Global behavior of solutions to an SI epidemic model with nonlinear diffusion in heterogeneous environment

1.   Introduction

In this paper, we study the following strongly coupled reaction-diffusion model

where $u(x, t)$ and $v(x, t)$ represent susceptible and infected individuals' density respectively at location $x$ and time $t$; the positive constants $d_{1}$ and $d_{2}$ denote the corresponding diffusion rates for the susceptible and infected populations; and $\beta(x)$ and $\lambda(x)$ are positive Hölder continuous functions on $\Omega\subset \mathbb{R}^{n}$ which account for the rates of disease transmission and disease recovery at $x$, respectively; $\rho$ and $\gamma$ are nonnegative constant, refers to the spatial influence of infectives, $\rho$ is called cross diffusion coefficient. The term positive cross diffusion coefficient denotes that the susceptible tends to diffuse in the direction of higher concentration of the infected. The density-dependent diffusion terms, given by $\gamma u$. This form of the diffusion term was experimentally motivated ^[1] and can be interpreted as a collective behavior for infected populations whose activity increases significantly if they are numerous at a spot. For more details on the biological background, see [1,p.172]. The system is strongly-coupled because of the coupling in the highest derivatives in the first equation. Strongly-coupled systems occur frequently in biological and chemical models and they are notoriously difficult to analyze.

The homogeneous Neumann boundary conditions mean there is no population flux across the boundary $\partial \Omega$ and both the infected and susceptible individuals live in the self-contained environment. From the biological point of view, the incidence function $h(u, v)$ is assumed to be continuously differentiable in $\mathbb{R}_{+}^{2}$ and satisfies the following hypotheses $\mathrm{(H)}$:

$\mathrm{(i)}$ $h(u, 0) = h(0, v) = 0$, for all $u, v\geq0$;

$\mathrm{(ii)}$ $h(u, v) > 0$, for all $u, v > 0$;

$\mathrm{(iii)}$ $\frac{\partial h(u, v)}{\partial u} > 0$, for all $u\geq0$, $v > 0$;

$\mathrm{(iv)}$ $\frac{\partial h(u, v)}{\partial v}\geq0$, for all $u, v\geq0$.

It is easy to check that class of functions $h(u, v)$ satisfying $\mathrm{(H)}$ include incidence functions such as

To the best of our knowledge, there are very few publications (see, for example, ^[1] and ^[7]) that consider a SI model with cross-diffusion and density-dependent diffusion. Their model is written by

In ^[7], Kirane and Kouachi showed the existence of global solutions of (1.2) when $d_{1}\geq d_{2}$. However, the structure of the nonlinear diffusion terms ($\rho\equiv\gamma = \alpha$) and the reaction terms in the system (1.2) in those works is different than in (1.1).

The coefficient in the model (1.2) are all spatially-independent. However, it has been shown that environmental heterogeneity can make a great difference to infections disease. There has been considerable an SIS epidemic model with heterogeneous environment^[21,22]. On the hand, pattern formation, anomalous diffusion, nonlocal dispersal and chemotaxis effect of the epidemic models are paid more and more attention^{[23,24,25,26,27]}. In recent years, the fractional order epidemic models has attracted great interests(see, for example, ^{[36,37,38,39,40,41,42,43]}).

Here we mention that global existence and boundedness of classical solutions to SKT competition systems with cross-diffusion

For the system (1.3), it is straightforward to find out that maximum principles can be applied to the second equation of (1.3) to obtain the boundedness of $v$. Then the key issue is to establish the boundedness for $u$. However, for the first equation of (1.3), the boundedness of $u$ cannot be obtained directly by using the maximum principle. This is the biggest obstacle in studying the global existence of system (1.3).

Global existence (in time) of (1.3) has recently received great attention^{[8,9,28,29,30,31,32,33,34,35]}, The global existence is proved for $n = 2$ by Lou-Ni-Wu^[32], thereafter for $n\leq5$ by Le-Nguyen-Nguyen^[28] and Choi-Liu-Yamada^[8], for $n \leq 9$ by phan^[9], and the uniform boundedness was asserted when $n \leq 9$ and $\Omega$ is convex by Tao-Winkler^[34]. In these papers, to get the boundedness of the solution of (1.3), authors first obtained $L^p$ -estimates of the solution and then used the Sobolev embeddings. Therefore, they have a restriction on the dimension $n$ of $\Omega$. Recently, the global existence is proved for arbitrary $n \geq 1$ by Hoang-Nguyen-Pan ^[29]. Their first obtained $L^p$ -estimates for $\nabla v$ for large $p$, and then obtained $L^p$ -estimates of $u$ for large $p$. In a different approach, Phan^[30] who proved the existence of global solutions of (1.3) without any restrictions on space dimension, but with some restrictions on the amplitude of cross-diffusion coefficient. the authors introduce a new function $w$ of the form $w = G(u, v)$ and then use maximum principles to obtain the boundedness of the solution $u$ (1.3). Using test function techniques, Le-Nguyen^[31] obtained some global existence results for $n\geq1$.

Here we should stress that the assumption $a_{11} > 0$ plays a crucial role in the analysis in the aforementioned works. When $a_{11} = 0$, whether the solution of the system (1.3) exists globally in time for $n \geq 1$ is still a well-known open problem made by Y. Yamada in ^[33]. Liu and Tao^[35] recently established the existence of global classical solutions for a simplified parabolic-elliptic system (1.3) when $a_{11} = 0$. However, parabolic-parabolic system (1.3) is still a open problem $a_{11} > 0$.

We also remark that while there have been many results on global solutions to cross-diffusion systems, such as ^{[8,9,10,11,12]}. However, in ^{[8,9,10,11,12]}, the authors utilize the fact that one component is 'trivially' uniformly bounded and use it to bound the other component(s).

To understand the global dynamics of the system (1.1), an crucial step is to establish the existence of classical solutions of (1.1). Since the dispersal includes cross-diffusion and density-dependent diffusion, the global well-posedness of system (1.1) is nontrivial.

We would like to mention that (1.1) is very similar to SKT system with cross-diffusion (1.3) when $a_{11} = 0$. Nevertheless, maximum principles can be not applied to the second equation of (1.1) to obtain the boundedness of $v$ We would like to stress that their approachs for SKT systems (1.3) cannot be applied to system (1.1). Our approach first obtain $L^2$ -estimates for $u$, $v$, $\nabla u$ and $\nabla v$, then introduce a new function $L(u, v)$, and this function allows us to use maximum principles to get the boundedness of the solution $u$ and $v$.

Main results. The purpose of this paper is to establish the global existence of classical solutions to (1.1) under heterogeneous environment and the large time behaviour of solution to (1.1) under homogeneous environment. Precisely, we prove the following results:

Theorem 1.1. Assume that $u_{0}, v_{0} > 0$satisfy the zero Neumann boundary condition and belong to$C^{2+\delta}(\overline{\Omega})$, and suppose $\beta(x), \lambda(x) \in C^{2+\delta}(\overline{\Omega})$ for some $0 < \delta < 1$.Then $(1.1)$ possesses a unique non-negative solution $u, v\in C^{2+\delta, 1+\frac{\delta}{2}}(\overline{\Omega}\times[0, \infty))$ if $d_{1}\geq d_{2}$ and $\rho\leq\gamma$.

Theorem 1.2. Assume that $d_{1}\geq d_{2}$, $\rho\leq\gamma$ and $\beta$, $\lambda$ are positive constants. Then, the problem $(1.1)$ possesses a unique non-negative global classical solutions $(u, v)$ which satisfies

Remark 1.3. Theorem 1.1 also valid for $(1.1)$ but with homogeneous Dirichlet boundary condition.

Remark 1.4. From Theorem 1.1, it is not difficult to see that the conditions $d_{1}\geq d_{2}$ and $\rho\leq\gamma$ play crucial roles in the study of global boundedness of solutions to problem $(1.1)$. It is an open question whether solutions of system $(1.1)$ with bounded non-negative initial data exist globally for $d_{1} < d_{2}$ or $\rho > \gamma$ ^[7]. We believe that the conditions $d_{1}\geq d_{2}$ and $\rho\leq\gamma$ of Theorem 1.1 are just the technical conditions. To drop these conditions, more new ideas and techniques must be developed, and we expect to completely solve it in the future.

Remark 1.5. The model presented by Kirane and Kouachi in ^[7] is a particular case of our model $(1.1)$ if we choose $\rho = \gamma = \alpha\beta, h(u, v) = \beta uv$.

The paper is organized as follows. In Section 2, we introduce some known results as priminaries. In Section 3, we prove Theorem 1.1. In Section 4, the large time behaviour of solution to (1.1) are studied. In Section 5, we give an example to illustrate our theoretical results. Conclusions are drawn in Section 6.

2.   Local existence and a priori estimate

2.1.   Local existence

For the time-dependent solutions of (1.1), the local existence of non-negative solutions is established by Amann in the seminal papers ^[15,16]. The results can be summarized as follows:

Theorem 2.1. Suppose that $u_{0}$, $v_{0}$ are in $W_p^1(\Omega)$ for some $p > n$.Then $(1.1)$ has a unique non-negative smooth solution$u, v$ in

with maximal existence time $T$.Moreover, if the solution $(u, v)$ satisfies the estimate

then $T = \infty$.

We denote

$T$ be the maximal existence time for the solution $(u, v)$ of (1.1).

Let $Z$ be a Banach space and $a\in R^+$, $C_{B}([a, +\infty), Z)$ denote the space of continuous functions such that remains bounded in $Z$ for $t > a$. In order to prove Theorem 1.1, we need the following some preliminary Lemmas.

2.2.   Apriori estimates

Lemma 2.2. Let $3 < p < \infty$. Suppose $w$ is a solution to the followingequation:

where $T < \infty$ and $\{a^{ij}(x, t)\}_{i, j = 1, \ldots, N}$ are boundedcontinuous functions on $\overline{Q}_{T}$ satisfying

where $\lambda, \Lambda$ are positive constants. Suppose $h\in L^{p}(\overline{Q}_{T})$. Then there exists a constant $C_{p}$depending on the bounds of $\{a^{ij}(x, t)\}_{i, j = 1, \ldots, N}$, $\lambda, \Lambda, \Omega, T$ and $p$ such that

where the constant $C_{p}$ remains bounded for finite values of $T$and $w_{0}(x)$ satisfies the compatibility condition$\frac{\partial w_{0}}{\partial\nu} = 0$ on $\partial{\Omega}$.

This lemma can be found in [17,Theorem 9.1 p.341 and Remark on p.351].

Lemma 2.3. Let $\beta, \lambda \in C^{2+\delta}(\overline{\Omega})$, $\gamma\geq\rho$. Thenthere exists a positive constant $C$ such that

for any $T > 0$.

Proof. By the first two equations in (1.1) we derive

Choosing $\delta_{1} = \frac{\gamma}{\rho}$, $\delta_{2} = \frac{\rho}{\gamma}$, we see from condition $\gamma\geq\rho$ that

Here from (2.4) and $\beta(x), \lambda(x) > 0$ to gain the estimate

Integrating the above inequality from $0$ to $t\; (t < T)$, we have

where the constant $C$ depends only on $d_{1}, \; d_{2}, \; \gamma, \; \rho$, $\|u_{0}\|_{L^2(\Omega)}$, $\|u_{0}\|_{L^1(\Omega)}$ and $\|v_{0}\|_{L^2(\Omega)}$.

Lemma 2.4. Let $\beta, \lambda \in C^{2+\delta}(\overline{\Omega})$. For any $0 < t < T$, we have

and

whenever $d_{1} > d_{2}$ and $\rho\leq\gamma$.

Proof. Define the function

where $d = \frac{d_{2}-d_{1}}{\gamma} < 0.$

Notice that $L(u, v) > 0$ for $u, v\in \mathbb{R}^+, \quad u\neq-d, v\neq0$ and $L(-d, 0) = 0.$ Now define $E(x, t): = L(u(x, t), v(x, t))$, we have

and

Therefore

$E_{\delta}(x)$ is bounded, where $0 < \delta < T$. Since $v\leq E$ and $v\in L^{\infty}((0, +\infty); L^{2}(\Omega))$. By the maximum principle ^[19] and the proposition 3.3 of ^[18], we have

As $u+H+H\log\left(-u/H\right) > 0$, we have

and

where $M$ depends only on $\|u_{\delta}\|_{L^{\infty}}$ and $\|v_{\delta}\|_{L^{\infty}}$, and $C_{0}(M)$ and $C_{1}(M)$ are the solutions of

By (1.1), we have

Multiplying the Eq (2.10) by $\frac{1}{p}(u+v)^{p-1}$ and integrating by parts, using the Young's inequality, $\lambda(x) \in C^{2+\delta}(\overline{\Omega})$ and (2.7), we have

which implies that (2.8) holds by the Lemma 2.3.

3.   Global existence

Proof of Theorem 1.1. Now, We will divide the proof of Theorem 1.1 into two cases according to $d_{1} > d_{2}$ and $d_{1} = d_{2}$.

Case $\bf{(a)}$. $d_{1} > d_{2}$.

The second equation of (1.1) can be written as the following form

where $d_{2}+\gamma u$ and $\beta(x)h(u, v)-\lambda(x) v$ are bounded in $\overline{Q}_T$ by Lemma 2.4, $\beta, \lambda \in C^{2+\delta}(\overline{\Omega})$ and the assumption $\mathrm{(H)}$. Applying the Lemma 2.2 to the Eq (3.1) ensures that $\|v\|_{W^{2, 1}_{p}(Q_{T})}$ is bounded, which implies

where $C_{3, 1}$ is a positive constant independent $t$. It follows from the first equation of (1.1), Lemma 2.4, $\beta, \lambda \in C^{2+\delta}(\overline{\Omega})$, the assumption $\mathrm{(H)}$ and (3.2) that

where $C_{3, 2}$ is a positive constant independent $t$. Therefore, $u, v\in W^{2, 1}(Q_{T})\hookrightarrow C^{\frac{\sigma}{2}, \sigma}(\overline{Q}_{T}).$ By the Schauder theory for parabolic equations and the bootstrap argument, we have

Case $\bf{(b)}$. $d_{1} = d_{2}$.

We next consider the case $d_{1} = d_{2}$. By (1.1), we have

Since $\gamma\geq\rho$, $\beta, \lambda > 0$ and $\beta, \lambda \in C^{2+\delta}(\overline{\Omega})$, using the maximum principle ^[19] yields

where $C_{3, 3} > 0$ only dependent $\|u_{0}\|_{L^{\infty}(\Omega)}$, $\|v_{0}\|_{L^{\infty}(\Omega)}$, $d_{1}$, $\gamma$ and $\rho$. The rest of the proof is same as in the case $d_{1} > d_{2}$. Finally, by Theorem 2.1 we have $(u, v)$ exists globally in time. The proof of Theorem 1.1 is now complete.

4.   Large time behaviour

Proof of Theorem 1.2. Define the Lyapunov functional

Then

Here we use the condition $\gamma\geq\rho$ and the assumption $\mathrm{(H)}$. $\psi(t)$ is bounded by Theorem 1.1. Applying [20,Lemma 1] to (4.1), we have

From (4.2) and the Poincaré inequality, we deduce that

where $\overline{g} = \frac{1}{|\Omega|}\int_{\Omega}gdx$ for a function $g\in L^{1}(\Omega)$.

On the other hand, we claim that

To achieve this, suppose that $\|v(t)\|_{L^{2}(\Omega)}$ does not converge to $0$. Then, there would exist a number $K > 0$ and a time sequence $\{t_{m}\}_{m = 1, 2, 3, \cdots}$ tending to $\infty$ such that $\|v(t_{m})\|_{L^{2}(\Omega)}\geq K$.

In the meantime, from (1.1) and Theorem 1.1, we have

So, consider for each $m$, a continuous function $\varphi_{m}(t)$ for $0 < t < \infty$ such that $\varphi(t)\equiv0$ for $|t-t_{m}|\geq\frac{K}{M_{0}}$, $\varphi_{m}(t_{m}) = K$ for $t = t_{m}$, and $\varphi_{m}(t)$ is linear for $t_{m}-\frac{K}{M_{0}}\leq t\leq t_{m}$ and for $t_{m}\leq t\leq t_{m}+\frac{K}{M_{0}}$. Then by the mean value theorem, it must hold that $\|v(t)\|_{L^{2}(\Omega)}^2\geq\varphi_{m}(t)$ for all $-\infty < t < \infty$. Furthermore, $|v(t)\|_{L^{2}(\Omega)}^2\geq\sup_{m}\varphi_{m}(t)$. But this contradicts $\int_{0}^{\infty}|v(t)\|_{L^{2}(\Omega)}^2dt < \infty$ by (4.5).

It follows from (4.3) and (4.4) that

Thus the proof of Theorem 1.2 is completed.

5.   Examples

5.1.   Example 1 (SI model with Beddington-DeAngelis type incidence rate)

Choose $h(u, v) = \frac{uv}{1+a u+b v}$, then hypotheses $\mathrm{(H)}$ hold. System (1.1) reduces to

According to Theorem 1.1 and Theorem 1.2, one can obtain the following.

Theorem 5.1. Assume that $u_{0}, v_{0} > 0$satisfy the zero Neumann boundary condition and belong to$C^{2+\delta}(\overline{\Omega})$, and suppose $\beta(x), \lambda(x) \in C^{2+\delta}(\overline{\Omega})$ for some $0 < \delta < 1$.Then $(5.1)$ possesses a unique non-negative solution $u, v\in C^{2+\delta, 1+\frac{\delta}{2}}(\overline{\Omega}\times[0, \infty))$ if $d_{1}\geq d_{2}$ and $\rho\leq\gamma$.

Theorem 5.2. Assume that $d_{1}\geq d_{2}$, $\rho\leq\gamma$ and $\beta$, $\lambda$ are positive constants. Then, the problem $(5.1)$ possesses a unique non-negative global classical solutions $(u, v)$ which satisfies

6.   Conclusions

This paper presents a mathematical study on the dynamical behavior of a nonlinear diffusion SI epidemic model with general nonlinear incidence rate of the form $h(u, v)$. The functions $h(u, v)$ includes a number of especial incidence rates. For instance, $h(u, v) = u^{p}v, p\geq1$, $h(u, v) = \frac{uv}{a+v^q}, 0 < q\leq1$, $h(u, v) = \frac{uv}{av+u}$, $h(u, v) = \frac{uv}{1+a u+b v}$ and $h(u, v) = \frac{uv}{(1+au)(1+bv)}$. The well-posedness of the model, including local existence, nonnegativity, global existence of solutions under heterogeneous environment and the large time behaviour of solution to (1.1) under homogeneous environment have been established if $d_{1}\geq d_{2}$ and $\rho\leq\gamma$. Our results cover and improve some known results. However, it is an open question whether solutions of system (1.1) with bounded non-negative initial data exist globally for $d_{1} < d_{2}$ or $\rho > \gamma$. We believe that the conditions $d_{1}\geq d_{2}$ and $\rho\leq\gamma$ of Theorem 1.1 are just the technical conditions.

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable comments and suggestions, which greatly improved the presentation of this work. The first author is partially supported by National Natural Science Foundation of China (Grants nos. 11661051), Sichuan province science and technology plan project (Grants nos. 2017JY0195), Research and innovation team of Neijiang Normal University (Grants nos. 17TD04), and the second author is partially supported by Scientific research fund of Sichuan province provincial education department (Grants nos. 18ZB0319).

Conflict of interest

The authors declare no conflicts of interest.