Theoretical Analysis on a Diffusive SIS Epidemic Model with Logistic Source, Saturated Incidence Rate and Spontaneous Infection Mechanism ^†

Abstract

In this paper, we study a diffusive SIS epidemic model with a logistic source, saturation rate, and spontaneous infection mechanism. Based on the uniform bounds of parabolic system, the global attractivity of the endemic equilibrium is established in a homogeneous environment. Moreover, we prove the existence of the endemic equilibrium via the topological degree theory. We mainly analyze the effects of dispersal, saturation, and spontaneous infection on the limiting behaviors of the endemic equilibrium. These results show that spontaneous infection and the logistic source can enhance the persistence of infectious disease, causing the disease to become more threatening.

1. Introduction

It has been commonly recognized in the study of epidemiology that reaction-diffusion equations are efficient and indispensable tools for understanding the spatial spread of infectious diseases in a heterogeneous environment and for predicting and controlling the transmission of diseases in these environments [1,2,3,4,5].

In the past few decades, more works have been devoted to investigating the following diffusive SIS epidemic model:

$$\left\{\begin{array}{cc}{\overline{u}}_t=d_1\Delta \overline{u}-\beta \left(x\right)\varphi (\overline{u},\overline{v})+\gamma \left(x\right)\overline{v},\hfill & x\in \Omega ,t>0,\hfill \\ {\overline{v}}_t=d_2\Delta \overline{I}+\beta \left(x\right)\varphi (\overline{u},\overline{v})-\gamma \left(x\right)\overline{v},\hfill & x\in \Omega ,t>0,\hfill \end{array}\right.$$

(1)

where $\overline{u}(x,t)$ and $\overline{v}(x,t)$ denote the density of the susceptible and infected populations at $(x,t)$; the positive coefficients $d_1$ and $d_2$, respectively, represent the motility of susceptible and infected individuals; the positive function $\beta \left(x\right)$ is the disease transmission rate; $\gamma \left(x\right)$ is the recovery rate; and $\varphi (\overline{u},\overline{v})$ indicates the infection mechanism. In order to make the infection mechanism easier to understand, we have grouped things into three general forms: $\left(\mathrm{i}\right)$ $\varphi (\overline{u},\overline{v})=\overline{u}\overline{v}/(\overline{u}+\overline{v})$ (standard incidence rate); $\left(\mathrm{ii}\right)$ $\varphi (\overline{u},\overline{v})=\overline{u}\overline{v}$ (bilinear infection mechanism); and $\left(\mathrm{iii}\right)$ $\varphi (\overline{u},\overline{v})=\overline{u}\overline{v}/(1+\kappa \overline{v})$ (saturated incidence). In the context of epidemiology, the impact of the spatial heterogeneity of the environment and the dispersal of individuals has been studied by many authors. For example, Allen et al. [6] and Peng et al. [7,8,9] considered the SIS model with a standard incidence mechanism. The main results of their work investigated the existence, uniqueness, and global asymptotic stability of the unique endemic equilibrium in two special cases, as well as the asymptotic profile with respect to the small/large diffusion coefficient of the susceptible or infected individuals. From their results, we can assert that limiting the migration of susceptible individuals is a better control strategy than limiting the migration of infected individuals. Let us mention that for the SIS epidemic model with a bilinear infection mechanism (i.e., $\varphi (\overline{u},\overline{v})=\overline{u}\overline{v}$), the dynamics and asymptotic behaviors of endemic equilibrium have been considered in [10,11,12]. In terms of a model with a saturation mechanism, one can find an example of such a SIS epidemic model in [13]. We also refer the readers to the SIS epidemic model in advective heterogeneous environments. The authors discussed some classes of the diffusion-advection SIS epidemic equation in [14,15,16,17,18,19,20,21,22,23]. These works focused on the effects of dispersal and advection on the dynamics of infectious disease and the asymptotic profile of endemic equilibrium.

Recently, further studies on the mathematical analysis of such an SIS diffusive equation with a varying total population have been conducted.

$$\left\{\begin{array}{cc}{\overline{u}}_t=d_1\Delta \overline{u}+F(x,\overline{u})-\beta \left(x\right)\varphi (\overline{u},\overline{v})+\gamma \left(x\right)\overline{v},\hfill & x\in \Omega ,t>0,\hfill \\ {\overline{v}}_t=d_2\Delta \overline{v}+\beta \left(x\right)\varphi (\overline{u},\overline{v})-\gamma \left(x\right)\overline{v},\hfill & x\in \Omega ,t>0,\hfill \end{array}\right.$$

(2)

where $F(x,\overline{u})$ represents the birth/death effect of the susceptible population, which takes a form similar to the linear source ($F(x,\overline{u})=\Lambda \left(x\right)-a\left(x\right)\overline{u}$), logistic source ($F(x,\overline{u})=a\left(x\right)\overline{u}-b\left(x\right){\overline{u}}^2$), and so on. The dynamics and asymptotic behavior of the system have been investigated by [24,25,26,27,28,29,30,31], and their results indicate that the birth/death effect can enhance disease persistence. In [32,33], Hill et al. adapted the classic epidemic model to describe the effect of a “spontaneous” infection mechanism by adding a term whereby uninfected populations can become infected independent of direct contacts. The “spontaneous factors” can occur when the pathogen transmission to susceptible hosts occurs via environmental reservoirs (e.g., air, water, soil) or fomites, independent of direct host-to-host contact. We note that in the class of epidemic model with a spontaneous infection mechanism, the rate of spontaneous infection is proportional to the number of susceptible populations, which in turn contributes to the migration of infected populations. Thus, the spontaneous infection can be adopted independently of direct contact between the infected and susceptible individual, which indicates that there is no longer a threshold for the extinction and persistence of the infectious disease. In recent works [34,35,36,37,38], the authors considered the diffusive SIS epidemic model with a spontaneous infection mechanism and studied the effects of spatial heterogeneity and the spontaneous infection mechanism on the dynamics and spatial behavior of the diseases.

Most existing diffusion-based SIS studies focus solely on single transmission mechanisms (e.g., contact transmission) without simultaneously incorporating the logistic source and spontaneous infection mechanisms, resulting in a lack of in-depth understanding of the coupled mechanisms between resource constraints and environmental transmission.

Motivated by these models, in this work we will explore a diffusive SIS epidemic model with spontaneous infection. The model is described by the following parabolic equations:

$$\left\{\begin{array}{cc}{\displaystyle \frac{\mathit{\partial }\overline{u}}{\mathit{\partial }t}}-d_1\Delta \overline{u}=a\left(x\right)\overline{u}-b\left(x\right){\overline{u}}^2-\beta \left(x\right){\displaystyle \frac{\overline{u}\overline{v}}{1+\kappa \overline{v}}}+\gamma \left(x\right)\overline{v}-\eta \left(x\right)\overline{u},\hfill & x\in \Omega ,t>0,\hfill \\ {\displaystyle \frac{\mathit{\partial }\overline{v}}{\mathit{\partial }t}}-d_2\Delta \overline{v}=\beta \left(x\right){\displaystyle \frac{\overline{u}\overline{v}}{1+\kappa \overline{v}}}-\gamma \left(x\right)\overline{v}+\eta \left(x\right)u,\hfill & x\in \Omega ,t>0,\hfill \\ {\displaystyle \frac{\mathit{\partial }\overline{u}}{\mathit{\partial }\nu }}={\displaystyle \frac{\mathit{\partial }\overline{v}}{\mathit{\partial }\nu }}=0,\hfill & x\in \mathit{\partial }\Omega ,t>0,\hfill \\ \overline{u}(x,0)={\overline{u}}_0\left(x\right),\overline{v}(x,0)={\overline{v}}_0\left(x\right),\hfill & x\in \Omega .\hfill \end{array}\right.$$

(3)

where $\overline{u},\overline{v},\beta$, and $\gamma$ have the same interpretation as in (1), and the constant $\kappa >0$ stands for the saturation coefficient; the nonlinear term $a\left(x\right)\overline{u}-b\left(x\right){\overline{u}}^2$ represents that the susceptible individual satisfies logistic growth; $\eta \left(x\right)$ denotes the spontaneous infection rate; the habitat $\Omega$ is a bounded domain in ${\mathbb{R}}^N(N\ge 1)$ with smooth boundary $\mathit{\partial }\Omega$; and the boundary conditions mean that the habitat is closed and no individuals flux crosses the boundary $\mathit{\partial }\Omega$. We assume that $\beta ,\gamma ,a,b$, and $\eta$ are positive and Hölder continuous functions on $\overline{\Omega }$, and the nonnegative initial data $u_0$ and $v_0$ are continuous functions on $\overline{\Omega }$ which satisfy ${\int }_{\Omega }{v}_0\left(x\right)\mathrm{d}x>0$.

Corresponding to (3), the steady state problem fulfills

$$\left\{\begin{array}{cc}-d_1\Delta u=a\left(x\right)u-b\left(x\right)u^2-\beta \left(x\right){\displaystyle \frac{uv}{1+\kappa v}}+\gamma \left(x\right)v-\eta \left(x\right)u,\hfill & x\in \Omega ,\hfill \\ -d_2\Delta v=\beta \left(x\right){\displaystyle \frac{uv}{1+\kappa v}}-\gamma \left(x\right)v+\eta \left(x\right)u,\hfill & x\in \Omega ,\hfill \\ {\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }\nu }}={\displaystyle \frac{\mathit{\partial }v}{\mathit{\partial }\nu }}=0,\hfill & x\in \mathit{\partial }\Omega ,\hfill \end{array}\right.$$

(4)

where $u\left(x\right)$ and $v\left(x\right)$ denote the density of susceptible and infected populations, respectively. Moreover, from the strong maximum principle, we know that $u\left(x\right)$ and $v\left(x\right)$ are positive on $\overline{\Omega }$. A positive steady state solution $\left(u\right(x),v(x\left)\right)$ of (4) is called an endemic equilibrium of (3).

This work focuses on characterizing the asymptotic behavior of the endemic equilibrium under diverse dispersal regimes with a small/large dispersal rate of susceptible/infected contacts and a large saturated coefficient. From the biological point of view, our purpose is to explore the influence of population dispersal, infection mechanism, and environmental heterogeneity on the spatial profiles of infectious diseases. These results may have some useful implications for disease prevention and control.

This work is organized as follows. In Section 2, based on the uniform bounds of the parabolic solution of (3), we obtain the global attractivity of the endemic equilibrium in a homogeneous environment. In Section 3, using the topological degree theory, we prove the existence of the endemic equilibrium. Section 4 is devoted to analyzing the asymptotic profiles of the endemic equilibrium as the dispersal rates $d_1$ or $d_2$ approach zero or infinity, and the saturated coefficient $\kappa$ goes to infinity.

2. Uniform Boundedness and Global Stability

In this section, we will study the uniform boundedness of parabolic solutions to (3), and then investigate the global stability of endemic equilibrium in the case that $\beta ,\gamma ,a,b$, and $\eta$ are positive constants. From now on, for convenience, we write

$${\Xi }^{\ast }=\underset{x\in \overline{\Omega }}{max}\Xi \left(x\right)\mathrm{and}{\Xi }_{\ast }=\underset{x\in \overline{\Omega }}{min}\Xi \left(x\right)\mathrm{for} g=\beta ,\gamma ,a,b,\eta .$$

An iterative method is a numerical algorithm used to approximate solutions to differential equations by progressively refining an initial guess through repeated application of a specific procedure. Iterative methods are often preferred for large-scale or computationally intensive problems due to their memory efficiency and adaptability. We first apply the iteration method in [39] to establish the ultimate uniform bounds of solutions to (3). We omit the details, as the proof is similar to that of Theorem 2.3 in [26] (see also Theorem 2.3 in [37]).

Proposition 1. 

There exists a positive constant K independent of initial value such that

$$\parallel \overline{u}{(x,t)\parallel }_{L^{\infty }(\Omega )}+{\parallel \overline{v}(x,t)\parallel }_{L^{\infty }(\Omega )}\le K,\forall (x,t)\in \overline{\Omega }\times [T_0,\infty ),$$

(5)

for some large $T_0>0$.

Next, we discuss the global stability of the endemic equilibrium of (3) in a special case: the homogeneous environment. In this case, the coefficients $\beta ,\gamma ,a,b$, and $\eta$ are assumed to be positive constants. It is not hard to see that (3) has a unique endemic equilibrium, denoted by

$$(\tilde{u},\tilde{v})=\left({\displaystyle \frac{a}{b}},{\displaystyle \frac{a(\beta +\kappa \eta )-b\gamma +\sqrt{{\left[\left(\beta +\kappa \eta \right)a-\gamma b\right]}^2+4\kappa ab\eta \gamma }}{2\kappa b\gamma }}\right).$$

Using Lyapunov’s theory, $\mathcal{E}\left(t\right)$ is a Lyapunov functional for system (3), on account of the uniform boundedness (5) in Proposition 1, from the standard parabolic regularity.

Theorem 1. 

For any $d_1$, $d_2,\kappa >0$, the endemic equilibrium $(\tilde{u},\tilde{v})$ is globally attractive.

Proof. 

For any solution $(\overline{u},\overline{v})$ of (3), we consider the following functional

$$\mathcal{E}\left(t\right)={\int }_{\Omega }\mathcal{L}\left[\overline{u}(x,t),\overline{v}(x,t)\right]\mathrm{d}x,$$

with

$$\mathcal{L}(\overline{u},\overline{v})=\gamma (\overline{u}-\tilde{u})-\gamma \tilde{u}ln{\displaystyle \frac{\overline{u}}{\tilde{u}}}+\gamma (\overline{v}-\tilde{v})-\eta \tilde{u}ln{\displaystyle \frac{\overline{v}}{\tilde{v}}}-{\displaystyle \frac{\beta \tilde{u}}{\kappa }}ln{\displaystyle \frac{1+\kappa \overline{v}}{1+\kappa \tilde{v}}}.$$

Then, for all $t>0$, it follows from the elementary computations that

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}\mathcal{E}\left(t\right)}{\mathrm{d}t}}=& {\int }_{\Omega }\left[L_{\overline{u}}(\overline{u},\overline{v}){\overline{u}}_t+L_{\overline{v}}(\overline{u},\overline{v}){\overline{v}}_t\right]\mathrm{d}x\hfill \\ \hfill =& {\int }_{\Omega }\left[\left(\gamma -{\displaystyle \frac{\gamma \tilde{u}}{\overline{u}}}\right)(d_1\Delta \overline{u})+\left(\gamma -{\displaystyle \frac{\eta \tilde{u}}{\overline{v}}}-{\displaystyle \frac{\beta \tilde{u}}{1+\kappa \overline{v}}}\right)(d_1\Delta \overline{v})\right]\mathrm{d}x\hfill \\ & +{\int }_{\Omega }\left(\gamma -{\displaystyle \frac{\gamma \tilde{u}}{\overline{u}}}\right)\left(a\overline{u}-b{\overline{u}}^2-\beta {\displaystyle \frac{\overline{u}\overline{v}}{1+\kappa \overline{v}}}+\gamma \overline{v}-\eta \overline{u}\right)\mathrm{d}x\hfill \\ & +{\int }_{\Omega }\left(\gamma -{\displaystyle \frac{\eta \tilde{u}}{\overline{v}}}-{\displaystyle \frac{\beta \tilde{u}}{1+m\overline{v}}}\right)\left(\beta {\displaystyle \frac{\overline{u}\overline{v}}{1+\kappa \overline{v}}}-\gamma \overline{v}+\eta \overline{u}\right)\mathrm{d}x\hfill \\ \hfill =& -{\int }_{\Omega }\left[d_1\gamma \tilde{u}{\displaystyle \frac{|\nabla \overline{u}{|}^2}{{\overline{u}}^2}}+d_2\left({\displaystyle \frac{\eta \tilde{u}}{{\overline{v}}^2}}+{\displaystyle \frac{m\beta \tilde{u}}{{(1+\kappa \overline{v})}^2}}\right){|\nabla \overline{v}|}^2\right]\mathrm{d}x\hfill \\ & +{\int }_{\Omega }\left[\left(\gamma -{\displaystyle \frac{\gamma \tilde{u}}{\overline{u}}}\right)(a\overline{u}-b{\overline{u}}^2)-\left(\beta {\displaystyle \frac{\overline{u}\overline{v}}{1+\kappa \overline{v}}}-\gamma \overline{v}+\eta \overline{u}\right)\left({\displaystyle \frac{\beta \tilde{u}}{1+\kappa \overline{v}}}-{\displaystyle \frac{\gamma \tilde{u}}{\overline{u}}}+{\displaystyle \frac{\eta \tilde{u}}{\overline{v}}}\right)\right]\mathrm{d}x\hfill \\ \hfill =& -{\int }_{\Omega }\left[d_1\gamma \tilde{u}{\displaystyle \frac{|\nabla \overline{u}{|}^2}{{\overline{u}}^2}}+d_2\left({\displaystyle \frac{\eta \tilde{u}}{{\overline{v}}^2}}+{\displaystyle \frac{m\beta \tilde{u}}{{(1+\kappa \overline{v})}^2}}\right){|\nabla \overline{v}|}^2\right]\mathrm{d}x\hfill \\ & -{\int }_{\Omega }\left\{b\gamma {(\overline{u}-\tilde{u})}^2+{\displaystyle \frac{\tilde{u}{\left[\beta \overline{u}\overline{v}-(\gamma \overline{v}-\eta \overline{u})(1+\kappa \overline{v})\right]}^2}{\overline{u}\overline{v}{(1+\kappa \overline{v})}^2}}\right\}\mathrm{d}x\hfill \\ \hfill \le & 0.\hfill \end{array}$$

Note that for any $t>0$, ${\mathcal{E}}^{\prime }\left(t\right)<0$ along all trajectories except at $(\tilde{u},\tilde{v})$ and ${\mathcal{E}}^{\prime }\left(t\right)=0$ if and only if $(\overline{u},\overline{v})=(\tilde{u},\tilde{v})$. From the Sobolev embedding theorem and some compactness arguments in [40] (see also similar arguments in Theorem 3.1 of [41]), we can deduce

$$(\overline{u}(x,t),\overline{v}(x,t))\to (\tilde{u},\tilde{v})in{\left[L^{\infty }(\Omega )\right]}^2,ast\to \infty .$$

Clearly, $(\tilde{u},\tilde{v})$ is globally attractive. □

3. The Existence of Endemic Equilibrium

In this section, we study the existence of the endemic equilibrium of (3) when $\beta \left(x\right)$, $\gamma \left(x\right)$, $a\left(x\right)$, $b\left(x\right)$, and $\eta \left(x\right)$ are positive Hölder continuous functions on $\overline{\Omega }$. By using the topological degree approach, we establish the existence of positive solutions to (4) and resort to the auxiliary problem:

$$\left\{\begin{array}{cc}\hfill -d_1\Delta u=& \left[(1-\theta )a_0+\theta a\right]u-\left[(1-\theta )b_0+\theta b\right]u^2-{\displaystyle \frac{\left[(1-\theta ){\beta }_0+\theta \beta \left(x\right)\right]uv}{1+\kappa v}}\hfill \\ \hfill & +\left[(1-\theta ){\gamma }_0+\theta \gamma \left(x\right)\right]v-\left[(1-\theta ){\eta }_0+\theta \eta \left(x\right)\right]u,\hfill & \hfill x\in \Omega ,\\ \hfill -d_2\Delta v=& {\displaystyle \frac{\left[(1-\theta ){\beta }_0+\theta \beta \left(x\right)\right]uv}{1+\kappa v}}-\left[(1-\theta ){\gamma }_0+\theta \gamma \left(x\right)\right]v\hfill \\ \hfill & +\left[(1-\theta ){\eta }_0+\theta \eta \left(x\right)\right]u,\hfill & \hfill x\in \Omega ,\\ \hfill {\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }\nu }}=& {\displaystyle \frac{\mathit{\partial }v}{\mathit{\partial }\nu }}=0,\hfill & \hfill x\in \mathit{\partial }\Omega ,\end{array}\right.$$

(6)

where ${\beta }_0,{\gamma }_0,a_0,b_0$, and ${\eta }_0$ are given positive constants satisfying ${\eta }_0\ge {\eta }_{\ast }$, ${\gamma }_0\le {\gamma }^{\ast }$, and $a_0+{min}_{x\in \overline{\Omega }}(\gamma /\eta ){\gamma }_0>{\beta }_0/m+{\eta }_0$ and the parameter $\theta \in [0,1]$. It is clear that the system (6) with $\theta =1$ becomes (4).

Theorem 2. 

Assume

$$\underset{x\in \overline{\Omega }}{min}\left(a\left(x\right)+\underset{x\in \overline{\Omega }}{min}\left({\displaystyle \frac{\eta \left(x\right)}{\gamma \left(x\right)}}\right)\gamma \left(x\right)-{\displaystyle \frac{\beta \left(x\right)}{m}}-\eta \left(x\right)\right)>0,$$

(7)

then, (4) possesses at least one positive solution.

Proof. 1. A priori bounds for u and v. 

Assume that $\left(u\right(x),v(x\left)\right)$ is a positive solution of (6). For convenience, we denote $a_{\theta }\left(x\right):=(1-\theta )a_0+\theta a\left(x\right),b_{\theta }\left(x\right):=(1-\theta )b_0+\theta b\left(x\right),{\beta }_{\theta }\left(x\right):=(1-\theta ){\beta }_0+\theta \beta \left(x\right),{\gamma }_{\theta }\left(x\right):=(1-\theta ){\gamma }_0+\theta \gamma \left(x\right),{\eta }_{\theta }\left(x\right):=(1-\theta ){\eta }_0+\theta \eta \left(x\right)$. In view of the maximum principle ([42], Proposition 2.2), letting $u\left(x_0\right)={max}_{x\in \overline{\Omega }}u\left(x\right),v\left(x_1\right)={max}_{x\in \overline{\Omega }}v\left(x\right)$, we have

$$a_{\theta }\left(x_0\right)u\left(x_0\right)-b_{\theta }\left(x_0\right)u^2\left(x_0\right)-{\beta }_{\theta }\left(x_0\right){\displaystyle \frac{u\left(x_0\right)v\left(x_0\right)}{1+\kappa I\left(x_0\right)}}+{\gamma }_{\theta }\left(x_0\right)v\left(x_0\right)-{\eta }_{\theta }\left(x_0\right)u\left(x_0\right)\ge 0$$

(8)

and

$${\beta }_{\theta }\left(x_1\right){\displaystyle \frac{u\left(x_1\right)v\left(x_1\right)}{1+\kappa v\left(x_1\right)}}-{\gamma }_{\theta }\left(x_1\right)v\left(x_1\right)+{\eta }_{\theta }\left(x_1\right)u\left(x_1\right)\ge 0.$$

(9)

It follows from (9) that

$${\gamma }_{\theta }\left(x_1\right)v\left(x_1\right)\le {\beta }_{\theta }\left(x_1\right){\displaystyle \frac{u\left(x_1\right)v\left(x_1\right)}{1+\kappa v\left(x_1\right)}}+{\eta }_{\theta }\left(x_1\right)u\left(x_1\right)\le {\displaystyle \frac{{\beta }_{\theta }\left(x_1\right)+\kappa {\eta }_{\theta }\left(x_1\right)}{\kappa }}u\left(x_1\right),$$

then

$$v\left(x\right)\le v\left(x_1\right)\le {\displaystyle \frac{max\{{\beta }_0+\kappa {\eta }_0,{\beta }^{\ast }+\kappa {\eta }^{\ast }\}}{\kappa min\{{\gamma }_0,{\gamma }_{\ast }\}}}\underset{x\in \overline{\Omega }}{max}u\left(x\right).$$

(10)

Combining (8) and (10), we have

$$\begin{array}{cc}\hfill b_{\theta }\left(x_0\right){\left[u\left(x_0\right)\right]}^2& \le b_{\theta }\left(x_0\right){\left[u\left(x_0\right)\right]}^2+{\beta }_{\theta }\left(x_0\right){\displaystyle \frac{Su\left(x_0\right)v\left(x_0\right)}{1+\kappa v\left(x_0\right)}}+{\eta }_{\theta }\left(x_0\right)u\left(x_0\right)\hfill \\ \hfill & \le a_{\theta }\left(x_0\right)u\left(x_0\right)+{\gamma }_{\theta }\left(x_0\right)v\left(x_0\right)\le a_{\theta }\left(x_0\right)u\left(x_0\right)+{\gamma }_{\theta }\left(x_0\right)v\left(x_1\right)\hfill \\ \hfill & \le a_{\theta }\left(x_0\right)u\left(x_0\right)+{\gamma }_{\theta }\left(x_0\right){\displaystyle \frac{max\{{\beta }_0+\kappa {\eta }_0,{\beta }^{\ast }+\kappa {\eta }^{\ast }\}}{\kappa min\{{\gamma }_0,{\gamma }_{\ast }\}}}u\left(x_0\right).\hfill \end{array}$$

Via straightforward calculation, one can obtain

$$\begin{array}{c}\hfill u\left(x\right)\le {\displaystyle \frac{max\{a_0,a^{\ast }\}}{min\{b_0,b_{\ast }\}}}+{\displaystyle \frac{max\{{\gamma }_0,{\gamma }^{\ast }\}max\{{\beta }_0+m{\eta }_0,{\beta }^{\ast }+\kappa {\eta }^{\ast }\}}{\kappa min\{{\gamma }_0,{\gamma }_{\ast }\}min\{b_0,b_{\ast }\}}}=:M_1,\forall x\in \overline{\Omega },\end{array}$$

and

$$\begin{array}{c}\hfill v\left(x\right)\le {\displaystyle \frac{max\{{\beta }_0+\kappa {\eta }_0,{\beta }^{\ast }+\kappa {\eta }^{\ast }\}}{mmin\{{\gamma }_0,{\gamma }_{\ast }\}}}{M}_1:=M_2,\forall x\in \overline{\Omega }.\end{array}$$

Similarly, set $v\left(x_2\right)={min}_{x\in \overline{\Omega }}v\left(x\right),u\left(x_3\right)={min}_{x\in \overline{\Omega }}u\left(x\right)$. We deduce from the maximum principle that

$$\begin{array}{c}\hfill {\gamma }_{\theta }\left(x_2\right)v\left(x_2\right)\ge {\eta }_{\theta }\left(x_2\right)u\left(x_2\right)+{\beta }_{\theta }\left(x_2\right){\displaystyle \frac{u\left(x_2\right)v\left(x_2\right)}{1+\kappa v\left(x_2\right)}}\ge {\eta }_{\theta }\left(x_2\right)u\left(x_2\right),\end{array}$$

which implies that

$$v\left(x\right)\ge v\left(x_2\right)\ge {\displaystyle \frac{{\eta }_{\theta }\left(x_2\right)}{{\gamma }_{\theta }\left(x_2\right)}}u\left(x_2\right)\ge \underset{x\in \overline{\Omega }}{min}\left({\displaystyle \frac{\eta }{\gamma }}\right)u\left(x_3\right),\forall x\in \overline{\Omega }.$$

(11)

In the same fashion, we have

$$\begin{array}{cc}\hfill a_{\theta }\left(x_3\right)u\left(x_3\right)+{\gamma }_{\theta }\left(x_3\right)\underset{x\in \overline{\Omega }}{min}\left({\displaystyle \frac{\eta }{\gamma }}\right)u\left(x_3\right)& \le a_{\theta }\left(x_3\right)u\left(x_3\right)+{\gamma }_{\theta }\left(x_3\right)v\left(x_3\right)\hfill \\ \hfill & \le b_{\theta }\left(x_3\right){\left[u\left(x_3\right)\right]}^2+{\beta }_{\theta }\left(x_3\right){\displaystyle \frac{u\left(x_3\right)v\left(x_3\right)}{1+\kappa I\left(x_3\right)}}+{\eta }_{\theta }\left(x_3\right)u\left(x_3\right)\hfill \\ \hfill & \le b_{\theta }\left(x_3\right){\left[u\left(x_3\right)\right]}^2+\left[{\displaystyle \frac{{\beta }_{\theta }\left(x_3\right)}{\kappa }}+\eta \left(x_3\right)\right]u\left(x_3\right).\hfill \end{array}$$

As a result,

$$u\left(x\right)\ge u\left(x_3\right)\ge {\displaystyle \frac{a_{\theta }\left(x_3\right)+{min}_{x\in \overline{\Omega }}\left(\frac{\eta }{\gamma }\right){\gamma }_{\theta }\left(x_3\right)-\frac{{\beta }_{\theta }\left(x_3\right)}{\kappa }-{\eta }_{\theta }\left(x_3\right)}{b_{\theta }\left(x_3\right)}},\forall x\in \overline{\Omega }.$$

(12)

It should be noted that

$$\begin{array}{cc}& a_{\theta }\left(x_3\right)+\underset{x\in \overline{\Omega }}{min}\left({\displaystyle \frac{\eta }{\gamma }}\right){\gamma }_{\theta }\left(x_3\right)-{\displaystyle \frac{{\beta }_{\theta }\left(x_3\right)}{\kappa }}-{\eta }_{\theta }\left(x_3\right)\hfill \\ \hfill =& (1-\theta )\left(a_0+\underset{x\in \overline{\Omega }}{min}\left({\displaystyle \frac{\eta }{\gamma }}\right){\gamma }_0-{\displaystyle \frac{{\beta }_0}{\kappa }}-{\eta }_0\right)+\theta \left(a\left(x_3\right)+\underset{x\in \overline{\Omega }}{min}\left({\displaystyle \frac{\eta }{\gamma }}\right)\gamma \left(x_3\right)-{\displaystyle \frac{\beta \left(x_3\right)}{\kappa }}-\eta \left(x_3\right)\right)\hfill \\ \hfill \ge & min\left\{a_0+\underset{x\in \overline{\Omega }}{min}\left({\displaystyle \frac{\eta }{\gamma }}\right){\gamma }_0-{\displaystyle \frac{{\beta }_0}{\kappa }}-{\eta }_0,\underset{x\in \overline{\Omega }}{min}\left(a\left(x\right)+\underset{x\in \overline{\Omega }}{min}\left({\displaystyle \frac{\eta }{\gamma }}\right)\gamma \left(x\right)-{\displaystyle \frac{\beta \left(x\right)}{\kappa }}-\eta \left(x\right)\right)\right\}.\hfill \end{array}$$

From the above inequality and (12), we know that

$$\begin{array}{cc}\hfill u\left(x\right)& \ge \underset{x\in \overline{\Omega }}{min}u\left(x\right)\hfill \\ \hfill & \ge {\displaystyle \frac{min\left\{a_0+{min}_{x\in \overline{\Omega }}\left(\frac{\eta }{\gamma }\right){\gamma }_0-\frac{{\beta }_0}{\kappa }-{\eta }_0,{min}_{x\in \overline{\Omega }}\left(a\left(x\right)+{min}_{x\in \overline{\Omega }}\left(\frac{\eta }{\gamma }\right)\gamma \left(x\right)-\frac{\beta \left(x\right)}{\kappa }-\eta \left(x\right)\right)\right\}}{max\left\{b^{\ast },b_0\right\}}}\hfill \\ \hfill & =:M_3>0.\hfill \end{array}$$

Substituting the above inequality into (11), we have

$$v\left(x\right)\ge v\left(x_1\right)\ge \underset{x\in \overline{\Omega }}{min}\left({\displaystyle \frac{\eta }{\gamma }}\right)\underset{x\in \overline{\Omega }}{min}u\left(x\right)=:M_4>0.$$

(13)

Consequently, by the above arguments, let $C_0=min\left\{M_1,M_2\right\}/2$ and ${\widehat{C}}_0=2max\left\{M_3,M_4\right\}$. Thus, we know that

$$C_0\le u\left(x\right),v\left(x\right)\le {\widehat{C}}_0,\forall x\in \overline{\Omega },$$

here, $C_0$ and ${\widehat{C}}_0$ are independent of $\theta$, $d_1$ and $d_2$.

2. Existence. 

For the above $C_1$ and $C_2$, we denote

$$\mathbb{X}=\left\{(u,v)\in C^2\left(\overline{\Omega }\right)\times C^2\left(\overline{\Omega }\right):{\displaystyle \frac{\mathit{\partial }u}{\mathit{\partial }\nu }}={\displaystyle \frac{\mathit{\partial }v}{\mathit{\partial }\nu }}=0\mathrm{on}\mathit{\partial }\Omega \right\},$$

and

$$\Pi =\left\{(S,I)\in \mathbb{X}:C_0\le u,v\le {\widehat{C}}_0,\forall x\in \overline{\Omega }\right\}.$$

Thus, it is evident that (6) has no positive solution $(u,v)\in \mathit{\partial }\Theta$. For any $\theta \in [0,1]$, we consider the operator

$$\mathcal{F}(\theta ,(u,v))={(-\Delta +\mathbf{I})}^{-1}\left(\begin{array}{c}u+d_1^{-1}\left(a_{\theta }u-b_{\theta }{u}^2-{\beta }_{\theta }{\displaystyle \frac{uv}{1+\kappa v}}+{\gamma }_{\theta }v-{\eta }_{\theta }S\right)\\ v+d_2^{-1}\left({\beta }_{\theta }{\displaystyle \frac{uv}{1+\kappa v}}-{\gamma }_{\theta }v+{\eta }_{\theta }u\right)\end{array}\right),$$

here ${(-\Delta +\mathbf{I})}^{-1}$ denotes the inverse operator in $\mathbb{X}$. The standard elliptic regularity theory implies that $\mathcal{F}:[0,1]\times \Pi \mapsto C^2\left(\overline{\Omega }\right)\times C^2\left(\overline{\Omega }\right)$ is a compact operator. Note that $(u,v)\ne \mathcal{F}(\theta ,(u,v\left)\right)$ for any $\theta \in [0,1]$ and $(u,v)\in \mathit{\partial }\Theta$, it turns out that $deg(\mathbf{I}-\mathcal{F}(\theta ,(·,·)),\Pi ,0)$ is well-defined and independent of $\theta \in [0,1]$. It follows from Theorem 1 that

$$\left({\tilde{u}}_0,{\tilde{v}}_0\right)=\left({\displaystyle \frac{a_0}{b_0}},{\displaystyle \frac{a_0({\beta }_0+\kappa {\eta }_0)-b_0{\gamma }_0+\sqrt{{\left[\left({\beta }_0+\kappa {\eta }_0\right)a_0-{\gamma }_0{b}_0\right]}^2+4\kappa a_0{b}_0{\eta }_0{\gamma }_0}}{2\kappa b_0{\gamma }_0}}\right),$$

is the unique fixed point of $\mathcal{F}(0,(·,·\left)\right)$ in $\Pi$, and thus

$$deg(\mathbf{I}-\mathcal{F}(0,(·,·)),\Pi ,0)=index\left(\mathbf{I}-\mathcal{F}(0,(·,·)),\left({\tilde{u}}_0,{\tilde{v}}_0\right)\right).$$

For the system (4) (i.e., the system (6) with $\theta =1$) from Theorem 1, it is clear that $({\tilde{u}}_0,{\tilde{v}}_0)$ is stable. Hence, it follows from the standard Leray–Schauder degree index formula ([43], Theorem 2.8.1) that

$$deg(\mathbf{I}-\mathcal{F}(0,(·,·)),\Pi ,0)=index\left(\mathbf{I}-\mathcal{F}(0,(·,·)),\left({\tilde{u}}_0,{\tilde{v}}_0\right)\right)=1.$$

Therefore, it follows from the homotopy invariance property that

$$deg(\mathbf{I}-\mathcal{F}(1,(·,·)),\Pi ,0)=deg(\mathbf{I}-\mathcal{F}(0,(·,·)),\Pi ,0)=1.$$

As a consequence, by the properties of the degree, we know that $\mathcal{F}(1,(·,·\left)\right)$ possesses at least one fixed point in $\Theta$, which implies the existence of the positive solution of (4). □

4. Asymptotic Profiles of Endemic Equilibrium

In this section, we are concerned with the asymptotic profiles of endemic equilibrium to (3) in the following cases: (i) small dispersal ($d_1\to 0$ or $d_2\to 0$); (ii) large dispersal ($d_2\to \infty$ or $d_2\to \infty$); and (iii) large saturation ($m\to \infty$).

4.1. Small Dispersal

In this subsection, we study the limiting profile of the positive solutions of (4) as the dispersal rate of the susceptible/infected individual is small. First we investigate the case of a small mobility rate in susceptible individuals.

Theorem 3. 

Assume (7) holds. Fix $d_2$, $\kappa >0$, and let $d_1\to 0$ (up to a subsequence), then every positive solution $\left(u\right(x),v(x\left)\right)$ of (4) fulfills

$$(u\left(x\right),v\left(x\right))\to \left({\Phi }_u\left(x\right),{\Phi }_v\left(x\right)\right)\mathit{uniformly}\mathit{on}\overline{\Omega },$$

where

$$\begin{array}{cc}\hfill {\Phi }_u\left(x\right)=& G(x,{\Phi }_v\left(x\right))\hfill \\ \hfill =& {\displaystyle \frac{(a-\eta )(1+\kappa {\Phi }_v)-\beta {\Phi }_v+\sqrt{{[(a-\eta )(1+\kappa {\Phi }_v)-\beta {\Phi }_v]}^2+4b\gamma {\Phi }_v{(1+\kappa {\Phi }_I)}^2}}{2b(1+\kappa {\Phi }_v)}},\hfill \end{array}$$

and ${\Phi }_v\left(x\right)$ is a positive solution of

$$\left\{\begin{array}{cc}-d_2\Delta {\Phi }_v=\beta \left(x\right){\displaystyle \frac{G(x,{\Phi }_v){\Phi }_v}{1+\kappa {\Phi }_v}}-\gamma \left(x\right){\Phi }_v+\eta \left(x\right)G(x,{\Phi }_v),\hfill & x\in \Omega ,\hfill \\ {\displaystyle \frac{\mathit{\partial }{\Phi }_v}{\mathit{\partial }\nu }}=0,\hfill & x\in \mathit{\partial }\Omega .\hfill \end{array}\right.$$

(14)

Proof. 

It follows from Theorem 2 that the steady state system (4) possesses at least one endemic equilibrium $(u,v)$.

We first consider the convergence of I. From the arguments in the proof of Theorem 2, we know that

$$C_0\le u\left(x\right),v\left(x\right)\le {\widehat{C}}_0,\forall x\in \overline{\Omega },$$

(15)

where the positive constants $C_0$ and ${\widehat{C}}_0$ are independent of $d_1$ and $d_2$. In light of the standard elliptic regularity theory, we can take a subsequence $\left\{d_{1,i}\right\}$ satisfying $d_{1,i}\to 0$ as $i\to \infty$ and the corresponding positive solution $(u_i\left(x\right),v_i\left(x\right)):=(u(x;d_{1,i}),v(x;d_{1,i}))$ with $d_i=d_{1,i}$, such that

$$v_i\left(x\right)\to {\Phi }_v\left(x\right)\mathrm{uniformly}\mathrm{on}\overline{\Omega },\mathrm{as}i\to \infty ,$$

(16)

where ${\Phi }_v\left(x\right)\in C^1\left(\overline{\Omega }\right)$ and ${\Phi }_v\left(x\right)>0$ on $\overline{\Omega }$ due to (15).

Next, we investigate the convergence of u and analyze the the following equation:

$$\left\{\begin{array}{cc}-d_{1,i}\Delta u_i=a\left(x\right)u_i-b\left(x\right)u_i^2-\beta \left(x\right){\displaystyle \frac{u_i{v}_i}{1+\kappa v_i}}+\gamma \left(x\right)v_i-\eta \left(x\right)u_i,\hfill & x\in \Omega ,\hfill \\ {\displaystyle \frac{\mathit{\partial }{u}_i}{\mathit{\partial }\nu }}=0,\hfill & x\in \mathit{\partial }\Omega .\hfill \end{array}\right.$$

(17)

In view of (16), for any small $ϵ>0$, there exists $i_1$ such that for all $i\ge i_1$,

$$0<{\Phi }_v\left(x\right)-ϵ\le v_i\left(x\right)\le {\Phi }_v\left(x\right)+ϵ,\forall x\in \overline{\Omega }.$$

(18)

Consequently, for all $i\ge i_1$, we have

$$\begin{array}{cc}\hfill & a\left(x\right)u_i-b\left(x\right)u_i^2-\beta \left(x\right){\displaystyle \frac{u_i{v}_i}{1+\kappa v_i}}+\gamma \left(x\right)v_i-\eta \left(x\right)u_i\hfill \\ \hfill \le & a\left(x\right)u_i-b\left(x\right)u_i^2-\beta \left(x\right){\displaystyle \frac{u_i\left({\Phi }_v-ϵ\right)}{1+\kappa \left({\Phi }_v+ϵ\right)}}+\gamma \left(x\right)\left({\Phi }_v+ϵ\right)-\eta \left(x\right)u_i\hfill \\ \hfill =& \left[g_{+}^{ϵ}(x,{\Phi }_v)-u_i\right]\left[u_i-g_{-}^{ϵ}(x,{\Phi }_v)\right],\hfill \end{array}$$

where

$$\begin{array}{c}\hfill g_{\pm }^{ϵ}(x,{\Phi }_v):={\displaystyle \frac{a-\eta }{2b}}-{\displaystyle \frac{\beta ({\Phi }_v-ϵ)}{2b[1+\kappa ({\Phi }_v+\epsilon )]}}\pm \sqrt{{\left[{\displaystyle \frac{a-\eta }{2b}}-{\displaystyle \frac{\beta ({\Phi }_v-ϵ)}{2b[1+\kappa ({\Phi }_v+\epsilon )]}}\right]}^2+{\displaystyle \frac{\gamma ({\Phi }_v+ϵ)}{b}}}.\end{array}$$

It is easy to see that $g_{+}^{ϵ}(x,{\Phi }_v)>0$ and $g_{-}^{ϵ}(x,{\Phi }_v)<0$ on $\overline{\Omega }$. Note that for any $i\ge i_1$, M and $u_i\left(x\right)$ is a pair of upper-lower solutions of the following auxiliary equation:

$$\left\{\begin{array}{cc}-d_{1,i}\Delta z=\left[g_{+}^{ϵ}(x,{\Phi }_v)-z_i\right]\left[z_i-g_{-}^{ϵ}(x,{\Phi }_v)\right],\hfill & x\in \Omega ,\hfill \\ {\displaystyle \frac{\mathit{\partial }z}{\mathit{\partial }\nu }}=0,\hfill & x\in \mathit{\partial }\Omega ,\hfill \end{array}\right.$$

(19)

where M is a positive constant satisfying $u_i\left(x\right)\le M$ on $\overline{\Omega }$. Then, for any $i\ge i_1$, (19) admits at least one positive solution, denoted by $z_i$, which satisfies $u_i\le z_i\le M$ on $\overline{\Omega }$. From the the maximum principle, we have

$$\underset{x\in \overline{\Omega }}{min}{g}_{+}^{ϵ}(x,{\Phi }_v\left(x\right))\le \underset{x\in \overline{\Omega }}{min}{z}_i\left(x\right)\le z_i\left(x\right)\le \underset{x\in \overline{\Omega }}{max}{z}_i\left(x\right)\le \underset{x\in \overline{\Omega }}{max}{g}_{+}^{ϵ}(x,{\Phi }_v\left(x\right)),\forall x\in \overline{\Omega }.$$

As an application of a singular perturbation technique ([44], Lemma 2.1; see also [45], Lemma 2.4), we know that any positive solution $z_i\left(x\right)$ of (19) satisfies

$$z_i\left(x\right)\to g_{+}^{ϵ}(x,{\Phi }_v\left(x\right))\mathrm{uniformly}\mathrm{on}\overline{\Omega },\mathrm{as}i\to \infty .$$

It follows from $u_i\left(x\right)\le z_i\left(x\right)$ on $\overline{\Omega }$ that we obtain

$$\underset{i\to \infty }{lim\; sup}{u}_i\left(x\right)\le g_{+}^{ϵ}(x,{\Phi }_v\left(x\right)).$$

(20)

At the same time, for all $i\ge i_1$, we have

$$\begin{array}{cc}\hfill & a\left(x\right)u_i-b\left(x\right)u_i^2-\beta \left(x\right){\displaystyle \frac{u_i{v}_i}{1+\kappa v_i}}+\gamma \left(x\right)v_i-\eta \left(x\right)u_i\hfill \\ \hfill \ge & a\left(x\right)u_i-b\left(x\right)u_i^2-\beta \left(x\right){\displaystyle \frac{u_i\left({\Phi }_v+ϵ\right)}{1+\kappa \left({\Phi }_v-ϵ\right)}}+\gamma \left(x\right)\left({\Phi }_v-ϵ\right)-\eta \left(x\right)u_i\hfill \\ \hfill =& \left[g_{ϵ}^{+}(x,{\Phi }_v)-u_i\right]\left[u_i-g_{ϵ}^{-}(x,{\Phi }_v)\right],\hfill \end{array}$$

where

$$\begin{array}{c}\hfill g_{ϵ}^{\pm }(x,{\Phi }_v):={\displaystyle \frac{a-\eta }{2b}}-{\displaystyle \frac{\beta ({\Phi }_v+ϵ)}{2b[1+\kappa ({\Phi }_v-\epsilon )]}}\pm \sqrt{{\left[{\displaystyle \frac{a-\eta }{2b}}-{\displaystyle \frac{\beta ({\Phi }_v+ϵ)}{2b[1+\kappa ({\Phi }_v-\epsilon )]}}\right]}^2+{\displaystyle \frac{\gamma ({\Phi }_v-ϵ)}{b}}},\end{array}$$

with $g_{ϵ}^{-}(x,{\Phi }_v)<0<g_{ϵ}^{+}(x,{\Phi }_v)$ on $\overline{\Omega }$. By a similar argument, one can obtain

$$\underset{i\to \infty }{lim\; inf}{u}_i\left(x\right)\ge g_{ϵ}^{+}(x,{\Phi }_v\left(x\right)).$$

(21)

We further obtain

$$\begin{array}{cc}\hfill \underset{ϵ\to 0}{lim}{g}_{+}^{ϵ}(x,{\Phi }_v\left(x\right))=& \underset{ϵ\to 0}{lim}{g}_{ϵ}^{+}(x,{\Phi }_v\left(x\right))\hfill \\ \hfill =& {\displaystyle \frac{(a\left(x\right)-\eta \left(x\right))(1+\kappa {\Phi }_v)-\beta \left(x\right){\Phi }_v}{2b\left(x\right)(1+\kappa {\Phi }_v)}}\hfill \\ \hfill & \pm {\displaystyle \frac{\sqrt{{[(a\left(x\right)-\eta \left(x\right))(1+\kappa {\Phi }_v)-\beta \left(x\right){\Phi }_v]}^2+4b\left(x\right)\gamma \left(x\right){\Phi }_v{(1+\kappa {\Phi }_v)}^2}}{2b\left(x\right)(1+\kappa {\Phi }_v)}}\hfill \\ \hfill :=& G(x,{\Phi }_v\left(x\right)).\hfill \end{array}$$

It follows from (20) and (21) that

$$u_i\left(x\right)\to G(x,{\Phi }_v\left(x\right))={\Phi }_u\left(x\right)\mathrm{uniformly}\mathrm{on}\overline{\Omega },\mathrm{as}i\to \infty .$$

(22)

As a result, it is clear that ${\Phi }_v\left(x\right)$ solves (14). This completes the proof. □

We next study the limiting behavior of the endemic equilibrium to (4) in the case of small dispersal of an infected individual.

Theorem 4. 

Assume (7) holds. Fix $d_1$, $\kappa >0$, and let $d_2\to 0$ (up to a subsequence), then every positive solution $\left(u\right(x),v(x\left)\right)$ of (4) satisfies

$$(u\left(x\right),v\left(x\right))\to \left({\Psi }_u\left(x\right),{\Psi }_v\left(x\right)\right)\mathit{uniformly}\mathit{on}\overline{\Omega },$$

where

$$\begin{array}{cc}\hfill {\Psi }_v\left(x\right)& =H\left(x,{\Psi }_u\left(x\right)\right)={\displaystyle \frac{\left(\beta +\kappa \eta \right){\Psi }_v-\gamma +\sqrt{{\left[\left(\beta +\kappa \eta \right){\Psi }_v-\gamma \right]}^2+4\kappa \eta \gamma {\Psi }_v}}{2\kappa \gamma }},\hfill \end{array}$$

and ${\Psi }_v\left(x\right)$ is a positive solution of

$$\left\{\begin{array}{cc}-d_1\Delta {\Psi }_u=a\left(x\right){\Psi }_u-b\left(x\right){\Psi }_u^2-{\displaystyle \frac{\beta \left(x\right){\Psi }_{u}H\left(x,{\Psi }_u\right)}{1+\kappa {\Psi }_v}}+\gamma \left(x\right)H\left(x,{\Psi }_u\right)-\eta \left(x\right){\Psi }_u,\hfill & x\in \Omega ,\hfill \\ {\displaystyle \frac{\mathit{\partial }{\Psi }_u}{\mathit{\partial }\nu }}=0,\hfill & x\in \mathit{\partial }\Omega .\hfill \end{array}\right.$$

(23)

Proof. 

It should be noted that the estimate (15) holds for any $d_2>0$. From the standard elliptic regularity theory and Sobolev embedding theorem, we know that there exists a subsequence $d_{2,j}$ with convergence to 0 as $j\to \infty$, and the corresponding positive solution $(u_j\left(x\right),v_j\left(x\right)):=(u(x;d_{2,j}),v(x,d_{2,j}))$ of (4) with $d_{2,j}$ such that

$$u_j\left(x\right)\to {\Psi }_u\left(x\right)\mathrm{in}{C}^1\left(\overline{\Omega }\right),\mathrm{as}j\to \infty ,$$

(24)

where ${\Phi }_u\left(x\right)>0$ on $\overline{\Omega }$ due to (15).

It is clear that $v_j$ fulfills:

$$\left\{\begin{array}{cc}-d_{2,j}\Delta v_j=\beta \left(x\right){\displaystyle \frac{u_j{v}_j}{1+\kappa v_j}}+\eta \left(x\right)u_j-\gamma \left(x\right)v_j,\hfill & x\in \Omega ,\hfill \\ {\displaystyle \frac{\mathit{\partial }{v}_j}{\mathit{\partial }\nu }}=0,\hfill & x\in \mathit{\partial }\Omega .\hfill \end{array}\right.$$

(25)

In view of (24), for any small $ϵ>0$, there exists $j_1$ such that for all $j\ge j_1$,

$$0<{\Psi }_u\left(x\right)-ϵ\le u_j\left(x\right)\le {\Psi }_u\left(x\right)+ϵ\mathrm{on} \overline{\Omega }.$$

(26)

Thus, for any $j\ge j_1$, it follows that

$$\begin{array}{cc}\hfill \beta \left(x\right){\displaystyle \frac{u_j{v}_j}{1+\kappa v_j}}+\eta \left(x\right)u_j-\gamma \left(x\right)v_j& \le \beta \left(x\right){\displaystyle \frac{v_j({\Psi }_u+ϵ)}{1+\kappa v_j}}+\eta \left(x\right)({\Psi }_u+ϵ)-\gamma \left(x\right)v_j\hfill \\ & ={\displaystyle \frac{\kappa \gamma \left(x\right)\left[h_{+}^{ϵ}\left(x,{\Psi }_u\left(x\right)\right)-v_j\right]\left[v_j-h_{-}^{ϵ}\left(x,{\Psi }_u\left(x\right)\right)\right]}{1+\kappa v_j}},\hfill \end{array}$$

where

$$h_{\pm }^{ϵ}\left(x,{\Psi }_u\left(x\right)\right)={\displaystyle \frac{\left(\beta +\kappa \eta \right)({\Psi }_u+ϵ)-\gamma \pm \sqrt{{\left[\left(\beta +\kappa \eta \right)({\Psi }_u+ϵ)-\gamma \right]}^2\pm 4\kappa \eta \gamma ({\Psi }_u+ϵ)}}{2\kappa \gamma }}.$$

We know that $h_{+}^{ϵ}(x,{\Psi }_u\left(x\right))>0$ and $h_{-}^{ϵ}(x,{\Psi }_u\left(x\right))<0$ on $\overline{\Omega }$. Note that for any $j\ge j_0$, M and $v_i\left(x\right)$ is a pair of upper-lower solutions of

$$\left\{\begin{array}{cc}-d_{2,j}\Delta w={\displaystyle \frac{\kappa \gamma \left(x\right)\left[h_{+}^{ϵ}\left(x,{\Psi }_u\left(x\right)\right)-w\right]\left[w-h_{-}^{ϵ}\left(x,{\Psi }_u\left(x\right)\right)\right]}{1+\kappa w}},\hfill & x\in \Omega ,\hfill \\ {\displaystyle \frac{\mathit{\partial }w}{\mathit{\partial }\nu }}=0,\hfill & x\in \mathit{\partial }\Omega .\hfill \end{array}\right.$$

(27)

where the positive constant M satisfies $v_j\left(x\right)\le M$ on $\overline{\Omega }$. Therefore, for all $j\ge j_1$, (27) admits at least one positive solution $w_j$, which satisfies $v_j\le w_j\le M$ on $\overline{\Omega }$. By similar arguments, we deduce

$$w_j\to h_{+}^{ϵ}\left(x,{\Psi }_u\left(x\right)\right)\mathrm{uniformly}\mathrm{on}\overline{\Omega },\mathrm{as}j\to \infty ,$$

and

$$\underset{j\to \infty }{lim\; sup}{v}_j\left(x\right)\le h_{+}^{ϵ}\left(x,{\Psi }_u\left(x\right)\right)\mathrm{on}\overline{\Omega }.$$

(28)

At the same time, for any $j\ge j_1$, we deduce from (26)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\beta \left(x\right)u_j{v}_j}{1+\kappa I{v}_j}}+\eta \left(x\right)u_j-\gamma \left(x\right)v_j& \ge {\displaystyle \frac{\beta \left(x\right)v_j({\Psi }_u-ϵ)}{1+\kappa v_j}}+\eta \left(x\right)({\Psi }_u-ϵ)-\gamma \left(x\right)v_j\hfill \\ & ={\displaystyle \frac{\kappa \gamma \left(x\right)\left[h_{ϵ}^{+}\left(x,{\Psi }_u\left(x\right)\right)-I_j\right]\left[v_j-h_{ϵ}^{-}\left(x,{\Psi }_u\left(x\right)\right)\right]}{1+\kappa v_j}},\hfill \end{array}$$

where

$$h_{ϵ}^{\pm }\left(x,{\Psi }_u\left(x\right)\right)={\displaystyle \frac{\left(\beta +\kappa \eta \right)({\Psi }_u-ϵ)-\gamma \pm \sqrt{{\left[\left(\beta +\kappa \eta \right)({\Psi }_u-ϵ)-\gamma \right]}^2\pm 4\kappa \eta \gamma ({\Psi }_u-ϵ)}}{2\kappa \gamma }},$$

with $h_{ϵ}^{-}\left(x,{\Psi }_u\left(x\right)\right)<0<h_{ϵ}^{+}\left(x,{\Psi }_u\left(x\right)\right)$ on $\overline{\Omega }$. In a similar fashion as above, we can conclude that

$$\underset{j\to \infty }{lim\; inf}{v}_j\left(x\right)\ge h_{ϵ}^{+}\left(x,{\Psi }_u\left(x\right)\right)\mathrm{uniformly}\mathrm{on}\overline{\Omega }.$$

(29)

Notice that

$$\begin{array}{cc}\hfill \underset{ϵ\to 0}{lim}{h}_{+}^{ϵ}\left(x,{\Psi }_u\left(x\right)\right)=& \underset{ϵ\to 0}{lim}{h}_{ϵ}^{+}\left(x,{\Psi }_u\left(x\right)\right)\hfill \\ \hfill =& {\displaystyle \frac{\left(\beta +\kappa \eta \right){\Psi }_u-\gamma +\sqrt{{\left[\left(\beta +\kappa \eta \right){\Psi }_u-\gamma \right]}^2+4\kappa \eta \gamma {\Psi }_u}}{2\kappa \gamma }}\hfill \\ \hfill :=& H(x,{\Psi }_u\left(x\right)).\hfill \end{array}$$

From (28) and (29), we know that

$$v_j\left(x\right)\to {\Psi }_v\left(x\right)\mathrm{uniformly}\mathrm{on}\overline{\Omega },\mathrm{as}j\to \infty .$$

According to the expression of ${\Psi }_v\left(x\right)$, it is easy to see that ${\Psi }_u\left(x\right)$ solves (23). The proof is thus complete. □

4.2. Large Dispersal

This subsection is devoted to discussing the limiting behavior of positive solutions in relation to a large dispersal rate of susceptible individuals in Theorem 5 and a large dispersal rate of infected individuals in Theorem 6.

Theorem 5. 

Assume (7) holds. Fix $d_2$, $\kappa >0$, and let $d_1\to \infty$ (up to a subsequence), then every positive solution $\left(u\right(x),v(x\left)\right)$ of (4) fulfills

$$(u\left(x\right),v\left(x\right))\to (u^{\infty }\left(x\right),v^{\infty }\left(x\right))\mathit{uniformly}\mathit{on}\overline{\Omega },$$

where the positive constant $u^{\infty }={\int }_{\Omega }a\left(x\right)\mathrm{d}x/{\int }_{\Omega }b\left(x\right)\mathrm{d}x$, and $v^{\infty }\left(x\right)$ is the unique positive solution of

$$\left\{\begin{array}{cc}-d_2\Delta v^{\infty }=\beta \left(x\right){\displaystyle \frac{u^{\infty }{v}^{\infty }}{1+\kappa v^{\infty }}}-\gamma \left(x\right)v^{\infty }+\eta \left(x\right)u^{\infty },\hfill & x\in \Omega ,\hfill \\ {\displaystyle \frac{\mathit{\partial }{v}^{\infty }}{\mathit{\partial }\nu }}=0,\hfill & x\in \mathit{\partial }\Omega .\hfill \end{array}\right.$$

(30)

Proof. 

Since the estimate (15) is valid for all sufficiently large $d_1$, from the standard elliptic regularity theory and Sobolev embedding theorem, we can take a subsequence $\left\{d_{1,k}\right\}$ satisfying $d_{1,k}\to \infty$ as $k\to \infty$ and a corresponding positive solution $(u_k\left(x\right),v_k\left(x\right)):=(u(x;d_{1,k}),v(x;d_{1,k}))$ with $d_1=d_{1,k}$, such that $u_k\left(x\right)\to u^{\infty }\left(x\right)$ in $C^1\left(\overline{\Omega }\right)$ as $k\to \infty$, where $u^{\infty }>0$ on $\overline{\Omega }$ due to (15). It is easy to see that $u^{\infty }$ solves

$$-\Delta u^{\infty }=0,x\in \Omega ;{\displaystyle \frac{\mathit{\partial }{u}^{\infty }}{\mathit{\partial }\nu }}=0,x\in \mathit{\partial }\Omega .$$

Consequently, the maximum principle ensures that $u^{\infty }$ is a positive constant. It follows from some direct calculation that $u^{\infty }={\int }_{\Omega }a\left(x\right)\mathrm{d}x/{\int }_{\Omega }b\left(x\right)\mathrm{d}x$. By a similar analysis, from the equation of v in (4), and by passing to a further subsequence if necessary, we know that

$$v_k\left(x\right)\to v^{\infty }\left(x\right)\mathrm{in}{C}^1\left(\overline{\Omega }\right),\mathrm{as}k\to \infty .$$

It follows from the similar technique explained in the Appendix in [46] that $v^{\infty }$ is the unique positive solution of (30). The proof is complete. □

The limiting behavior of the endemic equilibrium for the case of $d_2\to \infty$ can be deduced in the same fashion. Thus, we omit the details.

Theorem 6. 

Assume (7) holds. Fix $d_1$, $\kappa >0$, and let $d_2\to \infty$ (up to a subsequence). Then, every positive solution $\left(u\right(x),v(x\left)\right)$ of (4) fulfills

$$(u\left(x\right),v\left(x\right))\to (u_{\infty }\left(x\right),v_{\infty }\left(x\right))\mathit{uniformly}\mathit{on}\overline{\Omega },$$

where $v_{\infty }$ is a positive constant and the positive function $u_{\infty }>0$ solves

$$\left\{\begin{array}{cc}-d_1\Delta u_{\infty }=a\left(x\right)u_{\infty }-b\left(x\right)u_{\infty }^2-{\displaystyle \frac{\beta \left(x\right)u_{\infty }{v}_{\infty }}{1+\kappa I_{\infty }}}+\gamma \left(x\right)v_{\infty }-\eta \left(x\right)u_{\infty },\hfill & x\in \Omega ,\hfill \\ {\displaystyle \frac{\mathit{\partial }{u}_{\infty }}{\mathit{\partial }\nu }}=0,\hfill & x\in \mathit{\partial }\Omega ,\hfill \\ {\displaystyle {\int }_{\Omega }\left[{\displaystyle \frac{\beta \left(x\right)u_{\infty }{v}_{\infty }}{1+\kappa v_{\infty }}}-\gamma \left(x\right)v_{\infty }+\eta \left(x\right)u_{\infty }\right]\mathrm{d}x=0.}\hfill \end{array}\right.$$

(31)

4.3. Large Saturation

In this subsection, we investigate the limiting behavior of the endemic equilibrium of (3) with $d_1,d_2>0$ being fixed and $\kappa \to \infty$.

Theorem 7. 

Assume (7) holds. Fix $d_1$, $d_2>0$, and let $\kappa \to \infty$ (up to a subsequence), then every positive solution $\left(u\right(x),v(x\left)\right)$ of (4) fulfills

$$(u\left(x\right),v\left(x\right))\to (\widehat{u}\left(x\right),\widehat{v}\left(x\right))\mathit{uniformly}\mathit{on}\overline{\Omega },$$

where $(\widehat{u}\left(x\right),\widehat{v}\left(x\right))$ solves

$$\left\{\begin{array}{cc}-d_1\Delta \widehat{u}=a\left(x\right)\widehat{u}-b\left(x\right){\widehat{u}}^2+\gamma \left(x\right)\widehat{v}-\eta \left(x\right)\widehat{u},\hfill & x\in \Omega ,\hfill \\ -d_2\Delta \widehat{v}=-\gamma \left(x\right)\widehat{v}+\eta \left(x\right)\widehat{u},\hfill & x\in \Omega ,\hfill \\ {\displaystyle \frac{\mathit{\partial }\widehat{u}}{\mathit{\partial }\nu }}={\displaystyle \frac{\mathit{\partial }\widehat{v}}{\mathit{\partial }\nu }}=0,\hfill & x\in \mathit{\partial }\Omega .\hfill \end{array}\right.$$

(32)

Proof. 

Is is evident that the estimate (15) still holds true for any $\kappa \ge 1$. By the standard elliptic regularity and Sobolev embedding theorem, we can derive that there exists a subsequence $\left\{{\kappa }_n\right\}$ satisfying ${\kappa }_n\to \infty$ as $n\to \infty$, such that the corresponding positive solution $(u_n\left(x\right),v_n\left(x\right)):=(u(x;{\kappa }_n),v(x;{\kappa }_n))$ of (4) fulfills

$$(u_n\left(x\right),v_n\left(x\right))\to (\widehat{u}\left(x\right),\widehat{v}\left(x\right))\mathrm{in}{C}^1\left(\overline{\Omega }\right)\times C^1\left(\overline{\Omega }\right)asn\to \infty ,$$

where $\widehat{u}\left(x\right),\widehat{v}\left(x\right)>0$ on $\overline{\Omega }$ due to (15). We can deduce that $(\widehat{u}\left(x\right),\widehat{v}\left(x\right))$ solves (32). This proves Theorem 7. □

A comparative discussion of results with previous SIS models is given. First, the incidence rate mechanisms differ significantly between the two models, with the model studied in this paper exhibiting greater complexity. Second, the proposed model incorporates both natural host population growth and resource constraints, whereas the first model represents a simplified infectious disease framework that neglects host population dynamics. Third, the introduction of a spontaneous infection mechanism in the second model accounts for environmental transmission pathways absent in the first model, potentially leading to divergent transmission patterns and distinct equilibrium states.

There are many diseases governed by these mechanisms, such as COVID-19, dengue fever, and multidrug-resistant tuberculosis. The results in this paper can aid in determining ICU beds/vaccine production capacity aligned with Logistic model-predicted peaks and adjust social distancing policies to slow transmission rates, such as through the establishment of public campaigns to reduce contact frequency (e.g., staggered commuting).

5. Conclusions

By integrating multiple mechanisms such as diffusion, nonlinear infection rates, and spontaneous infection, this study extends the mathematical framework of the classical SIS model, addressing the limitations of single-mechanism studies. Through analyzing the stability, periodicity, and threshold conditions of model solutions, it reveals the emergent mechanisms underlying complex epidemiological behaviors. Future work will employ stochastic differential equations (SDEs) to model the stochastic dynamics of pathogen decay and reactivation in environmental reservoirs linked to spontaneous infection.