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Stationary Patterns of a Cross-Diffusion Epidemic Model
We investigate the complex dynamics of cross-diffusion 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 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 [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 .
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 epidemic model. And in , the authors presented Turing pattern selection in a cross-diffusion 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 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 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 is a bounded domain with smooth boundary (when ), and is the outward unit normal vector on . We consider the following cross-diffusion epidemic model: where and denote the density of susceptible and infected individuals at location 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 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 , and , which is the parabolic condition.
Throughout this paper, the positive solution satisfying (1) refers to a classical one with , on . Clearly, model (1) has a semitrivial constant solution (disease-free equilibrium) and a positive constant solution (endemic equilibrium) if .
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 of the form where Since is locally Lipschitz in , for every initial date , system (5) admits a unique local solution on , where is the maximal existence time for solution of system (5) .
Set . Then model (1) leads to
Applying the comparison principle we get that . 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 , from which it is sufficient to derive a uniform estimate for . 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 .
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 and . For this purpose, we need to introduce some notations.
Let be the eigenvalues of the operator- on with the homogeneous Neumann boundary conditions. Let , be an orthonormal basis of , and ; then where .
Theorem 2. For model (1),(a)if and , the positive constant steady state is locally asymptotically stable;(b)if , the semitrivial constant steady state is locally asymptotically stable.
Proof. (a) The linearization of model (1) at the positive constant steady state can be expressed by
For each , is invariant under the operator , and is an eigenvalue of if and only if is an eigenvalue of the matrix
for some . Thus the stability of the positive constant steady state is reduced to consider the characteristic equation:
It follows from that . Therefore, the eigenvalues of the matrix have negative real parts. It thus follows from the Routh-Hurwitz criterion that, for each , the two roots and of all have negative real parts.
In the following, we prove that there exists such that Let ; then . Since as , it follows that Clearly, has two negative roots: . Let . By continuity, we see that there exists such that the two roots , of satisfy , . In turn, , . Let . Then and (14) holds for .
Consequently, the spectrum of which consists of eigenvalues lies in . In the sense of , we obtain that the positive constant steady state solution of model (1) is uniformly asymptotically stable.
(b) The stability of the semitrivial constant steady state is reduced to consider the characteristic equation: where 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 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 .
Lemma 3. If , satisfies
Proof. For all , 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 It follows by a comparison argument that The proof is complete.
Now, we give the result of the global stability of and .
Theorem 4. For model (1),(a)if and , then positive constant steady state of model (1) globally asymptotically stable;(b)if , the semitrivial constant steady state of model (1) is globally asymptotically stable.
Proof. (a) We adopt the Lyapunov function:
where , , and . Then and if and only if . Then,
It follows from Lemma 3 that, for any , there exists , such that for all and . And by some computational analysis, we have
Hence, in view of the conditions of the theorem and the arbitrariness of , we have ; that is, for all and . 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 and if and only if . Then, if , we obtain It follows from and the second equation of model (1) that is a constant. As a consequence, from the first equation of model (1) and , we have . Hence, 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
Lemma 5 (maximum principle, see ). Let be a bounded Lipschitz domain in and .(a) Assume that and satisfies If , then .(b) Assume that and satisfies If , then .
Lemma 6 (Harnack inequality, see ). Let be a positive solution to , where , satisfying the homogeneous Neumann boundary conditions. Then there exists a positive constant , such that
For convenience, let us denote the constants , and collectively by . The positive constants , and so forth will depend only on the domains and .
Theorem 7. Assume that . Let be an arbitrary fixed positive number. Then there exist positive constants and , such that if , any positive solution of model (3) satisfies
Proof. By applying Lemma 5, we have for all . Let and set . Then, by applying (i) of Lemma 5 again, we have that
Let . Then, there exists a positive constant such that provided that . It follows from Lemma 6 that there exists a positive constant such that . On the other hand, satisfies Set . Then It follows from Lemma 6 that there exists a positive constant such that . As a consequence,
Now, it suffices to verify the lower bounds of . We will verify the conclusion by a contradiction argument.
On the contrary, suppose that the conclusion is not true; then there exist sequences , , and with , , and the positive solution of model (3) corresponding to , such that It follows from Lemma 6 that satisfies Integrating by parts, we obtain that, for , By the regularity theory for elliptic equations , we see that there exist a subsequence of , which we will still denote by and two nonnegative functions , such that in as . By (39), we have that . Letting in (41) we obtain that Since , the first equation of (42) becomes . As , 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 . Let be an arbitrary constant with and . Then model (3) has no positive nonconstant solution provided that and .
Proof. Let be any positive solution of model (3) and denote . Then, multiplying the first equation of model (3) by , integrating over , we have that In a similar manner, we multiply the second equation in model (3) by to have It follows from (43), (44), and the -Young inequality that Thanks to the well-known Poincaré inequality it follows that Since and from the assumption, one can conclude that and , which asserts our results.
5.3. Existence of Positive Nonconstant Steady States
Unless otherwise specified, in this section, we always require that , which guarantees that model (3) has one positive constant solution . From now on, let us denote We also define Let and . Then we write model (3) in the form where
Define a compact operator by where is the inverse operator of subject to the zero-flux boundary condition. Then is a positive solution of model (51) if and only if satisfies
To apply the index theory, we investigate the eigenvalue of the problem where and with If is not an eigenvalue of (55), by Theorem in , the index of at is given by where 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 for some and . That is to say, is an eigenvalue of (55), and so (58), if and only if is an eigenvalue of the matrix for any . Therefore, is invertible if and only if, for any the matrix is invertible.
Let be the multiplicity of . For the sake of convenience, we denote Then if is invertible for any , with the same arguments as in , we have
To compute , we have to consider the sign of . A straightforward computation yields where
If , then has two positive solutions given by
Theorem 9. Assume that . Then if and for some and 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 such that, (i)model (3) with diffusion coefficients , and has no nonconstant solutions;(ii) for all .
By virtue of Theorem 7, there exists a positive constant such that, for , any solution of model (3) with diffusion coefficients , and satisfies , .
Set and define by where
It is clear that finding the positive solution of model (51) becomes equivalent to finding the fixed point of in . has no fixed points in for all .
Since is compact, the Leray-Schauder topological degree is well defined. From the invariance of Leray-Schauder degree at the homotopy, we deduce
Clearly, . Thus, if model (3) has no other solutions except the constant one , we have
On the contrary, by the choice of , we have that is the only solution of and therefore 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 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 determines whether there is an endemic outbreak or not: the disease free dynamics occurs if while the unique endemic steady state is globally stable if and . 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.
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).
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