Stationary Patterns of a Cross-Diffusion Epidemic Model

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.

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  epidemic model.And in [19], the authors presented Turing pattern selection in a crossdiffusion  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 reactiondiffusion 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][26][27][28][29][30][31][32][33].In particular, in the field of epidemiology, there are many contributions to the existence of steady states in the diffusive epidemic models [34][35][36][37][38][39][40][41][42][43][44][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 crossdiffusion affects disease's dynamics through studying the existence of the constant and nonconstant steady states of a cross-diffusion  epidemic model.
The rest of this paper is organized as follows.In Section 2, we derive a cross-diffusion  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

Model Derivation
Assume that the habitat Ω ⊂ R  ( ≥ 1) is a bounded domain with smooth boundary Ω (when  > 1), and n is the outward unit normal vector on Ω.We consider the following cross-diffusion  epidemic model: where (, ) and (, ) denote the density of susceptible and infected individuals at location  ∈ Ω and time , respectively,   and   are the self-diffusion coefficients for the susceptible and infected individuals, and  is the cross-diffusion coefficient. stands for the susceptible population intrinsic growth rate,  the rate of transmission,  the death rate of the infected population , and  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   ,   , and  are such that   ,   > 0,  ≥ 0 and   >   ,  2 < 4    which is the parabolic condition.
A simple application of a comparison theorem to model ( 7) implies (see [47]) that for positive initial date  0 () > 0 and  0 () ≥ 0 we have that  (, ) > 0,  () ≥    −    (, ) , ∀ ∈ Ω,  > 0. ( Applying the comparison principle we get that (, ) ≤ max{‖ 0 ‖ ∞ , }.To establish the uniform boundedness of (, ), it is sufficient to show the uniform boundedness of (, ).This task is carried out using a result found in Henry [48], from which it is sufficient to derive a uniform estimate for ‖Υ(, )‖  .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.For the sake of simplicity, we omit the proof, and the interested readers may refer to [49,50] for details.

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

Local Stability of Nonnegative Steady States.
In this subsection, we will discuss the local stability of the constant steady states  0 = (, 0) and  * = ( * ,  * ).For this purpose, we need to introduce some notations.Let 0 =  0 <  1 <  2 < ⋅ ⋅ ⋅ be the eigenvalues of the operator-Δ on Ω with the homogeneous Neumann boundary conditions.
Proof.(a) The linearization of model ( 1) at the positive constant steady state  * = ( * ,  * ) can be expressed by For each  ≥ 0, X  is invariant under the operator £, and  is an eigenvalue of £ if and only if  is an eigenvalue of the matrix for some  ≥ 0. Thus the stability of the positive constant steady state is reduced to consider the characteristic equation: where It follows from   ( 2 0 −  0 − ) + ( 0 − 1) 2 <   ( 0 − 1) that det(  ) > 0. Therefore, the eigenvalues of the matrix   have negative real parts.It thus follows from the Routh-Hurwitz criterion that, for each  ≥ 0, the two roots  1 and  2 of   () = 0 all have negative real parts.
(b) The stability of the semitrivial constant steady state  0 = (, 0) is reduced to consider the characteristic equation: where The remaining arguments are rather similar as above.The proof is complete.

Global Stability of the Nonnegative Steady
States.This subsection is devoted to the global stability of  0 = (, 0) and  * = ( * ,  * ) 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  * = ( * ,  * ).
Proof.For all  ≥ 0, (, ) is an upper solution of the following problem: Let () be the unique positive solution of the problem Then () is a lower solution of (20).Since  <  we have lim It follows by a comparison argument that lim inf The proof is complete.
where  1 () = ∫ where It follows from Lemma 3 that, for any  > 0, there exists  0 > 0, such that  +  ≥  ≥ (1 − ( 0 /)) −  for all  ∈ Ω and  ≥  0 .And by some computational analysis, we have Hence, in view of the conditions of the theorem and the arbitrariness of , we have (, ) ≤ 0; that is, / ≤ 0 for all  ∈ Ω and  ≥  0 .So () decreases monotonically along a solution orbit and  * is globally asymptotically stable under the assumptions of the theorem.(b) We adopt the Lyapunov function: Then () ≥ 0 and () = 0 if and only if  = 0.Then, if  0 ≤ 1, we obtain It follows from  = 0 and the second equation of model ( 1) that  is a constant.As a consequence, from the first equation of model (1) and  ̸ = 0, we have  = .Hence,  0 = (, 0) is globally asymptotically stable.

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).

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

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.

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 <  0 < 1 + (/), which guarantees that model (3) has one positive constant solution  * .From now on, let us denote We also define where Define a compact operator F : X + → X + by 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 To apply the index theory, we investigate the eigenvalue of the problem where Ψ = (Ψ 1 , Ψ 2 )  and F w (w * ) = (I − Δ) −1 (I + A) with 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 where  = ∑ >0   and   is the algebraic multiplicity of the positive eigenvalue  of (55).
In fact, after calculation, (55) can be rewritten as Observe that (58) has a nontrivial solution if and only if det(I +(  +1) −1 (  I − A)) = 0 for some  ≥ 0 and  ≥ 0. That is to say,  is an eigenvalue of (55), and so (58), if and only if  is an eigenvalue of the matrix (  + 1) −1 (  I − A) for any  ≥ 0. Therefore, I − F u (u * ) is invertible if and only if, for any  ≥ 0 the matrix   I − A is invertible.
To compute index (F, u * ), we have to consider the sign of (  ).A straightforward computation yields where If trace(A) 2 −4 det(A) ≥ 0, then () = 0 has two positive solutions  ± given by By virtue of Theorem 7, there exists a positive constant  = (Λ, Ω) such that, for d ≥   , any solution ((), ()) of model ( 3) with diffusion coefficients d ,   , and  satisfies  −1 < ,  < . Set and define by ) . (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, ) has no fixed points in M for all 0 ≤  ≤ 1.

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  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  0 determines whether there is an endemic outbreak or not: the disease free dynamics occurs if  0 ≤ 1 while the unique endemic steady state is globally stable if  <  and 1 <  0 < ( + )/2.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 crossdiffusion, 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].