AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 852698 10.1155/2013/852698 852698 Research Article Stationary Patterns of a Cross-Diffusion Epidemic Model Cai Yongli 1,2 Chi Dongxuan 3 Liu Wenbin 4 http://orcid.org/0000-0002-7562-8260 Wang Weiming 2 Bianca Carlo 1 School of Mathematics and Computational Science Sun Yat-sen University Guangzhou 510275 China sysu.edu.cn 2 College of Mathematics and Information Science Wenzhou University Wenzhou 325035 China wzu.edu.cn 3 Department of Applied Mathematics Shanghai Finance University Shanghai 201209 China shfc.edu.cn 4 College of Physics and Electronic Information Engineering Wenzhou University Wenzhou 325035 China wzu.edu.cn 2013 12 11 2013 2013 16 09 2013 03 10 2013 2013 Copyright © 2013 Yongli Cai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 , are widely used for increasing the understanding of infectious disease dynamics and for determining preventive measures to control infection spread qualitatively and quantitatively . 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 . Spatial epidemiology with diffusion has become a principal scientific discipline aiming at understanding the causes and consequences of spatial heterogeneity in disease transmission .

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  and first applied in competitive population system by Shigesada et al. . In particular, Sun et al. , by using the standard linear analysis, studied the pattern formation in a cross-diffusion SI epidemic model. And in , 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 . In particular, in the field of epidemiology, there are many contributions to the existence of steady states in the diffusive epidemic models . 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 Ωm(m1) 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: (1)St-dSΔS=rS(1-SK)-βSIS+I,xΩ,t>0,It-DΔS-dIΔI=βSIS+I-μI,xΩ,t>0,Sn=In=0,xΩ,S(x,0)=S0(x)>0,I(x,0)=I0(x)0,xΩ, where S(x,t) and I(x,t) denote the density of susceptible and infected individuals at location xΩ and time t, respectively, dS and dI 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 dS, dI, and D are such that dS,dI>0,   D0 and dS>dI,  D2<4dSdI which is the parabolic condition.

For model (1), the basic reproduction number is defined as (2)R0=βμ. The steady states of model (1) satisfy (3)-dSΔS=rS(1-SK)-βSIS+I,xΩ,-DΔS-dIΔI=βSIS+I-μI,xΩ,Sn=In=0,xΩ,t>0.

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) (4)E*=(S*,I*)=(Kr(r-(R0-1)μ),Kr(r-(R0-1)μ)(R0-1)) 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 (5)U(t)=AU(t)+F(U(t)),t>0,U(0)=U0C(Ω-)×C(Ω-), where (6)AU(t)=(dSΔS,DΔS+dIΔI),F(U(t))=(rS(1-SK)-βSIS+I,βSIS+I-μI). Since F is locally Lipschitz in C(Ω-)×C(Ω-), for every initial date U0C(Ω-)×C(Ω-), system (5) admits a unique local solution on [0,Tmax), where Tmax is the maximal existence time for solution of system (5) .

Set Z=I-(D/(dS-dI))S. Then model (1) leads to (7)St-dSΔS=rS(1-SK)-βSIS+I:=Λ(S,Z),xΩ,t>0,Zt-dIΔZ=DdS-dI(rS(1-SK)-βSIS+I)+βSIS+I-μI:=Υ(S,Z),xΩ,t>0,Sn=Zn=0,xΩ,t>0,S(x,0)=S0(x)>0,Z(x,0)=Z0(x)=I0(x)-DdS-dIS0(x),xΩ.

A simple application of a comparison theorem to model (7) implies (see ) that for positive initial date S0(x)>0 and Z0(x)0 we have that (8)S(x,t)>0,I(x)DdS-dIS(x,t),xΩ,t>0.

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 , 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)]2S/n=I/n=0,xΩ}, {ϕijj=1,,dimE(λi)} be an orthonormal basis of E(λi), and Xij={cϕijc2}; then X=i=1Xi where Xi=j=1dimE(λi)Xij.

Theorem 2.

For model (1),

if 1<R0<1+(r/μ) and dI(μR02-rR0-μ)+Dμ(R0-1)2<dSμ(R0-1), the positive constant steady state E*=(S*,I*) is locally asymptotically stable;

if R01, 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 (9)ξt=dSΔξ+μR02-rR0-μR0ξ-μR0ζ,xΩ,t>0,ζt=DΔξ+dIΔζ+μ(R0-1)2R0ξ-μ(R0-1)R0ζ,xΩ,t>0,ξn=ζn=0,xΩ. Let (10)£=(dSΔ+μR02-rR0-μR0-μR0DΔ+μ(R0-1)2R0dIΔ-μ(R0-1)R0). For each i0, Xi is invariant under the operator £, and η is an eigenvalue of £ if and only if η is an eigenvalue of the matrix (11)Ai=(-dSλi+μR02-rR0-μR0-μR0-Dλi+μ(R0-1)2R0-dIλi-μ(R0-1)R0), for some i0. Thus the stability of the positive constant steady state is reduced to consider the characteristic equation: (12)det(ηI-Ai)=η2-trace(Ai)η+det(Ai):=φi(η), where (13)trace(Ai)=-λi(dS+dI)-(r-μ(R0-1))<0,det(Ai)=dSdIλi2+1R0((R0-1)2dSμ(R0-1)-dI(μR02-rR0-μ)-Dμ(R0-1)2)λi+μ(R0-1)(r-μ(R0-1))R0. It follows from dI(μR02-rR0-μ)+Dμ(R0-1)2<dSμ(R0-1) that det(Ai)>0. Therefore, the eigenvalues of the matrix Ai have negative real parts. It thus follows from the Routh-Hurwitz criterion that, for each i0, the two roots ηi1 and ηi2 of φi(η)=0 all have negative real parts.

In the following, we prove that there exists κ>0 such that (14){ηi1},{ηi2}-κ. Let η=λiξ; then φi(η)=λi2ξ2-tr(Ai)λiξ+det(Ai):=φ~i(η). Since λi as i, it follows that (15)limiφ~i(η)λi2=ξ2+(dS+dI)ξ+dSdI:=φ~(ξ). Clearly, φ~(ξ)=0 has two negative roots: -dS,-dI. Let d=min{dS,dI}. By continuity, we see that there exists i0 such that the two roots ξi1, ξi2 of φ~i(η)=0 satisfy {ξi1},{ξi2}-(d/2), ii0. In turn, {ηi1},{ηi2}-(λid/2)-(d/2), ii0. Let -κ~=max1ii0{{ηi1},{ηi2}}. Then κ~>0 and (14) holds for κ=min{κ~,d/2}.

Consequently, the spectrum of £ which consists of eigenvalues lies in {{η}-κ}. In the sense of , 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: (16)det(ηI-Ai)=η2-trace(Ai)η+det(Ai), where (17)trace(Ai)=-λi(dS+dI)-(r+μ(R0-1))<0,det(Ai)=dSdIλi2+(dSμ(1-R0)+dIr)λi+rμ(1-R0)>0. 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 (18)liminftminΩ-S(x,t)K(1-R0μr).

Proof.

For all t0, S(x,t) is an upper solution of the following problem: (19)zt-dSΔz=rz(1-βr-zK),xΩ,t>0,zn=0,xΩ,t>0,z(x,0)=S0(x)>0,xΩ-.

Let S(t) be the unique positive solution of the problem (20)wt=rw(1-βr-wK),t>0,w(0)=maxΩ-S0(x)>0. Then S(t) is a lower solution of (20). Since β<r we have (21)limtS(t)=K(1-βr)=K(1-R0μr). It follows by a comparison argument that (22)liminftminΩ-S(x,t)K(1-R0μr). The proof is complete.

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

Theorem 4.

For model (1),

if μ<r and 1<R0<(r+μ)/2μ, then positive constant steady state E*=(S*,I*) of model (1) globally asymptotically stable;

if R01, the semitrivial constant steady state E0=(K,0) of model (1) is globally asymptotically stable.

Proof.

(a) We adopt the Lyapunov function: (23)V(t)=Ω[V1(S(x,t))+νV2(I(x,t))]dx, where V1(S)=S*S((ξ-S*)/ξ)dξ, V2(I)=νI*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, (24)dVdt=Ω(S-S*SSt+ν(I-I*)IIt)dx=Ω(S-S*)(r-rSK-βIS+I+dSΔSS)dx+Ων(I-I*)(βSS+I-μ+dIΔII+DΔSI)dx=ΩP(S,I)dx-Ω(dSS*S2|S|2+dIνI*I2|I|2+DνI*I2|S||I|)dx, where (25)P(S,I)=(S-S*)(r-rSK-βIS+I)+ν(I-I*)(βSS+I-μ). It follows from Lemma 3 that, for any ε>0, there exists t0>0, such that S+ISK(1-(μR0/r))-ε for all xΩ- and tt0. And by some computational analysis, we have (26)P(S,I)=(S-S*)2(-rK+βI*(S*+I*)(S+I))-νS*(S*+I*)(S+I)(I-I*)2=-(S-S*)2(rK-μ(R0-1)S+I)-1R0(R0-1)(S+I)(I-I*)2-(S-S*)2(rK-μ(R0-1)K(1-μR0/r)  -ε)-1R0(R0-1)(S+I)(I-I*)2. Hence, in view of the conditions of the theorem and the arbitrariness of ε, we have P(S,I)0; that is,  dV/dt0 for all xΩ- and tt0. 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: (27)V(t)=ΩIdx. Then V(t)0 and V(t)=0 if and only if I=0. Then, if R01, we obtain (28)dVdt=ΩItdx=Ω(βSIS+I-μI+dIΔI+DΔS)dxΩμI(R0-1)dx0. 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 S0, 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 . 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  and Harnack inequality .

Lemma 5 (maximum principle, see [<xref ref-type="bibr" rid="B22">22</xref>]).

Let Ω be a bounded Lipschitz domain in m and gC(Ω-×).

Assume that wC2(Ω)C1(Ω-) and satisfies (29)Δw(x)+g(x,w(x))0inΩ,wn0onΩ.

If w(x0)=maxΩ-w(x), then g(x0,w(x0))0.

Assume that wC2(Ω)C1(Ω-) and satisfies (30)Δw(x)+g(x,w(x))0in  Ω,wn0onΩ.

If w(x0)=minΩ-w(x), then g(x0,w(x0))0.

Lemma 6 (Harnack inequality, see [<xref ref-type="bibr" rid="B52">52</xref>]).

Let wC2(Ω)C1(Ω-) be a positive solution to Δw(x)+c(x)w(x)=0, where cC(Ω-), satisfying the homogeneous Neumann boundary conditions. Then there exists a positive constant C*=C*(c,Ω), such that (31)maxΩ-wC*minΩ-w.

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 dS,dID1, any positive solution (S(x),I(x)) of model (3) satisfies (32)C_S(x),I(x)C¯.

Proof.

By applying Lemma 5, we have K(1-(μR0/r))<S(x)<K for all xΩ-. Let ϕ=DS+dII>0 and set ϕ(x0)=maxΩ-ϕ. Then, by applying (i) of Lemma 5 again, we have that (33)I(x0)(R0-1)S(x0)<(R0-1)K. Thus, (34)maxΩ-IdI-1maxΩ-ϕK(R0-1+DdI-1).

Let c1(x)=(1/dS)(rS(1-(S/K))-(βSI/(S+I))). Then, there exists a positive constant C=C(Λ) such that c1(x)C provided that dS>D1. It follows from Lemma 6 that there exists a positive constant C*=C*(Λ,D1) such that maxΩ-SC*minΩ-S. On the other hand, ϕ satisfies (35)-Δϕ=(β-μ)SI-μI2(S+I)(DS+dII)ϕ,xΩ,ϕn=0,xΩ. Set c(x)=((β-μ)SI-μI2)/(S+I)(DS+dII). Then (36)c(x)β-μdI+μdI=βdIβD1. It follows from Lemma 6 that there exists a positive constant C*=C*(Λ,D1) such that maxΩ-ϕC*minΩ-ϕ. As a consequence, (37)maxΩ-IminΩ-I=maxΩ-ϕ-DminΩ-SminΩ-ϕ-DmaxΩ-SC*minΩ-ϕminΩ-φ-DmaxΩ-SC.

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 {dS,i}i=1, {dI,i}i=1, and {Di}i=1 with dS,id,dI,i>D1, Di0, and the positive solution (Si,Ii) of model (3) corresponding to (dS,dI,D)=(dS,i,dI,i,Di), such that (38)minΩ¯Ii(x)0asi. It follows from Lemma 6 that (39)Ii(x)0uniformly  on  Ω-  as  i.(Si,Ii) satisfies (40)-dS,iΔSi=rSi(1-SiK)-βSiIiSi+Ii,xΩ,-DiΔSi-dI,iΔIi=βSiIiSi+Ii-μIi,xΩ,Sin=Iin=0,xΩ. Integrating by parts, we obtain that, for i=1,2,, (41)ΩSi(r(1-SiK)-βIiSi+Ii)dx=0,ΩIi(βSiSi+Ii-μ)dx=0. By the regularity theory for elliptic equations , 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 (39), we have that I~0. Letting i in (41) we obtain that (42)ΩS~(r-rKS~-βI~S~+I~)dx=0,ΩI~(βS~S~+I~-μ)dx=0. Since I~=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 4dSε>D2 and dI>ε. Then model (3) has no positive nonconstant solution provided that (4dSε-D2)λ1>4(rε+β2) and (dI-ε)λ1>μ(R0-1)+ε.

Proof.

Let (S(x),I(x)) be any positive solution of model (3) and denote g-=|Ω|-1Ωgdx. Then, multiplying the first equation of model (3) by (S-S-), integrating over Ω, we have that (43)dSΩ|S|2dx=Ω(S-S-)S(r-rSK-βIS+I)dx=Ω(S-S-)2(r-rK(S+S-)-βII-(S-+I-)(S+I))dx-ΩβSS-(S-+I-)(S+I)(S-S-)(I-I-)dxrΩ(S-S-)2+βΩ|S-S-||I-I-|dx. In a similar manner, we multiply the second equation in model (3) by (I-I-) to have (44)Ω(dI|I|2+DS·I)dx=Ω(I-I-)(βSIS+I-μI)Pdx=Ω(I-I-)2(-μ+βS-S(S-+I-)(S+I))dx+ΩβI-I(S-+I-)(S+I)(S-S-)(I-I-)dxμ(R0-1)Ω(I-I-)2dx+Ωβ|S-S-||I-I-|dx. It follows from (43), (44), and the ε-Young inequality that (45)Ω(dS|S|2+dI|I|2)dxΩ((r+β2ε)(S-S-)2+(μ(R0-1)+ε)(I-I-)2)dx+Ω(D24ε|S|2+ε|I|2)dx. Thanks to the well-known Poincaré inequality (46)λ1Ω(g-g¯)2dxΩ|(g-g¯)|2dx, it follows that (47)Ω(dS|S|2+dI|I|2)dxΩ4(rε+β2)+D2λ14ελ1|S|2dx+μ(R0-1)+ε(1+λ1)λ1Ω|I|2dx. Since 4dSελ1>4(rε+β2)+D2λ1 and dIλ1>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 (48)W=DS+dII,W*=DS*+dII*. We also define (49)u=(S,W),(50)u*=(S*,W*)=(Kr(r-μ(R0-1)),Kr(r-μ(R0-1))(dI(R0-1)+D)). Let X={u[C2(Ω)]2u/n=0,xΩ} and X+={uXS,W>0,xΩ-}. Then we write model (3) in the form (51)-Δu=G(u),xΩ,un=0,xΩ, where (52)G(u)=(SdS(r-rKS-β(W-DS)(dI-D)S+W)(W-DS)(βS(dI-D)S+W-μdI)).

Define a compact operator :X+X+ by (53)(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 (51) if and only if u satisfies (54)(I-)u=0,xΩ.

To apply the index theory, we investigate the eigenvalue of the problem (55)-(I-w(w*))Ψ=ξΨ,Ψ0, where Ψ=(Ψ1,Ψ2)T and w(w*)=(I-Δ)-1(I+A) with (56)A=(μdIR02-rdIR0-μdI+μDdSdIR0-μdSdIR0μ(dIR0-dI+D)(R0-1)dIR0-μ(R0-1)dIR0). If 0 is not an eigenvalue of (55), by Theorem  2.8.1 in , the index of I- at u* is given by (57)index(I-,u*)=(-1)γ, where γ=λ>0nλ and nλ is the algebraic multiplicity of the positive eigenvalue ξ of (55).

In fact, after calculation, (55) can be rewritten as (58)-(ξ+1)ΔΨ+(ξI-A)=0,xΩ,Ψn=0,xΩ. Observe that (58) has a nontrivial solution if and only if det(ξI+(λi+1)-1(λiI-A))=0 for some ξ0 and i0. That is to say, ξ is an eigenvalue of (55), and so (58), if and only if ξ is an eigenvalue of the matrix (λi+1)-1(λiI-A) for any i0. Therefore, I-u(u*) is invertible if and only if, for any i0 the matrix λiI-A is invertible.

Let m(λi) be the multiplicity of λi. For the sake of convenience, we denote (59)H(λi)=det(λiI-A). Then if λiI-A is invertible for any i0, with the same arguments as in , we have (60)index(,u*)=(-1)γ,where  γ=i0,H(λi)<0m(λi).

To compute index(,u*), we have to consider the sign of H(λi). A straightforward computation yields (61)H(λi)=λi2-trace(A)λi+det(A), where (62)trace(A)=μdIR02-(μdS+rdI)R0+μdS-μdI+μDdSdIR0,det(A)=μ(R0-1)(r-μ(R0-1))dIR0.

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

Theorem 9.

Assume that trace(A)2-4det(A)0. Then if λ-(λi,λi+1) and λ+(λj,λj+1) for some 0i<j and k=i+1jm(λk) 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->dS such that,

model (3) with diffusion coefficients D-,dI, and D has no nonconstant solutions;

H(λi)>0 for all λi0.

By virtue of Theorem 7, there exists a positive constant C=C(Λ,Ω) such that, for d^SdS, any solution (S(x),I(x)) of model (3) with diffusion coefficients d^S,dI, and D satisfies C-1<S,  I<C.

Set (64)={(S,W)C(Ω¯)×C(Ω¯):C-1<S,W<C,xΩ¯}, and define (65)Φ:×[0,1]C(Ω¯)×C(Ω¯) by (66)Φ(u,θ)=(I-Δ)-1{G(u,θ)+u}, where (67)G(u,θ)=((θdS+(1-θ)D-)-1(S(r-rKS-β(W-DS)(dI-D)S+W))(W-DS)(βS(dI-D)S+W-μdI)).

It is clear that finding the positive solution of model (51) becomes equivalent to finding the fixed point of Φ(u,1) in . Φ(u,θ) has no fixed points in for all 0θ1.

Since Φ(u,t) is compact, the Leray-Schauder topological degree deg(I-Φ(u,θ),,0) is well defined. From the invariance of Leray-Schauder degree at the homotopy, we deduce (68)deg(I-Φ(u,1),,0)=deg(I-Φ(u,0),,0).

Clearly, I-Φ(u,1)=I-. Thus, if model (3) has no other solutions except the constant one u*, we have (69)deg(I-Φ(u,1),,0)=index(I-,u*)=(-1)k=i+1jm(λk)=-1.

On the contrary, by the choice of D-, we have that u* is the only solution of Φ(u,0)=0 and therefore (70)deg(I-Φ(u,0),,0)=index(I-,u*)=(-1)0=1. 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 R01 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 . 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 .

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).