for some j≥1, there exists d0≫1 such that
(44)μ+(dS,dI)∈(μj,μj+1),0<μ-(dS,dI)<μ1,hhhhhhhhhhhhhhhhhhhhhhhhhhhh∀dI≥d0.
By Theorem5, we know that there exists d>d0 such that model (6) with diffusion coefficients dS=d and dI≥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
(45)0<μ-(dS,dI)<μ+(dS,dI)<μ1,∀dI≥d*.
We shall prove that, for any dI≥d*, model (6) has at least one nonconstant positive solution. On the contrary, suppose that this assertion is not true for some dI*>d*. In the following, we will derive a contradiction by using a homotopy argument.
By virtue of Theorems3 and 4, there exists a positive constant C=C(Λ,d,dS,d*,dI*) such that the positive solution (S(x),I(x)) of model (6) satisfies C-1<S,I<C.
Set(46)ℳ={(S,I)∈C(Ω¯)×C(Ω¯):C-1<S,I<C,x∈Ω¯},
and define
(47)Φ:ℳ×[0,1]⟶C(Ω¯)×C(Ω¯)
by
(48)Φ(u,θ)=(I-Δ)-1{G(u,θ)+u},
where
(49)G(u,θ)=((θdS+(1-θ)d)-1(νRd(S+I)(1-(S+I))-R0SIS+I-νS)(θdI+(1-θ)d*)-1(R0SIS+I-I)).
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.
SinceΦ(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
(50)deg(I-Φ(u,1),ℳ,0)=deg(I-Φ(u,0),ℳ,0).
In view ofμ-∈(μi,μi+1) and μ+∈(μj,μj+1), we have B(dS,dI)∩Sj={μi+1,μi+2,…,μp}. 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
(51)deg(I-Φ(u,1),ℳ,0)=index(I-ℱ,u*)=(-1)∑i=1jm(ui)=-1.
On the contrary, by the choice ofd and d*, we have that B(d-1,d-2)∩Sp=∅ and u* is the only fixed point of Φ(u,0). It therefore follows from Lemma 6 that
(52)deg(I-Φ(u,0),ℳ,0)=index(I-ℱ,u*)=(-1)0=1.
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 thatstationary 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.



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*Source: 921401-2013-12-22.xml*