Analysis on a diffusive SIS epidemic system with linear source and frequency-dependent incidence function in a heterogeneous environment

1.   Introduction

In the study of transmission of infectious disease, people have realised that environmental heterogeneity and individual motility are significant factors that should be incorporated into the mathematical models. In recent decades, more and more research works have been devoted to the investigation of the dynamics of infectious disease modelled by reaction-diffusion systems in which the migration of population and environmental heterogeneity are taken into account. One may refer to ^{[1,3,4,5,6,7,8,9]} and the references therein.

In a recent paper ^[1], Allen et al. investigated a frequency-dependent SIS (susceptible-infected-susceptible) epidemic reaction-diffusion model, which reads as

Here, $S$ and $I$ stand for the density of susceptible and infected population at location $x$ and time $t$ respectively; the positive constants $d_S$ and $d_I$ represent the motility of susceptible and infected individuals, respectively; the function $\beta(x)$ is the rate of disease transmission, and $\gamma(x)$ is the recovery rate of infected individuals, all of which are positive Hölder continuous functions on $\overline \Omega$. The habitat $\Omega\subset \mathbb R^N\; {\rm(}N\geq 1{\rm)}$ is a bounded domain with smooth boundary $\partial \Omega$, and the Neumann boundary conditions mean that no population flux crosses the boundary $\partial\Omega$. For realistic implication, the initial data $S_0$ and $I_0$ are assumed to be nonnegative continuous functions on $\overline\Omega$, and there is a positive number of infected individuals, i.e., $\int_{\Omega}I_0(x)dx > 0$.

It is easily seen from (1.1) that

which means that the total number of population is conserved.

Clearly, model (1.1) dose not account into account the birth rate of the susceptible population and the death rate induced by disease. Indeed, these factors are important in the evolution of disease transmission; see ^[7,10,11]. With such a consideration, in the paper ^[12], the authors studied a varying total population model in which the linear external source term $\Lambda(x)-S$ was introduced. That is, the model (1.1) becomes the following:

where $d_S, \, d_I, \, \beta, \, \gamma, \, S$ and $I$ have the same epidemiological interpretation as in (1.1). The parameter $\theta\in [0, 1]$ represent the number of infected population becomes susceptible. The positive functions $\beta, \, \gamma, \, \Lambda$ also be assumed to be Hölder continuous over $\overline\Omega$. In the linear external source term, $\Lambda(x)$ and $-S$, respectively, account for the birth rate of the susceptible population and the disease-induced death rate. It is worth mentioning that in some cases, people ignore the effect of external source on the infected population; one may see, for instance, ^[3,9,13] for related discussion.

Different from model (1.1), a new feature in (1.2) is that the total population of susceptible and infected individuals are varying with respect to time $t > 0$. On the other hand, the works in ^{[14,15,16,17]} have shown that, in certain circumstances, the frequency-dependent incidence function $\frac{SI}{S+I}$ used in models (1.1) and (1.2) may not be appropriate to describe the transmission process of disease; instead an alternate incidence function should be $\frac{SI}{c+S+I}$, where $c$ is a positive constant. Based on model (1.2), in this paper we are led to study the following SIS epidemic model:

In model (1.3), it should be noted that we also assume that the infected population allows the same natural death rate as for the susceptible population, which is represented by the function $\mu(x)$. With such a consideration, the system (1.3) is more realistic to describe the disease transmission in some cases as suggested in ^[15,16,17]. From now on, we always assume that $c$ is a nonnegative constant, the positive function $a(x)$ stands for the recruitment rate of the susceptible corresponding to births and immigration; $a(x)$ and $\mu(x)$ are also positive Hölder continuous functions on $\overline \Omega$. All the other parameters have the same assumptions as in (1.1) and (1.2).

Since $c > 0$, the term $\frac{SI}{c+S+I}$ is a smooth function of $S$ and $I$ in the first quadrant. By the standard theory for parabolic equations, combined with our assumption on the initial data, it is well known that (1.3) admits a unique classical solution $(S, I)$ (namely, $S, \, I\in C^{2, 1}(\overline\Omega\times(0, \infty))$. Moreover, it follows from the strong maximum principle and the Hopf boundary lemma for parabolic equations that both $S(x, t)$ and $I(x, t)$ are positive for $x\in\overline\Omega$ and $t\in(0, \infty)$.

The aim of the present paper is to provide the theoretical analysis of solution to (1.3) and its steady-state (i.e., equilibrium) problem. The extinction or persistence behavior of the infectious disease in the long run is one of our main focuses. Once the disease can persist, what we are particularly interested in is the spatial distribution of the disease when the diffusion (migration) rate (represented by $d_S$ or $d_I$ in our current context) is controlled to be small. Such information will be useful for decision-makers to understand and predict the pattern of disease occurrence and then to take more effective actions/measures to eradicate diseases; one may refer to Section 5 for further discussion.

We would like to mention that many research works have been devoted to the study of the related epidemic systems; one may see, for instance, ^{[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]}.

The rest of our paper is organized as follows. In section 2, we derive the uniform bounds of the solution of (1.3) and discuss the uniform persistence in terms of the basic reproduction number. In section 3, we study the global attractivity of the disease-free equilibrium and the endemic equilibrium in some special cases. In section 4, we analyze the asymptotic profile of the endemic equilibrium (if it exists) as the diffusion coefficient $d_S$ or $d_I$ goes to zero. Section 5 ends the paper with some discussion of the epidemiological implications of the theoretical results obtained in this paper.

2.   Uniform boundedness and uniform persistence

In this section, we will establish the uniform boundedness of solutions to (1.3), and then study the uniform persistence property. In the following, we use DFE and EE to represent the disease-free equilibrium and the endemic equilibrium, respectively.

From now on, for notational simplicity, we denote

for any given function $F\in C(\overline{\Omega})$.

Lemma 2.1. Assume that $d_{S} = d_{I}$. For any solution $(S, I)$ of (1.3), there holds

Here, $\varepsilon_{1}$ is any given positive constant so that $1-\varepsilon_{1}\beta^{*} > 0$ and

Proof. Let $d_{S} = d_{I} = d$. For any given positive constant $\varepsilon_{1}$ so that $1-\varepsilon_{1}\beta^{*} > 0$, we set

In view of (1.3), we then have

It is easily seen that $V_{1}$ and $\max\{\frac{a^{*}}{m_{1}}, S^{*}_{0}+(1+\varepsilon_{1})I^{*}_{0}\}$ is a pair of upper and lower solutions to the initial-boundary value problem:

By the comparison principle for parabolic equations, we obtain

For the general case, we also derive the uniform boundedness of solution to (1.3). Indeed, we can state the following result.

Lemma 2.2. For any solution $(S, I)$ of (1.3), there exists a positive constant $M_{1}$ depending on initial data such that

Moreover, there exists some positive constant $M_{2}$ independent of initial data such that

for some large $T > 0$.

Proof. For any fixed constant $\varepsilon_{2} > 0$ such that $\mu_{*}-\varepsilon_{2}\beta^{*} > 0$, we denote

Then we set

In view of (1.3), it can be easily shown that

from which it follows that

Hence, we have

Note that $S$ and $I$ are nonnegative. Thus together with (2.4), one can use [6,Lemma 2.1] (or ^[35]) with $\sigma = p_0 = 1$ to derive (2.2). Moreover, the inequality (2.4) infers that

which is independent of initial data. Making use of [6,Lemma 2.1] again, we can derive (2.3).

In what follows, we discuss the uniform persistence of solution. To this aim, we first consider the elliptic problem:

Obviously, (2.5) admits a uniform solution $\hat{S} > 0$, and $(\hat{S}, 0)$ is a unique disease-free equilibrium of (1.3), which we call as DFE.

We then define the basic reproduce number $\mathcal{R}_0$:

Indeed, one can follow the idea of next generation operators in ^[35] to introduce the basic reproduction number, which coincides with the value $\mathcal{R}_0$.

It should be noticed that the basic reproduction number $\mathcal R_{0}$ defined by (2.6) implicitly depends on the diffusion rate $d_{S}$ of the susceptible population; this qualitatively differs from the basic reproduction number $\mathcal R_{0}$ defined in ^[1] and ^[12].

Let $\lambda_{0}$ be the principal eigenvalue of the following eigenvalue problem

Then, we have the following proposition; the proof is the same as [1,Lemma 2.3] and is omitted here.

Proposition 2.3. The following statements hold.

${\rm (a)}$ $\mathcal{R}_0$ is a monotone decreasing function of $d_I$ with $\mathcal{R}_0\rightarrow\max_{x\in \overline{\Omega}}\frac{\beta\hat{S}}{(c+\hat{S})(\gamma+\mu)}$ as $d_I\rightarrow0$ and $\mathcal{R}_0\rightarrow\int_\Omega \frac{\beta\hat{S}}{(c+\hat{S})}/{\int_\Omega (\gamma+\mu)}$ as $d_I\rightarrow\infty$.

${\rm (b)}$ If $\int_\Omega \frac{\beta\hat{S}(x)}{c+\hat{S}(x)}\mathit{\mbox{d}}x < \int_\Omega (\gamma(x)+\mu(x))\mathit{\mbox{d}}x$, and $\frac{\beta\hat{S}}{c+\hat{S}}-(\gamma+\mu)$ changes sign, then there exists a threshold value $d_I^*\in(0, \infty)$ so that $\mathcal{R}_0 < 1$ for $d_I > d_I^*$ and $\mathcal{R}_0 > 1$ for $d_I < d_I^*$.

${\rm (c)}$ If $\int_\Omega \frac{\beta\hat{S}(x)}{c+\hat{S}(x)}\mathit{\mbox{d}}x\geq\int_\Omega (\gamma(x)+\mu(x))\mathit{\mbox{d}}x$, then $\mathcal{R}_0 > 1$ for all $d_I$.

${\rm (d)}$ $\mathcal{R}_0 > 1$ when $\lambda^{*} < 0$, $\mathcal{R}_0 = 1$ when $\lambda^{*} = 0$, and $\mathcal{R}_0 < 1$ when $\lambda^{*} > 0$.

${\rm (e)}$ $\mathcal{R}_0$ is a monotone decreasing function of $c$, and $\mathcal{R}_0 < 1$ if $c > c^*$ for some $c^*\geq0$.

Proposition 2.4. If $\mathcal{R}_0 > 1$ then the DFE $(\hat{S}, 0)$ is unstable, and if $\mathcal{R}_0 < 1,$ it is stable.

The proof of Proposition 2 is similar to the proof of [1,Lemma 2.4] and hence the details are omitted.

In view of (2.3) of Lemma 2.2, we can establish the uniform persistence of (1.3) when $\mathcal{R}_0 > 1$. In fact, according to the theory developed by Magal and Zhao (see [36,Theorem 4.5] or ^[37]), we can conclude the following theorem.

Theorem 2.5. If $\mathcal{R}_0 > 1$, then there exists some real number $\eta > 0$ independent of the initial data, such that any solution $(S, I)$ of (1.3) satisfies

and hence, the disease persists uniformly. Furthermore, (1.3) admits at least one $EE$ when $\mathcal{R}_0 > 1$.

3.   Global attractivity of the DFE and EE

This section is devoted to the study of the global attractivity of the DFE and EE of (1.3). In the first subsection, we obtain the global attractivity of the DFE in the the nonhomogeneous environment. In the second subsection, we will derive that the global attractivity of the EE in the homogeneous environment.

3.1.   Global attractivity of the DFE

For later purpose, we need a useful lemma; see [38,Lemma 2.5.1].

Lemma 3.1. Let $a_1$ and $a_2 > 0$ be any constants. Assume that $\varphi, \psi\in C^{1}([a_1, \infty)), \psi(t)\geq0$ in $[a_1, \infty)$ and $\varphi$ is bounded from below. If $\varphi'(t)\leq -a_2\psi(t)$ and $\psi'(t)\leq K$ in $[a_1, \infty)$ for some constant $K$, then $\lim_{t\rightarrow\infty}\psi(t) = 0$.

Our main result of this subsection reads as follows.

Theorem 3.2. The DFE $(\hat{S}, 0)$ is globally attractive if one of the following conditions holds:

(ⅰ) $\beta(x)\leq \gamma(x)+\mu(x), \ \forall x\in\overline{\Omega};$

(ⅱ) $c > c^*$ for some constant $c^* > 0$.

Proof. We first handle case (ⅰ). To verify our result, we construct the following Lyapunov function

Hereafter $(S, I)$ is the solution of (1.3).

Then some elementary calculation yields

This motivates us to define

Due to the Lemma 2.2, we know that both $\|S(\cdot, t)\|_{L^\infty(\Omega)}$ and $\|I(\cdot, t)\|_{L^\infty(\Omega)}$ are bounded. Thus, by [39,Theorem A2], we have

for some constant $C_{0} > 0$. In addition, using the second equation of (1.3), we can see that $\psi'(t)$ is bounded from above for $t\in[1, \infty)$. In view of Lemma 3 (by taking $\varphi(t) = V(t)$), we can conclude that

Moreover, in light of (3.1), it is clear that the set $\{I(\cdot, t):\ t\geq1\}$ is compact in $C^{2}(\overline{\Omega})$. Combining this fact with (3.2), we assert that

Thus, for any small $\epsilon > 0$, it follows that

for some large $T$.

Using the above fact, it is easy to find that $S$ is a lower solution of the following parabolic problem:

Let $w_{1}$ be the solution of (3.3). Then by the comparison principle, we have

Similarly, we can find that $S$ is an upper solution to

Thus, by letting $w_{2}$ be the solution of (3.4), we have

It is standard to show that problem (3.3) and problem (3.4) exist a unique positive steady state, denoted by $\hat{S}_{-}(\varepsilon, x)$ and $\hat{S}_{+}(\varepsilon, x)$, respectively. Moreover, we have

On the other hand, it is easy to check that

According to the arbitrariness of $\varepsilon$, we conclude that

This proves our assertion in case (ⅰ).

We next consider case (ⅱ). First of all, one can check the proof of Lemma 2.2 and claim that

for some positive constant $C_0$, depending on the initial data but independent of $c\geq0$. Hence, by (3.5), it is easily noticed that $I$ is a lower solution of the following parabolic problem:

Let $w_{3}$ be the solution of (3.6). Since $C_0$ does not depend on $c$, one can take $c$ to be large so that

Then, a simple analysis, together with the parabolic comparison principle, shows that

Now, in view of the above assertion, a similar argument as in case (ⅰ) allows us to conclude that

This proves our assertion in case (ⅱ). The proof is complete.

3.2.   Global attractivity of the EE

In this subsection, we will consider the global attractivity of the EE by assuming that all of the parameters $a, \beta, \mu$ and $\gamma$ are positive constant (that is, the environment is spatially homogeneous).

In this situation, in view of (3.1), we can see that the unique DFE is given by $(\hat{S}, 0) = (\frac{a}{\mu}, 0)$. And the unique EE $(\tilde{S}, \tilde{I})$ exists if and only if $\mathcal{R}_0 = \frac{\beta\frac{a}{\mu}}{(\gamma+\mu)(c+\frac{a}{\mu})} > 1$, where

Our result is stated as follows.

Theorem 3.3. Assume that the parameters $a, \beta, \mu$ and $\gamma$ are positive constant and $d_{S} = d_{I}$. If $\mathcal{R}_0 > 1$, then the EE $(\tilde{S}, \tilde{I})$ is globally attractive.

Proof. By setting $d_{S} = d_{I} = d$, we construct the following Lyapunov functional

where

For simplicity, we note

Then simple calculation gives

Moreover, integrating by parts, we have

It is easy to see that

Thus

In addition, by direct computations, we have

Here, we have used the fact:

According to (3.7), we thus derive that ${W}'(t)\leq0$, $\forall t > 0$ along all trajectories. By a standard argument, we can obtain

Due to the Lemma 2.2, we know that both $\|S(\cdot, t)\|_{L^\infty(\Omega)}$ and $\|I(\cdot, t)\|_{L^\infty(\Omega)}$ are bounded. As a consequence, by [39,Theorem A2], we have

Thus, $\{(S(\cdot, t), I(\cdot, t)):\ t\geq1\}$ is compact in $C^{2}(\overline{\Omega})\times C^{2}(\overline{\Omega})$. Then, combining this with the above $L^{2}$-convergence, we obtain that

Thus, the EE $(\tilde{S}, \tilde{I})$ is globally attractive.

4.   Asymptotic profiles of the EE

In this section, we consider the asymptotic behavior of the positive solution of the following elliptic system:

when $d_{S}$ or $d_{I}$ goes to zero.

4.1.   The case of $d_S\rightarrow0$

Via a singular perturbational argument, it is easily seen that (2.5) has a unique positive solution $\hat{S}$ which converges uniformly to $\hat{S}_{1}(x) = \frac{a(x)}{\mu(x)}$ as $d_S\rightarrow0$. We also recall that $\lambda_{0}$ is the principal eigenvalue of problem (2.7). Then, it follows that

where $\tilde{\lambda}$ is the principle eigenvalue of the following eigenvalue problem

In order to ensure that the elliptic system (4.1) admits a positive solution, we always assume $\tilde{\lambda} < 0$ in this subsection. Then, we will investigate the asymptotic behavior of positive solution of (4.1) as $d_{S}\rightarrow0$ while $d_{I} > 0$ is fixed.

Theorem 4.1. Fix $d_I > 0$ and assume that $\tilde{\lambda} < 0$. Let $d_S\rightarrow0$, then every positive solution $(S_{d_{S}}, I_{d_{S}})$ of (4.1) satisfies (up to a subsequence of $d_S\rightarrow0$)

where

and $W_{I}$ is a positive solution of

Proof. Mentioned as before, (4.1) has at least one EE for all small $d_{S}$ when $\tilde{\lambda} < 0$. In what follows, we divide the proof into three steps to derive the conclusion.

Step 1. A priori bounds for $S, I$. Integrating the first and the second equation of (4.1) over $\Omega$, respectively, we have

and

Inserting (4.5) into (4.4) gives

From (4.6), we get

It is obvious that the $L^1$-bounds of $S$ and $I$ are independent of both $d_S$ and $d_I$.

We write the $I$-equation as follows

Then, we apply the Harnack-type inequality (refer to ^[40] or [41,Lemma 2.2]) to (4.8) to assert that

Hereafter, the positive constant $C$ is independent of $d_{S} > 0$, and it may vary from place to place.

By (4.7) and (4.9), we can derive

Step 2. Convergence of $I$. From (4.10), we get

Applying the standard $L^p$-estimate for elliptic equations (^[42]), it follows that

Then, taking $p$ to be sufficiently large and using the embedding theorem ^[42], we can see that

Hence, there exist a subsequence of $d_S\to 0$, say $d_i: = d_{S, i}$, satisfying $d_i\to 0$ as $i\rightarrow\infty$, and a corresponding positive solution $(S_i, I_i): = (S_{d_{S, i}}, I_{d_{S, i}})$ of (4.1) with $d_S = d_i$, such that

where $W_{I}\in C^1(\overline{\Omega})$ and $W_{I}\geq 0$. By (4.9), we know that

Suppose that $W_{I}\equiv 0$. That is,

Then for sufficiently small $\varepsilon$ with $0 < \varepsilon < \min_{x\in\overline{\Omega}}a(x)$, we have $0\leq I_{i}(x)\leq\varepsilon, \forall x \in\overline{\Omega},$ for all large $i$. Combining this fact with the first equation of (4.1), for all large $i$, one sees that $(S_{i}, I_{i})$ satisfies

This leads us to consider the following auxiliary system:

It is clear that (4.13) admits a unique positive solution, denoted by $u_{i}$. A simple upper and lower solution argument guarantees that

Similarly, for all large $i$, $(S_{i}, I_{i})$ satisfies

We also consider the following auxiliary system:

Let $v_{i}$ denote the unique positive solution of (4.15). Similarly as before, we have

By a singular perturbation argument as in [43,Lemma 2.4], it is easy to show that

Hence, sending $i\rightarrow \infty$, by (4.14) and (4.16), we find

Due to the arbitrariness of $\varepsilon$, we obtain that

We now consider the second equation of (4.1) and then know that $I_{i}$ satisfies

Define $\hat{I}_i: = \frac{I_i}{\|I_i\|_{L^\infty(\Omega)}}$. Then $\|\hat{I}_i\|_{L^\infty(\Omega)} = 1$ for all $i\geq1$, and $\hat{I}_i$ solves

As before, using the standard compactness argument for the elliptic equation, after passing to a further subsequence if necessary, we assume that

where $\hat{I}\in C^1(\overline{\Omega})$ with $\hat{I}\geq0\ \mbox{on}\ \overline{\Omega}$ and $\|\hat{I}\|_{L^\infty(\Omega)} = 1$.

Together with the fact $I_i \rightarrow 0$ uniformly on $\overline{\Omega}$ as $i\rightarrow \infty$, from (4.17) and (4.19), one can show that $\hat{I}$ fulfills

Applying the Harnack-type inequality (^[40] or [41,Lemma 2.2]) to (4.20), we obtain $\hat{I} > 0$ on $\overline{\Omega}$. This implies that $\tilde{\lambda}$ of the eigenvalue problem (4.2) must be zero. It contradicts our assumption that $\tilde{\lambda} < 0$. Hence, the latter always holds in (4.12). That is

Step 3. Convergence of $S$. Observe that $S_{i}$ fulfills

From (4.21), given any small $\varepsilon > 0$, it holds

for all large $i$. For simplicity, denote $W_{\pm, I} = W_{I}\pm\varepsilon$. Hence, we have

where

for all large $i$.

Notice that $J_{+}^{1, \epsilon} > 0$ and $J_{-}^{1, \epsilon} < 0$ on $\overline\Omega$. Then, given large $i$, we consider the following auxiliary elliptic problem:

It is easily observed that $S_{i}$ is a lower solution of (4.23). And any sufficiently large constant $C > 0$ satisfies $S_{i}\leq C$ is an upper solution of (4.23). Therefore, (4.23) admits at least one positive solution, denoted by $Z_{i}$, which satisfies $S_{i}\leq Z_{i} \leq C$ on $\overline{\Omega}$. By similar arguments as in the proof of [43,Lemma 2.4], we find that

Since $S_{i}$ is a lower solution of (4.23), we have

On the other hand, from (4.22), for all large $i$, we have

where

with $J_{+}^{2, \epsilon}(x, W_{I}(x)) > 0$ and $J_{-}^{2, \epsilon}(x, W_{I}(x)) < 0$ on $\overline{\Omega}$.

As before, for any given large $i$, we also consider the following auxiliary elliptic problem:

Observe that $S_{i}$ and $0$ is a pair of upper and lower solution of (4.25). Hence, we can assert that (4.25) admits at least one positive solution. Using a similar argument as before, we further get

Notice that

Due to the arbitrariness of $\varepsilon$, by (4.24) and (4.26), we derive that

In addition, by (4.18), we can easily see that $W_{I}$ satisfies (4.3). The proof is complete.

4.2.   The case of $d_I\rightarrow0$

In this subsection, we will analyze the asymptotic behavior of positive solution of (4.1) as $d_{I}\rightarrow0$ with $d_{S} > 0$ being fixed. According to Proposition 2.3(a) and Theorem 2.5, we need to assume that $\{\beta(x) \hat{S}(x)/c+\hat{S}(x) > \gamma(x)+\mu(x):\ x\in\overline\Omega\}$ is nonempty so that (4.1) has positive solution for all small $d_{i}$.

As usual, we denote $g^{+} = \max\left\{g, 0\right\}$. Our main result can be stated as follows.

Theorem 4.2. Fix $d_S > 0$ and assume that $\{\beta(x)\hat{S}(x)/c+\hat{S}(x) > \gamma(x)+\mu(x):\ x\in\overline\Omega\}$ is nonempty. Let $d_I\rightarrow0$, then every positive solution $\left(S_{d_I}, I_{d_I}\right)$ of (4.1) fulfills

where $W^{S}$ is the unique positive solution of

and $W^{I}$ is a nonnegative function

Proof. We divide the proof into four steps for clarity. In the following, let us denote $m$ to be a positive constant which does not depend on $d_{I} > 0$ and may vary from line to line.

Step 1. Lower bound of $S$. Pick $x_{1}\in\overline{\Omega}$ so that $S(x_{1}) = \min_{\overline{\Omega}}S(x)$. In light of the first equation of (4.1), it follows from [44,Proposition 2.2] that

By (4.29), we obtain

from which we further have

Step 2. $W^{1, q}$-bound of $S$ for some $q\geq1$. We now write the $S$-equation as

From (4.7), we derive $\int_{\Omega}|a(x)+\gamma(x)I| dx\leq m.$ Hence, by the $L^1$-estimate theory for elliptic equations (see [45,Lemma 2.2] or ^[46]), we get

Step 3. $L^p$-bound of $S$ and $I$. In view of (4.32), by the Sobolev embedding theorem, we can see that $W^{1, q}(\Omega)$ is compactly embedded into $L^{p_{0}}(\Omega), \ \forall p_0\in[1, N/(N-q))$. This implies that

As $q$ is close to $N/(N-1)$, it is clear that

Notice that (4.33) holds for any $1 < p_0 < \infty$ when $N = 2$.

We now multiply the second equation of (4.1) by $I^{k}$ for any fixed $k > 0$ and then integrate by parts to obtain that

By direct calculation, we get

Taking $\frac{1}{p_{0}}+\frac{1}{q_{0}} = 1$ (notice $\frac{1}{q_{0}-1} = p_{0}-1$) and $k_{0} = \frac{1}{q_{0}}$, by (4.7), (4.33), (4.34) and Hölder inequality, we have

This means that

Then, we take $k_1 = (k_0+1)/q_0 = 1/q_0+1/q_0^2$. As before, by (4.34) and Hölder inequality, together (4.33) and (4.35), we infer that

that is,

Repeating the iteration as above, we can easily see that

where

Notice that (4.36) can be deduced through finitely many times of iterations with the help of (4.33). Thus, we deduce

Combining (4.33) and (4.37), from the equation (4.31) by using the well-known $L^{p}$-theory, one can assert that

By the Sobolev embedding theorem again, $W^{2, p_{0}}(\Omega)$ is compactly embedded into $L^{p_{1}}(\Omega), \ \forall p_{1}\in(1, Np_{0}/(N-2p_{0}))$. Observe that $\frac{Np_0}{N-2p_0}\to\frac{N}{N-4}$ as $p_0\to\frac{N}{N-2}$ (see (4.33)). Thus, we have

By a similar argument as in deducing (4.37), one gets

Making use of (4.31), the Sobolev embedding theorem and the well-known $L^{p}$-theory repeatedly, we can eventually conclude that

Step 4. Convergence of $S$ and $I$. According to (4.38), for the equation (4.31), it holds that

Then taking sufficiently large $p$, the standard embedding theorem enables us to conclude that, up to a sequence of $d_{I}\rightarrow 0$, denoted by $d_j: = d_{I, j}$, with $d_{j}\rightarrow 0$ as $j\rightarrow \infty$, the corresponding positive solution sequence $(S_j, I_j): = (S_{d_{I, j}}, I_{d_{I, j}})$ of (4.1) with $d_I = d_{I, j}$ fulfills

where $W^{S}\in C^{1}(\overline{\Omega})$ and $W^{S} > 0$ on $\overline{\Omega}$ due to (4.30). Observe that $I_{j}$ fulfills

In light of (4.39) and (4.40), using a simple upper and lower solution similarly as in step 3 of Theorem 4.1, we have

where $W^{I}$ is given by (4.28).

It is clear that $W^{S}$ satisfies (4.27). Moreover, by the expression of $W^{I}$, we can see that (4.27) admits a unique positive solution (refer to [47,Lemma A.1]). Thus, we can conclude that all the above limits hold without passing to a subsequence. This proof is complete.

5.   Discussion

In this paper, we have studied the SIS reaction-diffusion model (1.3) in which we have taken into account the natural mortality of the susceptible and infected populations. First of all, we have established the uniform bounds of solution to (1.3); see Lemma 2.1 and Lemma 2.2. Then, we define the basic reproduction number $\mathcal{R}_0$ associated with (1.3):

It is worth mentioning that $\mathcal{R}_0$ depends on the diffusion rates $d_{S}$ and $d_{I}$ when $c > 0$, while $\mathcal{R}_0$ depends only on the diffusion rate $d_{I}$ when $c = 0$. Thus, compared with the model (1.2) (i.e., $c = 0$), the parameters $d_S$ and $c$ play vital roles in the dynamics of the infectious disease in (1.3). In particular, we have proved that $\mathcal{R}_0$ is decreasing with respect to $c\in[0, \infty)$, and when $c$ is larger than a value, the basic reproduction number $\mathcal{R}_0 < 1$ so that the disease dies out in the long run; see Theorem 3.1(ⅱ).

In another special case that $\beta(x)\leq \gamma(x)+\mu(x), \ \forall x\in\overline{\Omega}$, we have proved the global stability of the disease-free equilibrium via a Lyapunov function method; see Theorem 3.1. On the other hand, when the spatial environment is homogeneous, that is, all the parameters in (1.3) are positive constants, we have shown the global stability of the endemic equilibrium provided that the diffusion rates are equal and the basic reproduction number $\mathcal{R}_0 > 1$; see Theorem 3.3. This result means that the disease will persist all the time. In the general situation of spatially heterogeneous environment, once $\mathcal{R}_0 > 1$, the uniform persistence property has been proved so that the disease exists eventually in the whole habitat; refer to Theorem 2.5. We suspect that the uniform persistence property holds if $\mathcal{R}_0 > 1$ whereas the disease extinction occurs if $\mathcal{R}_0\leq1$; this is a challenging problem and deserves future investigation.

According to Theorem 2.5 as well as the above discussion, once $\mathcal{R}_0 > 1$, (1.3) has an endemic equilibrium exists and the infectious disease will uniformly persist in space, and vice versa. Therefore, it becomes important to understand how the heterogeneity of spatial environment and the mobility of population dispersal (reflected by the change of the migration rates $d_I$ and $d_S$) prescribe the spatial profile of the endemic equilibrium, because this will help decision-makers to predict the pattern of disease occurrence and henceforth to conduct effective/optimal control strategies of disease eradication. This leads us to explore the asymptotic behavior of endemic equilibrium with respect to small diffusion rate $d_S$ or $d_I$, which in turn will tell us the spatial distribution of susceptible and infected population

Theorems 4.1 and 4.2 show that as the mobility of the susceptible or infected population goes to zero, the infectious disease will always exist in space at least in some region. Especially, it follows from Theorem 4.1 that the disease exists in the entire habitat even if the mobility of the susceptible is restricted to be small enough. This result is in sharp contrast with that of (1.1), as shown in ^[1] by Allen et al, where they proved that as the mobility of the susceptible goes to zero, the density of the infected population will vanish and so the disease dies out eventually. However, Peng in ^[12] showed that, for the model (1.2) which also includes the linear external source term $\Lambda(x)-S$, the density of the infected population will not vanish when the mobility of the susceptible or infected population goes to zero; that is, the infectious disease will always exist.

The above results suggest that simply controlling the migration rate of the susceptible or infected population can not eliminate the disease modelled by (1.2) and (1.3). In other words, the presence of the linear external source term $\Lambda(x)-S$ enhances the persistence of disease and the infectious disease will become more threatening and hard to control. As a consequence, in such a situation, more effective measures should be taken to eradicate diseases.

Acknowledgment

We would like to express our sincere thanks to the referees for their careful reading and valuable suggestions, which improve the presentation of the manuscript.

Conflict of interest

The authors declare there is no conflicts of interest.