#  On degenerate reaction-diffusion epidemic models with mass action or
standard incidence mechanism

Rachidi B. Salako
Department of Mathematical Sciences, University of Nevada Las Vegas,
Las Vegas, NV 89154
Yixiang Wu
Department of Mathematical Sciences, Middle Tennessee State University
Murfreesboro, Tennessee 37132, USA

###### Abstract

In this paper, we consider reaction-diffusion epidemic models with mass action
or standard incidence mechanism and study the impact of limiting population
movement on disease transmissions. We set either the dispersal rate of the
susceptible or infected people to zero and study the corresponding degenerate
reaction-diffusion model. Our main approach to study the global dynamics of
these models is to construct delicate Lyapunov functions. Our results show
that the consequences of limiting the movement of susceptible or infected
people depend on transmission mechanisms, model parameters, and population
size.

Keywords: SIS epidemic model, reaction-diffusion, dispersal rate, Lyapunov
function, global dynamics.
MSC 2010: 92D30, 37N25, 35K40, 35K57, 37L15.

##  1 Introduction

Various differential equation epidemic models have been proposed to study the
spread of infectious diseases [4, 7, 6, 15, 42], and it has been recognized
that population mobility [5, 53, 56] and the spatial heterogeneity of the
environment [20, 33] are key factors in disease transmissions. In order to
address these issues, many reaction-diffusion epidemic models with non-
constant coefficients have been proposed and studied [2, 18, 59, 60]. Specific
infectious diseases modeled by diffusive models include malaria [39, 40],
rabies [24, 43], dengue fever [54], West Nile virus [28, 32], influenza [41],
Covid-19 [19, 26, 52, 58], etc.

In this paper, we consider the following Susceptible-Infected-Susceptible
(SIS) diffusive epidemic model, which is a natural extension of the classic
ordinary differential equation epidemic model by Kermack and McKendrick [25]:

$\begin{cases}\displaystyle\partial_{t}S=d_{S}\Delta
S-f(x,S,I)+\gamma(x)I,&x\in\Omega,t>0,\\\
\displaystyle\partial_{t}I=d_{I}\Delta
I+f(x,S,I)-\gamma(x)I,&x\in\Omega,t>0,\\\
\displaystyle\partial_{\nu}S=\partial_{\nu}I=0,&x\in\partial\Omega,t>0,\\\
S(x,0)=S_{0}(x),\ I(x,0)=I_{0}(x),&x\in\Omega.\end{cases}$ (1.1)

Here, the individuals are assumed to live in a bounded domain
$\Omega\subset\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$; $S(x,t)$
and $I(x,t)$ are the density of susceptible and infected individuals at
position $x\in\Omega$ and time $t$, respectively; $d_{S}$ and $d_{I}$ are the
movement rates of susceptible and recovered individuals, respectively;
$\gamma$ is the disease recovery rate; $f(x,S,I)$ describes the interaction of
susceptible and infected people; $\nu$ is the unit outward normal of
$\partial\Omega$ and the homogeneous Neumann boundary conditions mean that the
individuals cannot cross the boundary.

In the pioneering work by Allen _et al_. [2], model (1.1) with standard
incidence mechanism, $f(x,S,I)=\beta(x)SI/(S+I)$, has been proposed and
studied ($\beta$ is called the disease recovery rate). In [2], the authors
define a basic reproduction number $\mathcal{R}_{0}$ and show that the model
has a unique endemic equilibrium (EE) (i.e. positive equilibrium) if
$\mathcal{R}_{0}>1$. Most importantly, they show that the disease component of
the EE approaches zero as $d_{S}$ approaches zero if $\beta-\gamma$ changes
sign. Biologically, assuming that the population eventually stabilizes at the
EE, this result indicates that the disease can be eliminated by controlling
the mobility of susceptible individuals if there are places that are of low
risk (i.e. $\beta(x)<\gamma(x)$). In contrast if the dispersal rate of
infected people is limited, [46] shows that the disease cannot be completely
eliminated. In [14, 63], the authors considered model (1.1) with mass action
mechanism, $f(x,S,I)=\beta(x)SI$, which is algebraically simpler but
mathematically more challenging. For this model, assuming again that the
population eventually stabilizes at EE solutions, it has been shown that
lowering the movement of susceptible people can eliminate the disease only
when the size of the population is below some critical number, solely
determined by $\gamma/\beta$, [10, 63], while infected individuals may
concentrate on certain hot spots when limiting their movement [10, 48, 63]. It
is important to note that stability of the EE solutions for models with
standard incidence or mass action mechanism is only known in a few cases :
either the population movement rate is uniform (i.e., $d_{S}=d_{I}$), or the
ratio $\beta/\gamma$ is constant [14, 47]. For more related works, we refer
the interested readers to [11, 12, 13, 23, 27, 29, 30, 31, 35, 36, 47, 49, 50,
51, 55, 57, 61] and the references therein.

For the corresponding ordinary differential equation epidemic model of (1.1):

$\begin{cases}S^{\prime}=-f(S,I)+\gamma I,\\\ I^{\prime}=f(S,I)-\gamma
I,\end{cases}$

the global dynamics is determined by the basic reproduction number, which can
be interpreted as the average number of infected individuals generated by one
infectious individual in an otherwise susceptible population: if it is greater
than one, the solution converges to an EE and the disease persists; if it is
less than one, the solution converges to a disease free equilibrium and the
infected individuals go to extinction. Here, the basic reproduction number is
$\mathcal{R}^{1}_{0}=N\beta/\gamma$; if $f(S,I)=\beta SI$ and
$\mathcal{R}_{0}^{2}=\beta/\gamma$ if $f(S,I)=\beta SI/(S+I)$.

In this paper, we revisit model (1.1) and study the impact of limiting
population movement on disease transmissions. Different from most of the
aforementioned studies, we will work on the global dynamics of the time
dependent model (1.1) with either $d_{S}=0$ or $d_{I}=0$ rather than consider
the asymptotic profiles of the EE as $d_{S}\to 0$ or $d_{I}\to 0$. Intuitively
if the disease evolution happens in a faster time scale than the control of
population movement and the solutions of model (1.1) establish at an EE, then
the asymptotic profiles of the EE will reflect the effect of the control.
However if the control of population movement happens in a faster time scale
and the solution of model (1.1) converges to that of the corresponding
degenerate system as $d_{S}\to 0$ or $d_{I}\to 0$, then the global dynamics of
the degenerate system will better tell the impact of the control strategies.

There are several recent efforts on degenerate reaction-diffusion population
models (see [16, 17, 34, 37, 38, 44, 64] and the references cited therein). In
particular, the two works [10, 34] have partial results on model (1.1) with
$d_{S}=0$, which was interpreted as the situation of a total lock down for the
susceptible population. Note that when either $d_{S}=0$ or $d_{I}=0$, system
(1.1) becomes degenerate and hence there is a lack of some compactness of the
solution operator, which induces many challenges in the study of the large
time behavior of the solutions.

Our main approach to study the global dynamics of the degenerate system (1.1)
(i.e $d_{S}=0$ or $d_{I}=0$) is to construct Lyapunov functions (except for
the case $d_{S}=0$ in subsection 3.1) and combine these with delicate
analysis. In particular, in subsection 3.2, we present a Lyapunov function
that one may easily draw a false conclusion using it and the convergence
result based on it is very quite unusual (see Remark 3.8 and Fig. 2). In the
proof of Theorem 4.6, we construct a Lyapunov function $V$ that does not
satisfy $\dot{V}\leq 0$ (see Eq. 4.22) but we are still able to conclude the
convergence of the solution.

The rest of our paper is organized as follows. In section 2, we list the
assumptions and terminology and present some useful results. In section 3, we
consider the model with mass action mechanism with $d_{S}=0$ or $d_{I}=0$. In
section 4, we consider the model with standard incidence mechanism. In section
5, we compare the results from the two different mechanisms and control
strategies and discuss the implications for disease control.

##  2 Preliminaries

Throughout the paper, we make the following assumptions on the parameters:

  1. (A1)

The functions $\beta,\gamma$ are positive and Hölder continuous on
$\bar{\Omega}$;

  2. (A2)

The functions $S_{0},I_{0}$ are nonnegative and continuous on $\bar{\Omega}$
with $I_{0}\not\equiv 0$. Moreover, $\int_{\Omega}(S_{0}+I_{0})dx=N$ for some
fixed positive constant $N$.

Integrating the first two equations of model (1.1) over $\Omega$ and summing
up them, we find that

$\frac{d}{dt}\int_{\Omega}(S(x,t)+I(x,t))dx=0,$

which means the total population $\int_{\Omega}(S(x,t)+I(x,t))dx$ remains a
constant for all $t\geq 0$. By assumption (A2), the total population is $N$.

For convenience, we introduce a few definitions and notations. Set
$r:=\gamma/\beta$ and $R:=\beta/\gamma$. For a real-valued integrable function
$h$ on $\overline{\Omega}$, let $h_{M}:=\sup_{x\in\Omega}h(x)$,
$h_{m}:=\inf_{x\in\Omega}h(x)$, and $\bar{h}=\int_{\Omega}h(x)dx/|\Omega|$.

Let $\\{e^{t\Delta}\\}_{t\geq 0}$ be the analytic $c_{0}$-semigroup on
$L^{p}(\Omega)$, $1\leq p<\infty$, generated by the Laplace operator $\Delta$
on $\Omega$ subject to the homogeneous Neumann boundary conditions on
$\partial\Omega$. Let ${\rm Dom}_{p}(\Delta)$ be the domain of the
infinitesimal generator $\Delta$ of $\\{e^{t\Delta}\\}_{t\geq 0}$ on
$L^{p}(\Omega)$. Then ${\rm Dom}_{p}(\Delta)=\\{u\in
W^{2,p}(\Omega)|\partial_{\nu}u=0\ \text{on}\ \partial\Omega\\}$ for
$p\in(1,\infty)$ and ${\rm Dom}_{1}(\Delta)\subset\\{u\in
W^{2,1}(\Omega)|\partial_{\nu}u=0\ \text{on}\ \partial\Omega\\}$ (see [3]).
When $\\{e^{t\Delta}\\}_{t\geq 0}$ is considered as an analytic
$c_{0}$-semigroup on $C(\bar{\Omega})$, the domain of its infinitesimal
generator $\Delta$ is given by

${\rm Dom}_{\infty}(\Delta):=\left\\{u\in\cap_{p\geq 1}{\rm Dom}_{p}(\Delta)|\
\Delta u\in C(\overline{\Omega})\right\\}.$

Let $\mathcal{Z}:=\\{u\in L^{1}(\Omega):\int_{\Omega}udx=0\\}$ be a Banach
subspace of $L^{1}(\Omega)$. Then $\\{e^{t\Delta}\\}_{t\geq 1}$ leaves
$\mathcal{Z}$ invariant and by [62, Lemma 1.3], there is a positive real
number $C_{0}>0$ such that

$\|e^{t\Delta}u\|_{L^{1}(\Omega)}\leq
C_{0}e^{-\sigma_{1}t}\|u\|_{L^{1}(\Omega)},\quad\forall\ u\in\mathcal{Z},$
(2.1)

where $\sigma_{1}$ is the first positive eigenvalue of $-\Delta$, subject to
the homogeneous boundary condition on $\partial\Omega$. By (2.1), the
restriction $\Delta_{|\mathcal{Z}}$ of $\Delta$ on $\mathcal{Z}\cap{\rm
Dom}_{1}(\Delta)$ is invertible. Hence, for any $0\leq\alpha<1$, the
fractional power spaces of $-\Delta_{|\mathcal{Z}}$ is well defined (see
[21]). Denote by $\\{e^{t\Delta_{|\mathcal{Z}}}\\}_{t\geq 0}$, the restriction
of the $\\{e^{t\Delta}\\}_{t\geq 0}$ on $\mathcal{Z}$, then it is also an
analytic $c_{0}$-semigroup. The following estimates on $\mathcal{Z}^{\alpha}$
will be needed:

######  Lemma 2.1.

[21, Theorem 1.4.3] For any $0<\alpha<1$, there is $C_{\alpha}>0$ such that

$\|e^{t\Delta_{|\mathcal{Z}}}z\|_{\mathcal{Z}^{\alpha}}\leq
C_{\alpha}t^{-\alpha}e^{-\sigma_{1}t}\|z\|_{L^{1}(\Omega)},\quad\forall\ t>0,\
z\in\mathcal{Z},$ (2.2)

and

$\|(e^{t\Delta_{|\mathcal{Z}}}-{\rm id})z\|_{L^{1}(\Omega)}\leq
C_{\alpha}t^{\alpha}\|z\|_{\mathcal{Z}^{\alpha}},\quad\forall\
z\in\mathcal{Z}^{\alpha}.$ (2.3)

The following well-known result will be used after the construction of
Lyapunov functions:

######  Lemma 2.2.

Suppose that $\phi:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is Hölder continuous and
$\int_{0}^{\infty}\phi(t)dt<\infty$. Then $\lim_{t\to\infty}\phi(t)=0$.

We will need the following Hanack type inequality (see [22]):

######  Lemma 2.3.

Let $u$ be a nonnegative solution of the following problem on
$\Omega\times(0,T)$:

$\begin{cases}\displaystyle\partial_{t}u=\Delta u+a(x,t)u,&x\in\Omega,t>0,\\\
\partial_{\nu}u=0,&x\in\partial\Omega,t>0,\end{cases}$

where $a\in L^{\infty}(\Omega\times(0,\infty))$. Then for any $0<\delta<T$,
there exists $C>0$ depending on $\delta$ and
$\|a\|_{L^{\infty}(\Omega\times(0,\infty))}$ such that

$\sup_{x\in\Omega}u(x,t)\leq C\inf_{x\in\Omega}u(x,t),\ \ \text{for all
}t\in[\delta,T).$

We also recall the following well known result about the elliptic eigenvalue
problem.

######  Lemma 2.4.

[8] Suppose that $d>0$ and $h\in C(\overline{\Omega})$. Let $\sigma(d,h)$ be
the principal eigenvalue of the following elliptic eigenvalue problem:

$\begin{cases}d\Delta\varphi+h\varphi=\sigma\varphi,&x\in\Omega,\cr\partial_{\nu}\varphi=0,&x\in\partial\Omega.\end{cases}$

Then $\sigma(d,h)$ is simple, associated with a positive eigenfunction, and
given by the variational formula

$\sigma(d,r)=\sup\Big{\\{}\int_{\Omega}\big{[}h\varphi^{2}-d|\nabla\varphi|^{2}\big{]}dx:\
\varphi\in W^{1,2}(\Omega)\ \text{and}\
\|\varphi\|_{L^{2}(\Omega)}=1\Big{\\}}.$ (2.4)

Furthermore, the following conclusions hold:

  * (i)

If $h(x)\equiv h$ is a constant function, then $\sigma(d,h)=h$ for all $d>0$.

  * (ii)

If $h(x)$ is not constant, then the map $(0,\infty)\ni d\mapsto\sigma(d,h)$ is
strictly decreasing with

$\lim_{d\to
0^{+}}\sigma(d,h)=h_{M}\quad\text{and}\quad{\lim_{d\to\infty}\sigma(d,h)=\overline{h}}.$
(2.5)

##  3 Model with mass action mechanism

###  3.1 Limiting the movement of susceptible people

First, we consider the impact of limiting the movement of susceptible people
on (1.1) with mass action mechanism, that is, the long time behavior of the
following degenerate system:

$\begin{cases}\partial_{t}S=-\beta(x)SI+\gamma(x)I,&x\in\bar{\Omega},t>0,\\\
\partial_{t}I=d_{I}\Delta I+\beta(x)SI-\gamma(x)I,&x\in\Omega,t>0,\\\
\partial_{\nu}I=0,&x\in\partial\Omega,t>0,\\\ S(x,0)=S_{0}(x),\
I(x,0)=I_{0}(x),&x\in\bar{\Omega}.\end{cases}$ (3.1)

The following result states that the solution of (3.1) exists and is bounded:

######  Proposition 3.1.

Suppose that (A1)-(A2) holds and $d_{I}>0$. Then (3.1) has a unique
nonnegative global solution $(S,I)$, where

$S\in C^{1}([0,\infty),C(\overline{\Omega}))\quad\text{and}\quad I\in
C([0,\infty),C(\overline{\Omega}))\cap C^{1}((0,\infty),{\rm
Dom}_{\infty}(\Delta)).$

Moreover, there exists $M>0$ depending on (the $L^{\infty}$ norm of) initial
data such that

$\|S(\cdot,t)\|_{L^{\infty}(\Omega)},\ \|I(\cdot,t)\|_{L^{\infty}(\Omega)}\leq
M,\ \forall t\geq 0.$ (3.2)

###### Proof.

First we suppose that the solution exists on some maximal interval
$[0,t_{max})$, $t_{\max}\in(0,\infty]$ and prove the boundedness of it. Let
$M_{1}=\max\\{\|S_{0}\|_{L^{\infty}(\Omega)},\ r_{M}\\}$. By the first
equation of (3.1), $S(x,t)\leq M_{1}$ for all $x\in\bar{\Omega}$ and $0\leq
t<t_{\max}$. Since $\int_{\Omega}Idx\leq\int_{\Omega}(S+I)dx\leq N$ for all
$t\in[0,t_{\max})$, by [1, Theorem 3.1] (or Lemma 2.3), there exists $M_{2}>0$
depending on $\|I_{0}\|_{L^{\infty}(\Omega)}$ and $M_{1}$ such that
$\|I(\cdot,t)\|_{L^{\infty}(\Omega)}\leq M_{2}$ for all $0\leq t<t_{\max}$.
This proves (3.2).

The integral form of (3.1) is

$\begin{cases}S(\cdot,t)=e^{-at}S_{0}+\int_{0}^{t}e^{-a(t-s)}(aS(\cdot,s)-\beta
S(\cdot,s)I(\cdot,s)+\gamma I(\cdot,s))ds,&t>0,\\\
I(\cdot,t)=e^{t(d_{I}\Delta-a)}I_{0}+\int_{0}^{t}e^{(t-s)(d_{I}\Delta-a)}(a+\beta
S(\cdot,s)-\gamma)I(\cdot,s)ds,&t>0,\end{cases}$ (3.3)

where $a>0$ is chosen to be sufficiently large. Using Banach fixed point
theory, one can show that (3.3) has a unique nonnegative solution
$(S,I)\in[C([0,T],C(\overline{\Omega}))]^{2}$ for some $T>0$. By the first
equation of (3.3), $S\in C^{1}([0,T],C(\overline{\Omega}))$. By [45, Theorem
4.3.1 and Corollary 4.3.3], $I\in C([0,T],C(\overline{\Omega}))\cap
C^{1}((0,T],{\rm Dom}_{\infty}(\Delta))$. The a priori bound (3.2) enables us
to extend the solution globally. ∎

We note that the case when the total population is small (i.e.,
$N<\int_{\Omega}rdx$) has been addressed in the literature:

######  Theorem 3.2.

[10, Theorem 2.8-(i)] Suppose that (A1)-(A2) holds and $d_{I}>0$. Let $(S,I)$
be the solution of (3.1). If $N<\int_{\Omega}rdx$, then
$(S(x,t),I(x,t))\to(S^{*}(x),0)$ uniformly in $x\in\bar{\Omega}$ as
$t\to\infty$, where $S^{*}\in C(\bar{\Omega})$ and $\int_{\Omega}S^{*}dx=N$.

We are ready to study the asymptotic behavior of the solution of (3.1).

######  Theorem 3.3.

Suppose that (A1)-(A2) holds and $d_{I}>0$. Let $(S,I)$ be the solution of
(3.1). Then $\|S(\cdot,t)-S^{*}\|_{L^{\infty}(\Omega)}\to 0$ as $t\to\infty$
for some $S^{*}\in C(\overline{\Omega})$, and exactly one of the following two
statements hold:

  1. (i)

$S^{*}=\lambda^{*}S_{0}+(1-\lambda^{*})r$ for some $\lambda_{*}\in
C(\bar{\Omega})$ with $0<\lambda_{*}<1$ and
$\sigma(d_{I},\beta\lambda^{*}(S_{0}-r))\leq 0$, $\int_{\Omega}S^{*}dx=N$, and
$\|I(\cdot,t)\|_{L^{\infty}(\Omega)}\to 0$ as $t\to\infty$;

  2. (ii)

$S^{*}=r$ and $\|I(\cdot,t)-I^{*}\|_{L^{\infty}(\Omega)}\to 0$ as
$t\to\infty$, where $I^{*}={(N-\int_{\Omega}rdx)}/{|\Omega|}$ is a positive
constant.

Moreover, (i) holds if $N\leq\int_{\Omega}rdx$ while (ii) holds if
$N>N^{*}_{S_{0},r}$, where $N^{*}_{S_{0},r}$ is defined by

$N^{*}_{r,S_{0}}:=\sup\Big{\\{}\int_{\Omega}(\lambda^{*}S_{0}+(1-\lambda^{*})r)dx\
:\ \lambda^{*}\in
C(\bar{\Omega};[0,1])\,\,\text{and}\,\,\sigma(d_{I},\beta\lambda^{*}(S_{0}-r))\leq
0\Big{\\}}.$ (3.4)

###### Proof.

Define

$J(x,t)=\int_{0}^{t}I(x,\tau)d\tau,\quad\forall\
(x,t)\in\bar{\Omega}\times[0,\infty).$

It is clear that $J(x,t)$ is strictly monotone increasing in $t$ for each
$x\in\bar{\Omega}$. Hence we can define

$\hat{J}(x):=\int_{0}^{\infty}I(x,\tau)d\tau=\lim_{t\to\infty}J(x,t)\in(0,\infty],\quad\forall\
x\in\bar{\Omega}.$

Now, we distinguish two cases.

Case 1. $\hat{J}(x_{0})<\infty$ for some $x_{0}\in\overline{\Omega}$. By
(3.2), we have $\sup_{t\geq 0}\|\beta S(\cdot,t)-\gamma\|_{\infty}<\infty$. So
by Lemma 2.3, there is a positive constant $c_{1}$ such that

$I_{M}(\cdot,t)\leq c_{1}I_{m}(\cdot,t),\quad\forall\ t\geq 1.$ (3.5)

It follows that

$\hat{J}(x)=J(x,1)+\int_{1}^{\infty}I(x,\tau)d\tau\leq
J(x,1)+c_{1}\int_{1}^{\infty}I(x_{0},\tau)d\tau\leq
J(x,1)+c_{1}J(x_{0}),\quad\forall\ x\in\bar{\Omega}.$

This shows that $\hat{J}(x)<\infty$ for all $x\in\bar{\Omega}$ and
$\|\hat{J}\|_{L^{\infty}(\Omega)}<\infty$. Moreover, by (3.5),

$\|\hat{J}-J(\cdot,t)\|_{L^{\infty}(\Omega)}\leq
c_{1}\int_{t}^{\infty}I(x_{0},\tau)d\tau\to 0\quad\text{as}\quad t\to\infty.$

Hence, $\hat{J}\in C(\bar{\Omega})$. Next, observing that

$\partial_{t}(S-r)=\partial_{t}S=-\beta(S-r)I,\qquad\forall\
(x,t)\in\bar{\Omega}\times(0,\infty),$

we have

$S(x,t)-r(x)=(S_{0}(x)-r(x))e^{-\beta(x)J(x,t)},\quad\forall\
(x,t)\in\bar{\Omega}\times(0,\infty).$ (3.6)

This implies

$S(\cdot,t)\to S^{*}:=r+e^{-\beta\hat{J}}(S_{0}-r)\ \text{uniformly on}\
\bar{\Omega}\ \text{as}\ t\to\infty.$

Let $\lambda^{*}=e^{-\beta\hat{J}}$. Then $0<\lambda^{*}<1$, $\lambda^{*}\in
C(\bar{\Omega})$, and $S^{*}=\lambda^{*}S_{0}+(1-\lambda^{*})r$. Let
$\varphi^{*}>0$ be the eigenfunction associated with
$\sigma(d_{I},\beta(S^{*}-r))$ satisfying $\varphi^{*}_{M}=1$. Then it holds
that

$\displaystyle\frac{d}{dt}\int_{\Omega}\varphi^{*}Idx$
$\displaystyle=d_{I}\int_{\Omega}\varphi^{*}\Delta
Idx+\int_{\Omega}\beta(S-r)\varphi^{*}Idx$ $\displaystyle=$
$\displaystyle\sigma(d_{I},\beta(S^{*}-r))\int_{\Omega}\varphi^{*}Idx+\int_{\Omega}\beta(S-S^{*})\varphi^{*}Idx$
$\displaystyle\geq$
$\displaystyle\Big{(}\sigma(d_{I},\beta(S^{*}-r))-\beta_{M}\|S-S^{*}\|_{L^{\infty}(\Omega)}\Big{)}\int_{\Omega}\varphi^{*}Idx.$

Hence, we have

$N\geq\int_{\Omega}\varphi^{*}I(x,t)dx\geq
e^{\int_{0}^{t}(\sigma(d_{I},\beta(S^{*}-r))-\beta_{M}\|S-S^{*}\|_{L^{\infty}(\Omega)})d\tau}\int_{\Omega}\varphi^{*}I_{0}dx,\quad\forall\
t>0.$

This implies that

$\frac{\beta_{M}}{t}\int_{0}^{t}\|S(\cdot,\tau)-S^{*}\|_{L^{\infty}(\Omega)}d\tau+\frac{\ln(N)-\ln(\|\varphi^{*}I_{0}\|_{L^{1}(\Omega)})}{t}\geq\sigma(d_{I},\beta(S^{*}-r)),\quad
t>0.$

Observing that the left-hand side of this inequality tends zeo as
$t\to\infty$, we conclude that $\sigma(d_{I},\beta(S^{*}-r))\leq 0$.

Finally, by (3.2) and the parabolic estimates and the Sobolev emdedding
theorem, $I$ is Hölder continuous on $\bar{\Omega}\times[1,\infty)$. Hence, by
Lemma 2.2 and the fact that $\hat{J}(x_{0})<\infty$, we obtain from (3.5) that

$\|I(\cdot,t)\|_{L^{\infty}(\Omega)}\leq c_{1}I(x_{0},t)\to
0\quad\text{as}\quad t\to\infty,$

which in turn implies that

$\int_{\Omega}S^{*}dx=\lim_{t\to\infty}\int_{\Omega}Sdx=\lim_{t\to\infty}\int_{\Omega}(S+I)dx=N.$

This completes the proof of (i).

Case 2. $\hat{J}(x)=\infty$ for all $x\in\bar{\Omega}$. Fix
$x_{1}\in\bar{\Omega}$. Then $\int_{1}^{t}I(x_{1},\tau)d\tau\to\infty$ as
$t\to\infty$. This together with (3.5)-(3.6) implies that

$\|S(\cdot,t)-r\|_{L^{\infty}(\Omega)}\leq\|S_{0}-r\|_{L^{\infty}(\Omega)}e^{-\frac{\beta_{m}}{c_{1}}\int_{1}^{t}I(x_{1},\tau)d\tau}\to
0\quad\text{as}\quad t\to\infty.$

As a result,

$\lim_{t\to\infty}\int_{\Omega}Idx=N-\int_{\Omega}rdx.$ (3.7)

This shows that, in the current case, we must have that
$N\geq\int_{\Omega}rdx$. By (3.2), the parabolic estimates and the Sobolev
embedding theorem, the orbit $\\{I(\cdot,t)\\}_{t\geq 1}$ is precompact in
$C(\bar{\Omega})$. Noticing $S(\cdot,t)\to r$ in $C(\bar{\Omega})$ as
$t\to\infty$, the $\omega$ limit set $\omega(S_{0},I_{0})=\cap_{t\geq
1}\overline{\cup_{s\geq t}\\{(S(\cdot,t),I(\cdot,s))\\}}$ is well-defined,
where the completion is in $[C(\bar{\Omega})]^{2}$. By the invariant property
of $\omega(S_{0},I_{0})$, for any $(S^{*},I^{*})\in\omega(S_{0},I_{0})$, there
exists a complete orbit of (3.1) through $(S^{*},I^{*})$. It is clear that the
$S-$component of the complete orbit is constant function $r$. Hence, there
exists a bounded entire solution of
$\partial_{t}\tilde{I}=d_{I}\Delta\tilde{I}$ with homogeneous Neumann boundary
condition such that $\tilde{I}(\cdot,0)=I^{*}$. So $I^{*}$ is a constant and
$I^{*}=(N-\int_{\Omega}rdx)/|\Omega|$ by (3.7). This shows

$\lim_{t\to\infty}\Big{\|}I(\cdot,t)-\frac{N-\int_{\Omega}rdx}{|\Omega|}\Big{\|}_{L^{\infty}(\Omega)}=0,$
(3.8)

which completes the proof of (ii).

It is easy to see that if $N\leq\int_{\Omega}rdx$, then (i) holds. Note that
if (i) holds then $N=\int_{\Omega}(\lambda^{*}S_{0}+(1-\lambda^{*})r)dx$ for
some $\lambda^{*}\in C(\overline{\Omega})$ satisfying $0<\lambda^{*}<1$ and
$\sigma(d_{I},\beta\lambda^{*}(S_{0}-r))\leq 0$. Hence, we must have $N\leq
N^{*}_{S_{0},r}$. So alternative (ii) must hold whenever $N>N^{*}_{S_{0},r}$.
∎

######  Remark 3.4.

  * (1)

It always holds that $N^{*}_{S_{0},r}\geq\int_{\Omega}rdx$, which can be seen
by taking $\lambda^{*}\equiv 0$.

  * (2)

It is easy to see that $N^{*}_{S_{0},r}\leq\int_{\Omega}\max\\{S_{0},r\\}dx$.
Hence if $N>\int_{\Omega}\max\\{S_{0},r\\}dx$, then alternative (ii) of
Theorem 3.3 holds.

  * (3)

The strict inequality $N^{*}_{S_{0},r}<\int_{\Omega}\max\\{S_{0},r\\}dx$ holds
if $\|(S_{0}-r)_{+}\|_{L^{\infty}(\Omega)}>0$ (see a proof right after the
remark). Therefore, in general, the condition $N>N^{*}_{S_{0},r}$ is weaker
than $N>\int_{\Omega}\max\\{S_{0},r\\}dx$.

###### Proof.

Suppose that $\|(S_{0}-r)_{+}\|_{L^{\infty}(\Omega)}>0$ and we prove
$N^{*}_{S_{0},r}<\int_{\Omega}\max\\{S_{0},r\\}dx$ by contradiction. Suppose
to the contrary that there is $\lambda^{*}_{k}\in C(\overline{\Omega};[0,1])$
satisfying $\lambda(d_{I},\beta\lambda^{*}_{k}(S_{0}-r))\leq 0$ for each
$k\geq 1$ such that

$\int_{\Omega}(\lambda^{*}_{k}S_{0}+(1-\lambda^{*}_{k})r)dx\to\int_{\Omega}\max\\{S_{0},r\\}dx\quad\text{as}\quad
k\to\infty.$

Since $\|\lambda^{*}_{k}\|_{L^{\infty}(\Omega)}\leq 1$, possibly after passing
to a subsequence and using the Banach-Alaoglu theorem, there is
$\lambda^{*}\in L^{\infty}(\Omega)$ satisfying $0\leq\lambda^{*}\leq 1$ almost
everywhere on $\Omega$, such that $\lambda^{*}_{k}\to\lambda^{*}$ weakly-star
in $L^{\infty}(\Omega)$ as $k\to\infty$. So we have

$\int_{\Omega}(\lambda^{*}_{k}S_{0}+(1-\lambda^{*}_{k})r)dx\to\int_{\Omega}(\lambda^{*}S_{0}+(1-\lambda^{*})r)dx\quad\text{as}\quad
k\to\infty.$

It follows that

$\int_{\Omega}(\lambda^{*}S_{0}+(1-\lambda^{*})r)dx=\int_{\Omega}\max\\{S_{0},r\\}dx,$

which yields that $\max\\{S_{0},r\\}=\lambda^{*}S_{0}+(1-\lambda^{*})r$ almost
everywhere on $\Omega$. So,
$\beta\lambda^{*}(S_{0}-r)=\beta(\max\\{S_{0},r\\}-r)$ almost everywhere on
$\Omega$. Therefore, by the assumption $\|(S_{0}-r)_{+}\|_{\infty}>0$,

$\int_{\Omega}\beta\lambda^{*}(S_{0}-r)dx=\int_{\Omega}\beta(\max\\{S_{0},r\\}-r)dx=\int_{\\{S_{0}>r\\}}\beta(S_{0}-r)dx>0.$
(3.9)

However since $\lambda(d_{I},\beta\lambda^{*}_{k}(S_{0}-r))\leq 0$, we have
$\int_{\Omega}\beta\lambda^{*}_{k}(S_{0}-r))dx\leq 0$ for any $k\geq 1$ by
Lemma 2.4. Letting $k\to\infty$, we get
$\int_{\Omega}\beta\lambda^{*}(S_{0}-r)dx\leq 0$, which contradicts with
(3.9). ∎

We complement Theorem 3.3 with a corollary:

######  Corollary 3.5.

Suppose that (A1)-(A2) holds, $N>\int_{\Omega}rdx$, and $d_{I}>0$. If either
$\beta$ is constant or $S_{0}-r$ has a constant sign, then alternative (ii) of
Theorem 3.3 holds for any solution of (3.1).

###### Proof.

Let $N^{*}_{S_{0},r}$ be defined as in (3.4). (1) If $\beta$ is constant, then
for any $\lambda^{*}\in C(\overline{\Omega};[0,1])$,
$\sigma(d_{I},\beta\lambda^{*}(S_{0}-r))\leq 0$ implies that
$\int_{\Omega}\lambda^{*}(S_{0}-r)dx\leq 0$. In this case,

$\int_{\Omega}(\lambda^{*}S_{0}+(1-\lambda^{*})r)dx=\int_{\Omega}\lambda^{*}(S_{0}-r)dx+\int_{\Omega}rdx\leq\int_{\Omega}rdx,\quad\forall\
\lambda^{*}\in C(\overline{\Omega};[0,1]).$

So, we have $N^{*}_{S_{0},r}=\int_{\Omega}rdx$.

(2) If $S_{0}\leq r$, then $N^{*}_{S_{0},r}=\int_{\Omega}rdx$.

(3) If $S_{0}\geq r$, then $N^{*}_{S_{0},r}\leq\int_{\Omega}S_{0}dx$.

In cases (1)-(3), we have
$N^{*}_{S_{0},r}\leq\max\\{\int_{\Omega}rdx,\int_{\Omega}S_{0}dx\\}$. By
hypothesis (A2), we always have $N>\int_{\Omega}S_{0}dx$. Therefore if either
of these scenarios holds, $N>\int_{\Omega}rdx$ implies $N>N^{*}_{S_{0},r}$.
The conclusion now follows from Theorem 3.3. ∎

######  Remark 3.6.

Let $N^{*}_{S_{0},r}$ be defined as in (3.4). Alternative (ii) of Theorem 3.3
always holds when $N>\int_{\Omega}rdx$ if
$N^{*}_{S_{0},r}\leq\max\\{\int_{\Omega}S_{0}dx,\int_{\Omega}rdx\\}$.
Sufficient conditions ensuring the validity of the latter inequality are given
in Corollary 3.5. It remains an open problem to know whether alternative (i)
of Theorem 3.3 may hold for some initial data satisfying
$\max\\{\int_{\Omega}S_{0}dx,\int_{\Omega}rdx\\}<N<N^{*}_{S_{0},r}$.
Proposition 6.2 of the Appendix provides an example of a positive and
continuous function $S_{0}$ satisfying
$\max\\{\int_{\Omega}S_{0}dx,\int_{\Omega}rdx\\}<N^{*}_{S_{0},r}$. Whenever
such $S_{0}$ is fixed, we can always select $I_{0}$ to be small enough such
that $N=\int_{\Omega}(S_{0}+I_{0})dx$ satisfies
$\max\\{\int_{\Omega}S_{0}dx,\int_{\Omega}rdx\\}<N<N^{*}_{S_{0},r}$.

###  3.2 Limiting the movement of infected people

We consider the impact of limiting the movement of infected people on (1.1)
with mass action mechanism, that is, the long time behavior of the following
degenerate system:

$\begin{cases}\partial_{t}S=d_{S}\Delta
S-\beta(x)SI+\gamma(x)I,&x\in\Omega,t>0,\\\
\partial_{t}I=\beta(x)SI-\gamma(x)I,&x\in\bar{\Omega},t>0,\\\
\partial_{\nu}S=0,&x\in\partial\Omega,t>0,\\\ S(x,0)=S_{0}(x),\
I(x,0)=I_{0}(x),&x\in\bar{\Omega}.\end{cases}$ (3.10)

In the following result, we show the global existence of the solution of
(3.10). We remark that the $S$ component of the solution is globally bounded
while the $I$ component may blow up at $t=\infty$.

######  Proposition 3.7.

Suppose that (A1)-(A2) holds and $d_{S}>0$. Then (3.10) has a unique
nonnegative global solution $(S,I)$, where

$S\in C([0,\infty),C(\overline{\Omega}))\cap C^{1}((0,\infty),{\rm
Dom}_{\infty}(\Delta))\quad\text{and}\quad I\in
C^{1}([0,\infty),C(\overline{\Omega})).$

Moreover,
$\|S(\cdot,t)\|_{L^{\infty}(\Omega)}\leq\max\\{\|S_{0}\|_{L^{\infty}(\Omega)},\
r_{M}\\}$ for all $t\geq 0$.

###### Proof.

Suppose that a nonnegative solution exists. Let
$M_{1}=\max\\{\|S_{0}\|_{L^{\infty}(\Omega)},\ r_{M}\\}$. By the first
equation of (3.10) and the comparison principle, $0\leq S(x,t)\leq M_{1}$ for
all $x\in\bar{\Omega}$ and $t>0$, which implies that

$0\leq I(x,t)=I_{0}(x)e^{\int_{0}^{t}(\beta(x)S(x,s)-\gamma(x))ds}\leq
I_{0}(x)e^{\beta_{M}M_{1}t},\ \ \forall x\in\bar{\Omega},t\geq 0.$

This gives
$\|I(\cdot,t)\|_{L^{\infty}(\Omega)}\leq\|I_{0}\|_{L^{\infty}(\Omega)}e^{\beta_{M}M_{1}t}$
for all $t\geq 0$. The local existence of the solution of (3.10) can be proved
similar to Proposition 3.1. The a prior bound of the solution ensures the
global existence of it. ∎

To study the asymptotic behavior of the solutions of (3.10), we use the
following Lyapunov function

$V(S,I)=\int_{\Omega}\left(\frac{1}{2}S^{2}+rI\right)dx.$

If $(S,I)$ is the solution of (3.10), it is easy to check that

$\frac{d}{dt}V(S,I)=-d_{S}\int_{\Omega}|\triangledown
S|^{2}dx-\int_{\Omega}\beta(S-r)^{2}Idx.$ (3.11)

######  Remark 3.8.

We point out that it is very easy to draw a false conclusion using the above
Lyapunov function: firstly by the term $\int_{\Omega}\beta(S-r)^{2}Idx$ in
(3.11), one may conclude that either $S\to r$ or $I\to 0$ as $t\to\infty$;
then by the term $\int_{\Omega}|\triangledown S|^{2}dx$, $\triangledown S\to
0$ and so $I\to 0$ if $r$ is not constant. We will show that this intuition is
indeed false in Theorem 3.11. Actually, it is possible that $S$ converges to
some constant $\bar{S}$ with $r_{m}\leq\bar{S}\leq r_{M}$ and $I$ converges to
some measure supported at $\\{x\in\bar{\Omega}:r(x)=\bar{S}\\}$.

To conclude that $\int_{\Omega}|\triangledown S|^{2}dx\to 0$ or
$\int_{\Omega}\beta(S-r)^{2}Idx\to 0$ as $t\to\infty$, we will need the
following lemma.

######  Lemma 3.9.

Suppose that (A1)-(A2) holds and $d_{S}>0$. Let $(S,I)$ be the solution of
(3.10). Then the following conclusions hold:

  * (i)

The mapping $[1,\infty)\ni t\mapsto
S(\cdot,t)-\overline{S(\cdot,t)}\in{L^{1}(\Omega)}$ is Hölder continuous.
Furthermore if $n=1$, then the mapping $[1,\infty)\ni t\mapsto
S(\cdot,t)-\overline{S(\cdot,t)}\in{W^{2,1}(\Omega)}$ is also Hölder
continuous.

  * (ii)

If $n=1$, the mapping $[1,\infty)\ni t\mapsto\int_{\Omega}\beta(S-r)^{2}Idx$
is Hölder continuous.

###### Proof.

Setting $Z=S(\cdot,t)-\overline{S(\cdot,t)}$ and
$F(\cdot,t)=\beta(r-S(\cdot,t))I(\cdot,t)-\overline{\beta(r-S(\cdot,t))I(\cdot,t)}$,
it holds that $Z(\cdot,t),F(\cdot,t)\in\mathcal{Z}$ for all $t\geq 0$ and

$\begin{cases}\partial_{t}Z=d_{S}\Delta Z+F(\cdot,t),&t\geq 0,\
x\in\Omega,\cr\partial_{\nu}Z=0,\ &t>0,\ x\in\partial\Omega.\end{cases}$

Hence by the variation of constant formula, we have

$Z(\cdot,t)=e^{td_{S}\Delta_{|\mathcal{Z}}}Z(\cdot,0)+\int_{0}^{t}e^{(t-\tau)\Delta_{|\mathcal{Z}}}F(\cdot,\tau)d\tau,\quad\forall\
t>0.$

By Lemma 3.7 with $M:=\max\\{\|S_{0}\|_{L^{\infty}(\Omega)},r_{M}\\}$, it
holds that

$\|F(\cdot,t)\|_{L^{1}(\Omega)}\leq
2\beta_{M}M\|I(\cdot,t)\|_{L^{1}(\Omega)}\leq
M_{1}:=2NM\beta_{M},\quad\forall\ t\geq 0.$ (3.12)

(i) Fix $0\leq\tilde{\alpha}<\tilde{\alpha}+\alpha<1$. For any $t\geq 1$ and
$h>0$, by (2.1)-(2.3) and (3.12),

$\displaystyle\|Z(t+h)-Z(t)\|_{\mathcal{Z}^{\tilde{\alpha}}}$ (3.13)
$\displaystyle\leq$ $\displaystyle\|(e^{hd_{S}\Delta_{|\mathcal{Z}}}-{\rm
id})e^{td_{S}\Delta_{|\mathcal{Z}}}Z_{0}\|_{\mathcal{Z}^{\tilde{\alpha}}}+\int_{0}^{t}\|(e^{hd_{S}\Delta_{|\mathcal{Z}}}-{\rm
id})e^{(t-\tau)d_{S}\Delta_{|\mathcal{Z}}}F(\cdot,\tau)\|_{\mathcal{Z}^{\tilde{\alpha}}}d\tau$
(3.14)
$\displaystyle+\int_{0}^{h}\|e^{(h-\tau)d_{S}\Delta_{|\mathcal{Z}}}F(\cdot,t+\tau)\|_{\mathcal{Z}^{\tilde{\alpha}}}d\tau$
(3.15) $\displaystyle\leq$ $\displaystyle
C_{\alpha}d_{S}^{\alpha}h^{\alpha}\Big{(}\|e^{td_{S}\Delta_{|\mathcal{Z}}}Z_{0}\|_{\mathcal{Z}^{\tilde{\alpha}+\alpha}}+\int_{0}^{t}\|e^{(t-\tau)d_{S}\Delta_{|\mathcal{Z}}}F(\cdot,\tau)\|_{\mathcal{Z}^{\tilde{\alpha}+\alpha}}d\tau\Big{)}$
(3.16)
$\displaystyle+C_{\tilde{\alpha}}\int_{0}^{h}(d_{S}(h-\tau))^{-\tilde{\alpha}}e^{-d_{S}\sigma(h-\tau)}\|F(\cdot,t+\tau)\|_{L^{1}(\Omega)}d\tau$
(3.17) $\displaystyle\leq$ $\displaystyle
C_{\alpha}C_{\tilde{\alpha}+\alpha}d_{S}^{-\tilde{\alpha}}h^{\alpha}\Big{(}t^{-(\tilde{\alpha}+\alpha)}e^{-\sigma
td_{S}}\|Z_{0}\|_{L^{1}(\Omega)}+\int_{0}^{t}(t-\tau)^{-(\tilde{\alpha}+\alpha)}e^{-d_{S}\sigma(t-\tau)}\|F(\cdot,\tau)\|_{L^{1}(\Omega)}d\tau\Big{)}$
(3.18)
$\displaystyle+C_{\tilde{\alpha}}d_{S}^{-\tilde{\alpha}}M_{1}\int_{0}^{h}\tau^{-\tilde{\alpha}}e^{-d_{S}\sigma\tau}d\tau$
(3.19) $\displaystyle\leq$ $\displaystyle
C_{\alpha}C_{\alpha+\tilde{\alpha}}d_{S}^{-\tilde{\alpha}}h^{\alpha}\Big{(}\|Z_{0}\|_{L^{1}(\Omega)}+M_{1}\int_{0}^{\infty}\tau^{-(\tilde{\alpha}+\alpha)}e^{-d_{S}\sigma\tau}d\tau\Big{)}+C_{\tilde{\alpha}}d_{S}^{-\tilde{\alpha}}M_{1}\int_{0}^{h}\tau^{-\tilde{\alpha}}d\tau$
(3.20) $\displaystyle\leq$
$\displaystyle{M}_{\alpha,\tilde{\alpha}}(h^{\alpha}+h^{1-\tilde{\alpha}}).$
(3.21)

In particular, if $\tilde{\alpha}=0$, we get that

$\|Z(t+h)-Z(t)\|_{L^{1}(\Omega)}\leq{M}_{\alpha,\tilde{\alpha}}(h^{\alpha}+h),\quad\forall\
t\geq 1,\ h>0,$

which yields the first assertion of (i). On the other, if $n=1$, choosing
$\alpha\in(\frac{3}{4},1)$, it follows from [21, Theorem 1.6.1] that
$\mathcal{Z}^{\alpha}$ is continuously embedded in $W^{1,2}(\Omega)$. Hence
the last assertion of (i) also follows from (3.13).

(ii) Suppose $n=1$. Let $G(t):=\int_{\Omega}\beta(S-r)^{2}Idx$ for $t\geq 0$.
Since $n=1$, $W^{1,2}(\Omega)$ is compactly embbedded in
$C(\overline{\Omega})$. Since the mapping $[1,\infty)\ni t\mapsto
Z(\cdot,t)\in{W^{1,2}(\Omega)}$ is Hölder continuous by (i), the mapping
$[1,\infty)\ni t\mapsto Z(\cdot,t)\in{C(\overline{\Omega})}$ is also Hölder
continuous. This together with the fact that

$\sup_{t>0}\Big{|}\frac{d}{dt}\int_{\Omega}S(x,t)dx\Big{|}=\sup_{t>0}\Big{|}\int_{\Omega}\beta(S(x,t)-r(x))I(x,t)dx\Big{|}\leq
MN\beta_{M}$

implies that the mapping $[1,\infty)\ni t\mapsto S(\cdot,t)\in
C(\overline{\Omega})$ is also Hölder continuous. Thus there exist $0<\tau<1$
and $c>0$ such that

$\|S(\cdot,t+h)-S(\cdot,t)\|_{L^{\infty}(\Omega)}\leq c|h|^{\tau},\quad t\geq
1.$

So for any $t\geq 1$ and $h>0$, we have

$\displaystyle|G(t+h)-G(t)|\leq$
$\displaystyle\beta_{M}\|(S(\cdot,t+h)-r)^{2}-(S(\cdot,t)-r)^{2}\|_{\infty}\int_{\Omega}I(x,t+h)dx$
$\displaystyle+\beta_{M}\|(S(\cdot,t)-r)^{2}\|_{L^{\infty}(\Omega)}\int_{\Omega}|I(x,t+h)-I(x,t)|dx$
$\displaystyle\leq$ $\displaystyle
2cMNh^{\tau}\beta_{M}+M^{2}\beta_{M}\int_{\Omega}\int_{0}^{h}\beta|S(x,t+s)-r(x)|I(x,t+s)dsdx$
$\displaystyle\leq$ $\displaystyle
2cMNh^{\tau}\beta_{M}+M^{3}\beta_{M}^{2}\int_{\Omega}\int_{0}^{h}I(x,t+s)dsdx$
$\displaystyle=$ $\displaystyle
2cMNh^{\tau}\beta_{M}+M^{3}\beta_{M}^{2}\int_{0}^{h}\int_{\Omega}I(x,t+s)dxds$
$\displaystyle\leq$ $\displaystyle
2cMNh^{\tau}\beta_{M}+M^{3}\beta_{M}^{2}Nh\leq(c+M^{2}\beta_{M})M_{1}(h^{\tau}+h),$

which yields the desired result. ∎

######  Lemma 3.10.

Suppose that $n=1$, (A1)-(A2) holds and $d_{S}>0$. Let $(S,I)$ be the solution
of (3.10). Then, $\|\triangledown S(\cdot,t)\|_{L^{2}(\Omega)}\to 0$ and
$\int_{\Omega}\beta(S(x,t)-r)^{2}I(x,t)dx\to 0$ as $t\to\infty$.

###### Proof.

Integrating (3.11) over $(0,\infty)$ and by $\int_{\Omega}Idx\leq N$ and
Proposition 3.7, we have

$\int_{0}^{\infty}\|\triangledown S(\cdot,t)\|_{L^{2}(\Omega)}^{2}dt<\infty\
\text{and}\
\int_{0}^{\infty}\int_{\Omega}\beta(S(x,t)-r)^{2}I(x,t)dxdt<\infty.$ (3.22)

Then the claim follows from Lemmas 2.2 and 3.9. ∎

We are ready to state the main result concerning the global dynamics of
(3.10). Motivated by the biological meaning and expression of
$\mathcal{R}^{1}_{0}$, as in [2, 63], we call $H^{+}$ and $H^{-}$ as the high-
risk and low-risk sites, respectively, where

$H^{+}=\left\\{x\in\bar{\Omega}:\
\frac{N}{|\Omega|}\beta(x)-\gamma(x)>0\right\\}\ \ \text{and}\ \
H^{-}=\left\\{x\in\bar{\Omega}:\
\frac{N}{|\Omega|}\beta(x)-\gamma(x)<0\right\\}.$

Define $\tilde{r}_{m}=\inf_{x\in\\{I_{0}>0\\}}r(x)$ and
$\mathcal{M}:=\\{x\in\overline{\\{I_{0}>0\\}}\ :\ r(x)=\tilde{r}_{m}\\}$.
Biologically, $\mathcal{M}$ consists with all the points of the highest risk
relative to $I_{0}$. We will show that the infected people will concentrate on
$\mathcal{M}$ when limiting their movement.

######  Theorem 3.11.

Suppose that (A1)-(A2) holds and $d_{S}>0$. Let $(S,I)$ be the solution of
(3.10). Then, we have

$\lim_{t\to\infty}\|S(\cdot,t)-\overline{S(\cdot,t)}\|_{L^{p}(\Omega)}=0,\quad\forall\
p\in[1,\infty).$ (3.23)

If in addition $n=1$, then the limit in (3.23) also holds for $p=\infty$,

$\lim_{t\to\infty}\|I(\cdot,t)\|_{L^{\infty}(K\cup\\{I_{0}=0\\})}=0$ (3.24)

for any compact set $K\subset H^{-}$ if $H^{-}$ is not empty, and the
following conclusions hold:

  * (i)

If $H^{+}\cap\\{I_{0}>0\\}=\emptyset$, then
$\|S(\cdot,t)-N/{|\Omega|}\|_{L^{\infty}(\Omega)}\to 0$ and
$\int_{\Omega}I(x,t)dx\to 0$ as $t\to\infty$;

  * (ii)

If $H^{+}\cap\\{I_{0}>0\\}\neq\emptyset$, then there is a sequence
$\\{t_{k}\\}_{k\geq 1}$ converging to infinity such that
$\|S(\cdot,t_{k})-\tilde{r}_{m}\|_{L^{\infty}(\Omega)}\to 0$,
$\int_{\Omega}I(x,t_{k})dx\to N-|\Omega|\tilde{r}_{m}$ and $I(\cdot,t_{k})\to
0$ as $k\to\infty$ almost everywhere on $\\{x\in\Omega:\
r(x)\neq\tilde{r}_{m}\\}$. In particular, if
$\mathcal{M}=\\{x_{1},\cdots,x_{L}\\}\subset\\{I_{0}>0\\}$, then
$I(\cdot,t_{k})\to(N-|\Omega|\tilde{r}_{m})\sum_{i=1}^{L}c_{i}\delta_{x_{i}}$
weakly as $k\to\infty$, where $0\leq c_{i}\leq 1$, $\sum_{i=1}^{L}c_{i}=1$,
and $\delta_{x_{i}}$ is the Dirac measure centered at $x_{i}$.

###### Proof.

By the Poincaré inequality, there is a positive constant $\lambda_{0}>0$ such
that

$\|S(\cdot,t)-\overline{S(\cdot,t)}\|_{L^{2}(\Omega)}\leq\lambda_{0}\|\nabla
S(\cdot,t)\|_{L^{2}(\Omega)},\quad\forall\ t>0.$ (3.25)

Hence using Hölder’s inequality and recalling (3.11), we get that

$\displaystyle\int_{0}^{\infty}\|S(\cdot,t)-\overline{S(\cdot,t)}\|_{L^{1}(\Omega)}^{2}dt\leq$
$\displaystyle|\Omega|\int_{0}^{\infty}\|S(\cdot,t)-\overline{S(\cdot,t)}\|_{L^{2}(\Omega)}^{2}dt$
$\displaystyle\leq$
$\displaystyle\lambda_{0}^{2}|\Omega|\int_{0}^{\infty}\|\nabla
S(\cdot,t)\|_{L^{2}(\Omega)}^{2}dt<\infty.$

Therefore by Lemmas 2.2 and 3.9, we obtain
$\|S(\cdot,t)-\overline{S(\cdot,t)}\|_{L^{1}(\Omega)}^{2}\to 0$ as
$t\to\infty$. By $\sup_{t\geq 1}\|S(\cdot,t)\|_{\infty}<\infty$ and Hölder’s
inequality, (3.23) holds.

From this point, we shall suppose that $n=1$ and complete the proof of the
theorem. In view of Lemma 3.10 and inequality (3.25), we have

$\lim_{t\to\infty}\|S(\cdot,t)-\overline{S(\cdot,t)}\|_{W^{1,2}(\Omega)}=0.$

Since $n=1$, $W^{1,2}(\Omega)$ is compactly embedded into $C(\bar{\Omega})$.
Therefore,

$\lim_{t\to\infty}\big{\|}S(\cdot,t)-\overline{S(\cdot,t)}\big{\|}_{L^{\infty}(\Omega)}=0.$
(3.26)

Fix $\varepsilon>0$. By $N=\int_{\Omega}(S+I)dx$ for all $t\geq 0$ and (3.26),
there is $t_{\varepsilon}>0$ such that

$S(x,t)\leq\frac{1}{|\Omega|}\Big{(}N+\varepsilon-\int_{\Omega}I(x,t)dx\Big{)},\quad\forall\
(x,t)\in\bar{\Omega}\times[t_{\varepsilon},\infty).$ (3.27)

Let $K\subset H^{-}$ be a compact set if $H^{-}$ is not empty. By the
definition of $H^{-}$, we have $\min_{x\in K}r(x)>N/|\Omega|$. If
$0<\varepsilon\ll 1$ is chosen such that
$\eta_{\varepsilon}:=(N+\varepsilon)/|\Omega|-\min_{x\in K}r(x)<0$, then

$\partial_{t}I(x,t)\leq\beta(x)\Big{(}\frac{N+\varepsilon}{|\Omega|}-\min_{x\in
K}r(x)\Big{)}I(x,t)\leq\beta_{m}\eta_{\varepsilon}I(x,t),\quad(x,t)\in
K\times[t_{\varepsilon},\infty).$

It follows that

$\|I(\cdot,t)\|_{L^{\infty}(K)}\leq
e^{(t-t_{\varepsilon})\beta_{m}\eta_{\varepsilon}}\|I(\cdot,t_{\varepsilon})\|_{L^{\infty}(K)}\to
0,\quad\text{as}\ t\to\infty.$

This together with the fact that $I(x,t)=0$ for all $t\geq 0$ and
$x\in\\{I_{0}=0\\}$ completes the proof of (3.24).

To prove (i)-(ii), we claim that

$\limsup_{t\to\infty}\|I(\cdot,t)\|_{L^{1}(\Omega)}\leq(N-|\Omega|\tilde{r}_{m})_{+}.$
(3.28)

Since $\int_{\\{I_{0}=0\\}}I(x,t)dx=0$ for all $t\geq 0$, it suffices to show
$\limsup_{t\to\infty}\|I(\cdot,t)\|_{L^{1}(\\{I_{0}>0\\})}\leq(N-|\Omega|\tilde{r}_{m})_{+}$.
To see this, observe from (3.27) that

$\partial_{t}I\leq\frac{\beta}{|\Omega|}\Big{(}(N-|\Omega|\tilde{r}_{m})_{+}+\varepsilon-\|I(\cdot,t)\|_{L^{1}(\\{I_{0}>0\\})}\Big{)}I,\quad\forall\
t\geq t_{\varepsilon},\ x\in\\{I_{0}>0\\},$ (3.29)

where $\tilde{r}_{m}=\inf_{x\in\\{I_{0}>0\\}}r(x)$. Let

$F(t):=\frac{\int_{\\{I_{0}>0\\}}\beta
I(x,t)dx}{\int_{\\{I_{0}>0\\}}I(x,t)dx},\ \ \forall\ t\geq 1.$ (3.30)

Then $F(t):[1,\infty)\to\mathbb{R}_{+}$ is Locally Lipschitz continuous with
$\beta_{m}\leq F(t)\leq\beta_{M}$, $t\geq 1$. Integrating (3.29) over
$\\{I_{0}>0\\}$, for any $t>t_{\varepsilon}$, we get

$\displaystyle\frac{d}{dt}\|I\|_{L^{1}(\\{I_{0}>0\\})}\leq$
$\displaystyle\frac{1}{|\Omega|}\Big{(}(N-|\Omega|\tilde{r}_{m})_{+}+\varepsilon-\|I\|_{L^{1}(\\{I_{0}>0\\})}\Big{)}\int_{\\{I_{0}>0\\}}\beta
Idx$ $\displaystyle=$
$\displaystyle\frac{F(t)}{|\Omega|}\Big{(}(N-|\Omega|\tilde{r}_{m})_{+}+\varepsilon-\|I\|_{L^{1}(\\{I_{0}>0\\})}\Big{)}\|I\|_{L^{1}(\\{I_{0}>0\\})},$

where $F$ is defined by (3.30). Let $v(t)$ be the solution of

$\left\\{\begin{array}[]{lll}v^{\prime}(t)&=&\displaystyle\frac{F(t)}{|\Omega|}\Big{(}(N-|\Omega|\tilde{r}_{m})_{+}+\varepsilon-v(t)\Big{)}v(t),\
\ t>t_{\varepsilon},\\\
v(t_{\varepsilon})&=&\|I(\cdot,t_{\varepsilon})\|_{L^{1}(\\{I_{0}>0\\})}.\end{array}\right.$

By the comparison principle, we know that

$\|I(\cdot,t)\|_{L^{1}(\\{I_{0}>0\\})}\leq v(t),\quad\forall t\geq
t_{\varepsilon}.$

Since $v(t)\to(N-|\Omega|\tilde{r}_{m})_{+}+\varepsilon$ as $t\to\infty$
(because $F\geq\beta_{m}>0$ and $v(t_{\varepsilon})>0$) and $\varepsilon$ is
arbitrary chosen, (3.28) holds.

(i) Suppose that $H^{+}\cap\\{I_{0}>0\\}=\emptyset$. Then we have
$(N-|\Omega|\tilde{r}_{m})_{+}=0$. By (3.28),
$\|I(\cdot,t)\|_{L^{1}(\Omega)}\to 0$ as $t\to\infty$. This together with the
fact that $\int_{\Omega}Sdx=N-\int_{\Omega}Idx$ for all $t>0$, yields
$\int_{\Omega}S(x,t)dx\to N$ as $t\to\infty$. It then follows from (3.26) that
$\|S(\cdot,t)-{N}/{|\Omega|}\|_{L^{\infty}(\Omega)}\to 0$ as $t\to\infty$.

(ii) Suppose that $H^{+}\cap\\{I_{0}>0\\}\neq\emptyset$. Then we have
$N/|\Omega|>\tilde{r}_{m}$. Since $\int_{\Omega}Sdx=N-\int_{\Omega}Idx$, we
conclude from (3.28) that

$\liminf_{t\to\infty}\int_{\Omega}S(x,t)dx\geq|\Omega|\tilde{r}_{\min},$

which in view of (3.26) implies that

$\liminf_{t\to\infty}\min_{x\in\overline{\Omega}}S(x,t)\geq\tilde{r}_{m}.$
(3.31)

Now, we claim that

$\liminf_{t\to\infty}\min_{x\in\overline{\Omega}}S(x,t)=\tilde{r}_{m}.$ (3.32)

We proceed by contradiction. Suppose to the contrary that (3.32) is false.
Thanks to (3.31), there exist $0<\tilde{\varepsilon}\ll 1$ and
$\tilde{t}_{0}>0$ such that

$S(x,t)\geq\tilde{r}_{m}+\tilde{\varepsilon},\quad\forall\
t\geq\tilde{t}_{0}.$

Since $r$ is continuous on $\bar{\Omega}$ and
$\tilde{r}_{m}=\inf_{x\in\\{I_{0}>0\\}}r(x)$, there is an open set
$\mathcal{O}\subset\\{I_{0}>0\\}$ such that
$r(x)<\tilde{r}_{\min}+\tilde{\varepsilon}/{2}$ for all $x\in\mathcal{O}$.
Hence, we have

$\partial_{t}I(x,t)=\beta(S-r)I\geq\frac{\tilde{\varepsilon}}{2}\beta_{m}I(x,t),\quad
t>\tilde{t}_{0},\ x\in\mathcal{O}.$

An integration of this inequality yields

$N\geq\int_{\mathcal{O}}I(x,t)dx\geq
e^{\frac{\tilde{\varepsilon}}{2}\beta_{m}(t-\tilde{t}_{0})}\int_{\mathcal{O}}I(x,\tilde{t}_{0})dx,\quad
t>\tilde{t}_{0}.$

This is clearly impossible since $\int_{\mathcal{O}}I(x,\tilde{t}_{0})dx>0$.
Therefore, (3.32) must hold.

By (3.32), there is a sequence $\\{t_{k}\\}_{k\geq 1}$ converging to infinity
such that $\min_{x\in\overline{\Omega}}S(x,t_{k})\to\tilde{r}_{m}$ as
$k\to\infty$. Hence by (3.26),
$\|S(\cdot,t_{k})-\tilde{r}_{m}\|_{L^{\infty}(\Omega)}\to 0$ and
$\int_{\Omega}I(x,t_{k})dx\to N-|\Omega|\tilde{r}_{m}$ as $k\to\infty$.
However, by Lemma 3.10, we know that
$\int_{\\{I_{0}>0\\}}(S(x,t_{k})-r)^{2}I(x,t_{k})dx=\int_{\Omega}(S(x,t_{k})-r)^{2}I(x,t_{k})dx\to
0$ as $k\to\infty$. Therefore, possibly after passing to a subsequence,
$I(\cdot,t_{k})\to 0$ as $k\to\infty$ almost everywhere on $\\{x\in\Omega:\
r(x)\neq\tilde{r}_{m}\\}$. Finally since $\\{I(\cdot,t_{k})\\}$ is bounded in
$L^{1}(\Omega)$ and by Riesz representation theorem, passing to a subsequence
if necessary, $I(\cdot,t_{k})\to(N-|\Omega|\tilde{r}_{m})\mu$ weakly as
$k\to\infty$ for some probability Radon measure $\mu$. Since

$\int_{\\{I_{0}>0\\}}\beta(S(x,t_{k})-r)^{2}I(x,t_{k})dx\to(N-|\Omega|\tilde{r}_{m})\int_{\\{I_{0}>0\\}}\beta(r_{m}-r)^{2}d\mu=0\
\ as\ {k\to\infty},$

$\mu$ is supported in $\overline{\\{I_{0}>0\\}}\cap\mathcal{M}$. In particular
if $\mathcal{M}=\\{x_{1},\cdots,x_{L}\\}\subset\\{I_{0}>0\\}$, then
$\mu=\sum_{i=1}^{L}c_{i}\delta_{x_{i}}$ for some $0\leq c_{i}\leq 1$ with
$\sum_{i=1}^{L}c_{i}=1$. ∎

######  Remark 3.12.

We conjecture that Theorem 3.11 holds for any $N\geq 1$ and
$\|S(\cdot,t)-\tilde{r}_{m}\|_{L^{\infty}(\Omega)}\to 0$ as $t\to\infty$ in
(ii).

##  4 Model with standard incidence mechanism

###  4.1 Limiting the movement of susceptible people

First, we consider the impact of limiting the movement of susceptible people
on (1.1) with standard incidence mechanism by setting $d_{S}=0$, i.e.,

$\begin{cases}\displaystyle\partial_{t}S=-\beta(x)\frac{SI}{S+I}+\gamma(x)I,&x\in\bar{\Omega},t>0,\\\
\displaystyle\partial_{t}I=d_{I}\Delta
I+\beta(x)\frac{SI}{S+I}-\gamma(x)I,&x\in\Omega,t>0,\\\
\partial_{\nu}I=0,&x\in\partial\Omega,t>0,\\\ S(x,0)=S_{0}(x),\
I(x,0)=I_{0}(x),&x\in\bar{\Omega}.\end{cases}$ (4.1)

######  Proposition 4.1.

Suppose that (A1)-(A2) holds and $d_{I}>0$. Then (4.1) has a unique
nonnegative global solution $(S,I)$, where

$S\in C^{1}([0,\infty),C(\overline{\Omega}))\quad\text{and}\quad I\in
C([0,\infty),C(\overline{\Omega}))\cap C^{1}((0,\infty),{\rm
Dom}_{\infty}(\Delta)).$

Moreover, there exists $M>0$ depending on (the $L^{\infty}$ norm of) initial
data such that

$\|I(\cdot,t)\|_{L^{\infty}(\Omega)}\leq M,\ \forall t\geq 0.$ (4.2)

###### Proof.

If we define $SI/(S+I)=0$ when $(S,I)=(0,0)$, then $SI/(S+I)$ is Lipschitz in
the first quadrant. So the existence and uniqueness of nonnegative local
solution $(S,I)$ can be proved using the Banach fixed point theorem, where
$S\in C^{1}([0,T),C(\overline{\Omega}))$ and $I\in
C([0,T),C(\overline{\Omega}))\cap C^{1}((0,T),{\rm Dom}_{\infty}(\Delta))$ for
some $T>0$.

Since the right hand side of the second equation of (4.1) has linear growth
rate (i.e. $\beta SI/(S+I)-\gamma I\leq CI$ for some constant $C>0$) and
$\int_{\Omega}Idx\leq N$ for all $t>0$, by [1, Theorem 3.1], there exists
$M>0$ such that $\|I(\cdot,t)\|_{L^{\infty}(\Omega)}\leq M$ for all $t>0$. By
the first equation of (4.1), we have $\partial_{t}S\leq\gamma I$, which
implies

$\|S(\cdot,t)\|_{L^{\infty}(\Omega)}\leq\|S_{0}\|_{L^{\infty}(\Omega)}+M\gamma_{M}t,\
\ t\geq 0.$

Hence, we can extend the solution globally. ∎

We define $H^{+}$, $H^{0}$, and $H^{-}$ as the high-risk, moderate-risk, and
low-risk sites, respectively, where $H^{+}=\left\\{x\in\bar{\Omega}:\
\beta(x)-\gamma(x)>0\right\\}$, $H^{0}=\left\\{x\in\bar{\Omega}:\
\beta(x)-\gamma(x)=0\right\\}$ and $H^{-}=\left\\{x\in\bar{\Omega}:\
\beta(x)-\gamma(x)<0\right\\}$. The following result has appeared in the
literature.

######  Theorem 4.2.

[35, Theorem 2.5, Lemma 5.6] Suppose that (A1)-(A2) holds and $d_{I}>0$. Let
$(S,I)$ be the solution of (4.1). Then the following conclusions hold:

  1. (i)

If $H^{-}$ is nonempty, then
$\lim_{t\to\infty}\|S(\cdot,t)-S^{*}\|_{L^{\infty}(\Omega)}=0$ and
$\lim_{t\to\infty}\|I(\cdot,t)\|_{L^{\infty}(\Omega)}=0$, where $S^{*}\in
C(\bar{\Omega})$ satisfies $S^{*}>0$ on $H^{-}\cup H^{0}$.

  2. (ii)

If $\beta(x)>\gamma(x)$ for all $x\in\bar{\Omega}$, then
$\lim_{t\to\infty}\|S(t,\cdot)-\gamma
I^{*}/(\beta-\gamma)\|_{L^{\infty}(\Omega)}=0$ and
$\lim_{t\to\infty}\|I(t,\cdot)-I^{*}\|_{L^{\infty}(\Omega)}=0$, where $I^{*}$
is a constant given by

$I^{*}:=\frac{N}{\int_{\Omega}\frac{\beta}{\beta-\gamma}dx}.$ (4.3)

######  Remark 4.3.

The only case not covered by Theorem 4.2 is when $\beta\geq\gamma$ and $H^{0}$
is not empty, which we will deal with later in this section. In the case
$H^{-}\neq\emptyset$ and $\mathcal{R}_{0}>1$, it has been shown in [35,
Theorem 2.5] that $J^{*}:=\\{x\in\bar{\Omega}:S^{*}(x)=0\\}$ is a subset of
$H^{+}$ such that both $J^{*}$ and $\Omega\backslash J^{*}$ have positive
measure, where $\mathcal{R}_{0}$, defined as

$\mathcal{R}_{0}=\sup\left\\{\frac{\int_{\Omega}\beta\phi^{2}dx}{\int_{\Omega}(d_{I}|\triangledown\phi|^{2}+\gamma\phi^{2})dx}:\
\ \phi\in H^{1}(\Omega)\backslash\\{0\\}\right\\},$ (4.4)

is the the basic reproduction number of the diffusive epidemic model (1.1)
with standard infection incidence mechanism and diffusion rates $d_{S}>0$ and
$d_{I}>0$.

######  Theorem 4.4.

Suppose that (A1)-(A2) holds and $d_{I}>0$. Let $(S,I)$ be the solution of
(4.1). If $\beta(x)\geq\gamma(x)$ for all $x\in\bar{\Omega}$ and $H^{0}$ is
nontrivial, then the following conclusions hold:

  1. (i)

If $H^{0}$ has positive measure, then there exists $M>0$ depending on (the
$L^{\infty}$ norm of) initial data such that

$\|S(\cdot,t)\|_{L^{\infty}(\Omega)}\leq M,\ \ \text{for all }t\geq 0.$ (4.5)

Moreover, there is $S^{*}\in C(H^{0})$ satisfying $S^{*}>0$ on $H^{0}$ such
that $\lim_{t\to\infty}\|S(\cdot,t)-S^{*}\|_{L^{\infty}(H^{0})}=0$, and
$\lim_{t\to\infty}\|I(\cdot,t)\|_{L^{\infty}(\Omega)}=0$.

  2. (ii)

If $H^{0}$ has zero measure and $\int_{\Omega}1/(\beta-\gamma)dx=\infty$, then
there exists $\\{t_{k}\\}$ converging to infinity such that $I(\cdot,t_{k})\to
0$ in $C(\bar{\Omega})$ as $n\to\infty$.

###### Proof.

(i) By Lemma 2.3, there exists $C>1$ such that

$\max_{x\in\bar{\Omega}}I(x,t)\leq C\min_{x\in\bar{\Omega}}I(x,t),\ \
x\in\bar{\Omega},t\geq 1.$ (4.6)

Fix $x_{0}\in H^{0}$. Let $x\in\bar{\Omega}$. Then
$\partial_{t}S=\frac{(-\beta+\gamma)S+\gamma I}{S+I}I\leq\gamma I^{2}/(S+I)$.
We consider the following problem:

$\begin{cases}\displaystyle\bar{S}^{\prime}=\gamma(x)\frac{I^{2}(x,t)}{\bar{S}+I(x,t)},&t>1,\\\
\bar{S}(1)=S(x,1).&\end{cases}$ (4.7)

Then we have $S(x,t)\leq\bar{S}(t)$ for all $t\geq 1$. We claim that
$KS(x_{0},t)$ is an upper solution of (4.7) if $K>0$ is large enough. To see
it, it suffices to check

$K\partial_{t}S(x_{0},t)=K\gamma(x_{0})\frac{I^{2}(x_{0},t)}{S(x_{0},t)+I(x_{0},t)}\geq\gamma(x)\frac{I^{2}(x,t)}{KS(x_{0},t)+I(x,t)}.$

Noticing (4.6), we only need to check

$K\gamma(x_{0})\frac{I^{2}(x,t)/C^{2}}{S(x_{0},t)+{\frac{I(x,t)}{C}}}\geq\gamma(x)\frac{I^{2}(x,t)}{KS(x_{0},t)+I(x,t)},$

which is equivalent to
$(K^{2}\gamma(x_{0})-C^{2}\gamma(x))S(x_{0},t)+(K\gamma(x_{0})-{C}\gamma(x))I(x,t)\geq
0$. So we can choose $K$ large independent of $x\in\bar{\Omega}$ such that the
inequality holds. Hence, we have $KS(x_{0},t)\geq\bar{S}(t)\geq S(x,t)$ for
all $t\geq 1$. Moreover, interchanging the role of $x_{0}$ and $x$, we have

$S(x_{0},t)/K\leq S(x,t)\leq KS(x_{0},t),\ \ \ \forall x\in H^{0},\ t\geq 1.$
(4.8)

By (4.8),
$N\geq\int_{\Omega}S(x,t)dx\geq\int_{H^{0}}S(x,t)dx\geq\int_{H^{0}}S(x_{0},t)/Kdx=|H^{0}|S(x_{0},t)/K$
for all $t\geq 1$. Therefore, we have $S(x_{0},t)\leq KN/|H^{0}|$ and
$S(x,t)\leq K^{2}N/|H^{0}|$ for all $x\in\bar{\Omega}$ and $t\geq 1$.

The convergence of $(S,I)$ can be proved similar to [35, Lemma 5.6], and we
include it for completeness. By Proposition 4.1, we have $0\leq
S(x,t),I(x,t)\leq M$ for all $x\in\bar{\Omega}$ and $t\geq 0$. It follows from
the equation of $S$ that

$\partial_{t}S=\frac{\gamma
I^{2}}{S+I}\geq\frac{\gamma_{m}}{2M}\min_{x\in\Omega^{0}}I^{2}(x,t),\ \
\forall\ x\in H^{0}.$

Integrating the above inequality over $H^{0}\times(0,\infty)$ and noticing
that $H^{0}$ has positive measure, we see that $\int_{0}^{\infty}\min_{x\in
H^{0}}I^{2}(x,t)dx<\infty$. By (4.6), it holds that

$\int_{0}^{\infty}\|I(\cdot,t)\|^{2}_{L^{\infty}(\Omega)}dt<\infty.$ (4.9)

By (4.2) and the $L^{p}$ estimate, $I$ is Hölder continuous on
$\bar{\Omega}\times[1,\infty)$. Therefore, Lemma 2.2 and (4.9) imply that
$I(x,t)\to 0$ uniformly on $\bar{\Omega}$ as $t\to\infty$.

Since $\partial_{t}S=\gamma I^{2}/(S+I)$, $S(x,t)$ is strictly increasing in
$t\in(0,\infty)$ for every $x\in H^{0}$ and

$\displaystyle\int_{1}^{\infty}\|S_{t}(\cdot,t)\|_{L^{\infty}(H^{0})}dt\leq$
$\displaystyle\gamma_{M}\int_{1}^{\infty}\frac{\|I(\cdot,t)\|_{L^{\infty}(\Omega)}^{2}}{\min_{x\in
H^{0}}S(x,t)}dt$ (4.10) $\displaystyle\leq$
$\displaystyle\frac{\gamma_{M}}{\min_{x\in
H^{0}}S(x,1)}\int_{1}^{\infty}\|I(\cdot,t)\|^{2}_{L^{\infty}(\Omega)}dt<\infty.$
(4.11)

Whence, $S(\cdot,t)\to S^{*}:=S(\cdot,0)+\int_{0}^{\infty}S_{t}(\cdot,t)dt\in
C(H^{0})$ uniformly on $H^{0}$ as $t\to\infty$.

(ii) Let $\kappa=(\beta-\gamma)/\gamma$. Define

$V(S,I)=\frac{1}{2}\int_{\Omega}\left(\kappa S^{2}+I^{2}\right)dx.$

It is easy to check that

$\frac{d}{dt}V(S,I)=-d_{I}\int_{\Omega}|\triangledown
I|^{2}dx-\int_{\Omega}\gamma\frac{(\kappa S-I)^{2}}{S+I}Idx.$ (4.12)

Integrating (4.12) over $(0,t)$ and taking $t\to\infty$, we find that

$\int_{0}^{\infty}\int_{\Omega}|\triangledown I|^{2}dxdt<\infty.$ (4.13)

By (4.2), the parabolic estimates and the Sobolev embedding theorem, $I\in
C^{1+\alpha,1+\alpha/2}(\bar{\Omega}\times[1,\infty)])$. So by (4.13) and
Lemma 2.2, $\|\triangledown I(\cdot,t)\|_{L^{2}(\Omega)}\to 0$ as
$t\to\infty$. Moreover, let $\omega_{I}:=\cap_{t\geq 1}\overline{\cup_{s\geq
t}I(\cdot,s))}$, where the completion is in $C(\bar{\Omega})$. Then
$\omega_{I}$ is well-defined, compact, and consists with constants.

Suppose to the contrary that $0\not\in\omega_{I}$. By the compactness of
$\omega_{I}$, there exists $\varepsilon_{0}>0$ such that

$\liminf_{t\to\infty}\min_{x\in\bar{\Omega}}I(x,t)>\varepsilon_{0}.$ (4.14)

By the first equation of (4.1), we have

$\partial_{t}S=\frac{(\gamma I-(\beta-\gamma)S)I}{S+I}.$ (4.15)

This combined with (4.14) implies that
$\liminf_{t\to\infty}S(x,t)\geq\varepsilon_{0}\gamma/(\beta-\gamma)$ pointwise
for $x\in\Omega\backslash H^{0}$ as $t\to\infty$. By Fatou’s Lemma, we have

$\int_{\Omega\backslash
H^{0}}\left(\frac{\varepsilon_{0}\gamma}{\beta-\gamma}+\varepsilon_{0}\right)dx\leq\int_{\Omega}\liminf_{t\to\infty}(S+I)dx\leq\liminf_{t\to\infty}\int_{\Omega}(S+I)dx=N.$

Since $H^{0}$ has measure zero and $\int_{\Omega}1/(\beta-\gamma)dx=\infty$,
the first term in the above inequality equals infinity. This is a
contradiction, and therefore $0\in\omega_{I}$. Hence there exists
$\\{t_{k}\\}$ converging to infinity such that $I(\cdot,t_{k})\to 0$ in
$C(\bar{\Omega})$ as $n\to\infty$.

∎

###  4.2 Limiting the movement of infected people

Then, we consider the impact of limiting the movement of infected people on
(1.1) with standard incidence mechanism by setting $d_{I}=0$, i.e.,

$\begin{cases}\displaystyle\partial_{t}S=d_{S}\Delta
S-\beta(x)\frac{SI}{S+I}+\gamma(x)I,&x\in\Omega,t>0,\\\
\displaystyle\partial_{t}I=\beta(x)\frac{SI}{S+I}-\gamma(x)I,&x\in\bar{\Omega},t>0,\\\
\partial_{\nu}S=0,&x\in\partial\Omega,t>0,\\\ S(x,0)=S_{0}(x),\
I(x,0)=I_{0}(x),&x\in\bar{\Omega}.\end{cases}$ (4.16)

We will establish a prior bound for the solution of (4.16) first.

######  Proposition 4.5.

Suppose that (A1)-(A2) holds and $d_{S}>0$. Then (4.16) has a unique
nonnegative global solution $(S,I)$, where

$S\in C([0,\infty),C(\overline{\Omega}))\cap C^{1}((0,\infty),{\rm
Dom}_{\infty}(\Delta))\quad\text{and}\quad I\in
C^{1}([0,\infty),C(\overline{\Omega})).$

Moreover, there exists $M>0$ depending on (the $L^{\infty}$ norm of) initial
data such that

$\|S(\cdot,t)\|_{L^{\infty}(\Omega)},\|I(\cdot,t)\|_{L^{\infty}(\Omega)}\leq
M,\ \ \text{for all }t\geq 0.$ (4.17)

###### Proof.

Similar to Proposition 4.1, (4.16) has a unique local nonnegative solution
$(S,I)$, where $S\in C([0,T),C(\overline{\Omega}))\cap C^{1}((0,T),{\rm
Dom}_{\infty}(\Delta))$ and $I\in C^{1}([0,T),C(\overline{\Omega}))$ for some
$T>0$. It remains to show the boundedness of the solution.

We claim that for any nonnegative integer $k$ there exists $C>0$ depending on
initial data such that
$\|S(\cdot,t)\|_{L^{2^{k}}(\Omega)},\|I(\cdot,t)\|_{L^{2^{k}}(\Omega)}\leq C$
for all $t\geq 0$. We prove this claim by induction. It is easy to see that
the claim holds for $k=0$. Now we assume that the claim holds for $k$ and will
show that it holds for $k+1$. To see it, multiplying both sides of the second
equation of (4.16) by $I^{2^{k+1}-1}$ and integrating over $\Omega$, we obtain

$\displaystyle\frac{1}{2^{k+1}}\frac{d}{dt}\int_{\Omega}I^{2^{k+1}}dx$
$\displaystyle\leq$
$\displaystyle\beta_{M}\int_{\Omega}\frac{SI^{{2^{k+1}}-1}I}{S+I}dx-\gamma_{m}\int_{\Omega}I^{2^{k+1}}dx$
(4.18) $\displaystyle\leq$ $\displaystyle
C_{0}\int_{\Omega}S^{2^{k+1}}dx-\frac{\gamma_{m}}{2}\int_{\Omega}I^{2^{k+1}}dx,$

where we have used Young’s inequality in the last step.

Multiplying both sides of the first equation of (4.16) by $S^{2^{k+1}-1}$ and
integrating over $\Omega$, we obtain

$\displaystyle\frac{1}{2^{k+1}}\frac{d}{dt}\int_{\Omega}S^{2^{k+1}}dx$
$\displaystyle=$
$\displaystyle-\frac{2^{k+1}-1}{2^{k+1}}d_{S}\int_{\Omega}|\triangledown
S^{2^{k}}|^{2}dx-\int_{\Omega}\frac{\beta
S^{2^{k+1}}I}{S+I}pdx+\int_{\Omega}\gamma S^{2^{k+1}-1}Idx$ (4.19)
$\displaystyle\leq$ $\displaystyle-C_{1}\int_{\Omega}|\triangledown
S^{2^{k}}|^{2}dx+C_{2}\left(\int_{\Omega}S^{2^{k+1}}dx+\int_{\Omega}I^{2^{k+1}}dx\right),$

where we have used Young’s inequality in last step.

Multiplying (4.19) by $\displaystyle C_{3}:=\frac{\gamma_{m}}{4C_{2}}$ and
summing up with (4.18), we have

$\frac{d}{dt}\int_{\Omega}\left(C_{3}S^{2^{k+1}}+I^{2^{k+1}}\right)dx\leq-
C_{4}\int_{\Omega}|\triangledown
S^{2^{k}}|^{2}dx+C_{5}\int_{\Omega}S^{2^{k+1}}dx-
C_{6}\int_{\Omega}I^{2^{k+1}}dx.$

By the following interpolation inequality

$\displaystyle\|u\|_{L^{2}(\Omega)}^{2}\leq\epsilon\|\triangledown
u\|^{2}_{L^{2}(\Omega)}+C_{\epsilon}\|u\|_{L^{1}(\Omega)}^{2},\ \ \forall u\in
H^{1}(\Omega),$

we obtain

$\frac{d}{dt}\int_{\Omega}\left(C_{3}S^{2^{k+1}}+I^{2^{k+1}}\right)dx\leq
C_{7}\left(\int_{\Omega}S^{2^{k}}dx\right)^{2}-C_{8}\int_{\Omega}\left(C_{3}S^{2^{k+1}}+I^{2^{k+1}}\right)dx.$
(4.20)

By the assumption that the claim holds for $k$ and (4.20) there exists $C>0$
such that
$\|S(\cdot,t)\|_{L^{2^{k+1}}(\Omega)},\|I(\cdot,t)\|_{L^{2^{k+1}}(\Omega)}\leq
C$. This proves the claim.

Fixing $p>N+2$, by the claim and the parabolic $L^{p}$ estimate, there exists
$C>0$ such that $\|S\|_{W^{2,1}_{p}(\Omega\times(\tau,\tau+1))}<C$ for any
$\tau>0$. Since $W^{2,1}_{p}(\Omega\times(\tau,\tau+1))$ can be embedded into
$C(\bar{\Omega}\times[\tau,\tau+1])$, we obtain the boundedness of $S$. We
rewrite the equation of $I$ as

$\partial_{t}I=\frac{((\beta-\gamma)S-\gamma I)I}{S+I}.$ (4.21)

Hence, $\|I(\cdot,t)\|_{L^{\infty}(\Omega)}\leq\max_{0\leq s\leq
t}\\{\|I_{0}\|_{L^{\infty}(\Omega)},\
\|(\beta+\gamma)S(\cdot,s)\|_{L^{\infty}(\Omega)}/\gamma_{m}\\}$ for any
$t\geq 0$. This proves the boundedness of $I$. ∎

We are ready to study the asymptotic behavior of the solution of (4.16). Let

$\kappa=\frac{\gamma}{(\beta-\gamma)}\chi_{H^{+}\cap\\{I_{0}>0\\}}$

and define

$V(S,I)=\frac{1}{2}\int_{\Omega}\left(S^{2}+\kappa I^{2}\right)dx.$

It is easy to check that

$\displaystyle\displaystyle\dot{V}(S,I)=$ $\displaystyle-
d_{S}\int_{\Omega}|\triangledown S|^{2}dx+\int_{H^{-}\cup
H^{0}\cup\\{I_{0}=0\\}}S\left(-{\beta}\frac{SI}{S+I}+\gamma I\right)dx$ (4.22)
$\displaystyle-\int_{H^{+}\cap\\{I_{0}>0\\}}(\beta-\gamma)_{+}\frac{(S-\kappa
I)^{2}}{S+I}Idx.$ (4.23)

The function $V$ does not satisfy $\dot{V}\leq 0$ (the second term on the
right hand side of (4.22) is positive), but it still enables us to conclude
the convergence of the solution.

######  Theorem 4.6.

Suppose that (A1)-(A2) holds and $d_{S}>0$. Let $(S,I)$ be the solution of
(4.16). Then the following statements hold:

  * (i)

There is a positive number $\overline{S}$ such that

$\sup_{t\geq
0}\left(\|S(\cdot,t)\|_{L^{\infty}(\Omega)}+\|I(\cdot,t)\|_{L^{\infty}(\Omega)}\right)\leq\overline{S}.$
(4.24)

Furthermore, there exist $t_{1}>0$ and $\underline{S}>0$, independent of
initial data, such that

$\underline{S}\leq\min_{x\in\overline{\Omega}}S(x,t)\leq\max_{x\in\overline{\Omega}}S(x,t)\leq\overline{S},\quad\forall\
t\geq t_{1},$ (4.25)

and

$\displaystyle(R-1)_{+}\underline{S}\chi_{\\{I_{0}>0\\}}\leq$
$\displaystyle\liminf_{t\to\infty}I(x,t)$ (4.26) $\displaystyle\leq$
$\displaystyle\limsup_{t\to\infty}I(x,t)\leq(R-1)_{+}\overline{S}\chi_{\\{I_{0}>0\\}},\quad\forall\
x\in\bar{\Omega}.$ (4.27)

  * (ii)

If $1/(\beta-\gamma)\in L^{1}(H^{+}\cap\\{I_{0}>0\\})$, then

$\lim_{t\to\infty}(S(x,t),I(x,t)=(S^{*},I^{*})),\ \ \text{uniformly for}\
x\in\bar{\Omega},$

where

$I^{*}=\frac{(\beta-\gamma)_{+}S^{*}}{\gamma}\chi_{H^{+}\cap\\{I_{0}>0\\}}$

and $S^{*}$ is a positive constant given by

$S^{*}=\frac{N}{|\Omega|+\int_{H^{+}\cap\\{I_{0}>0\\}}\frac{(\beta-\gamma)_{+}}{\gamma}dx}.$
(4.29)

###### Proof.

(i) Note that (4.24) follows from Proposition 4.5. Next, observe that

$\displaystyle\frac{d}{dt}\int_{\Omega}Sdx=$ $\displaystyle\int_{\Omega}\gamma
Idx-\int_{\Omega}\beta\frac{I}{I+S}Sdx$ $\displaystyle\geq$
$\displaystyle\gamma_{m}\int_{\Omega}Idx-\beta_{M}\int_{\Omega}Sdx$
$\displaystyle=$
$\displaystyle\gamma_{m}\left(N-\int_{\Omega}Sdx\right)-\beta_{M}\int_{\Omega}Sdx$
$\displaystyle=$
$\displaystyle\gamma_{m}N-(\beta_{M}+\gamma_{m})\int_{\Omega}Sdx.$

Therefore, by $\int_{\Omega}S_{0}dx\geq 0$ and the comparison principle, we
have that

$\int_{\Omega}S(x,t)dx\geq\frac{\gamma_{m}N}{\beta_{M}+\gamma_{m}}(1-e^{-t(\beta_{M}+\gamma_{m})}),\quad\forall\
t\geq 0.$ (4.30)

Next by (4.16), we see that

$\begin{cases}\partial_{t}S\geq d_{S}\Delta
S-\beta_{M}S,&x\in\Omega,t>0,\cr\partial_{\nu}S=0,&x\in\partial\Omega,\
t>0.\end{cases}$

For any $t_{0}>0$, it follows from the comparison principle for parabolic
equations that

$S(\cdot,t+t_{0})\geq
e^{-t\beta_{M}}e^{td_{S}\Delta}S(\cdot,t_{0}),\quad\forall\ t>0.$ (4.31)

Thanks to the Harnack’s inequality (see Lemma 2.3), there is a positive
constant $c_{0}$ such that

$e^{td_{S}\Delta}S(\cdot,t_{0})(x)\geq
c_{0}e^{td_{S}\Delta}S(\cdot,t_{0})(y),\quad\forall\ td_{S}\geq 1,\
x,y\in\bar{\Omega},\ t_{0}>0.$ (4.32)

This in turn together with (4.30) implies that

$\displaystyle\min_{x\in\overline{\Omega}}e^{d_{S}t\Delta}S(\cdot,t_{0})(x)\geq$
$\displaystyle\frac{c_{0}}{|\Omega|}\int_{\Omega}e^{td_{S}\Delta}S(\cdot,t_{0})(y)dy$
$\displaystyle=$
$\displaystyle\frac{c_{0}}{|\Omega|}\int_{\Omega}S(y,t_{0})dy$
$\displaystyle\geq$
$\displaystyle\frac{\gamma_{m}Nc_{0}}{|\Omega|(\beta_{M}+\gamma_{m})}(1-e^{-t_{0}(\beta_{M}+\gamma_{m})}),\quad\quad
t\geq\frac{1}{d_{S}},\ t_{0}>0.$

As a result, it follows from (4.31) that

$S\left(x,\frac{1}{d_{S}}+t_{0}\right)\geq\frac{\gamma_{m}Nc_{0}e^{-\frac{\beta_{M}}{d_{S}}}}{|\Omega|(\beta_{M}+\gamma_{m})}(1-e^{-t_{0}(\beta_{M}+\gamma_{m})}),\quad\quad\
x\in\bar{\Omega},t_{0}>0.$

Therefore taking $t_{1}=\frac{1}{d_{S}}+\frac{\ln(2)}{\beta_{M}+\gamma_{m}}$,
(4.25) holds with
$\underline{S}:=\frac{\gamma_{m}Nc_{0}e^{-\frac{\beta_{M}}{d_{S}}}}{2|\Omega|(\beta_{M}+N\gamma_{m})}>0$,
where $\underline{S}$ and $t_{1}$ are independent of the initial data.

Next, we show that (4.26) holds. To this end, we first note that

$\displaystyle I_{t}=\beta\Big{(}(1-r)-\frac{I}{I+S}\Big{)}I\leq$
$\displaystyle\beta\Big{(}(1-r)-\frac{I}{I+\overline{S}}\Big{)}I$
$\displaystyle=$
$\displaystyle\gamma\Big{(}(R-1)\overline{S}-I\Big{)}\frac{I}{I+\overline{S}},\
\quad t>0,$

which in view of the comparison principle for ordinary differential equations
implies that $\limsup_{t\to\infty}I(x,t)\leq(R-1)_{+}\overline{S}$. Recalling
that $I(x,t)=0$ for all $t>0$ whenever $I_{0}(x)=0$, we then conclude that
$\limsup_{t\to\infty}I(x,t)\leq(R-1)_{+}\overline{S}\chi_{\\{I_{0}>0\\}}$.

Similarly, observing that

$\displaystyle
I_{t}=\beta\Big{(}(1-\frac{\gamma}{\beta})-\frac{I}{I+S}\Big{)}I\geq$
$\displaystyle\beta\Big{(}(1-r)-\frac{I}{I+\underline{S}}\Big{)}I$
$\displaystyle=$
$\displaystyle\gamma\Big{(}(R-1)\underline{S}-I\Big{)}\frac{I}{I+\underline{S}},\quad
t>t_{1},$

we can proceed as in the previous case to establish that
$\liminf_{t\to\infty}I(x,t)\geq(R-1)_{+}\underline{S}\chi_{\\{I_{0}>0\\}}$,
which completes the proof of (i).

(ii) By the equation of $I$, we have

$\int_{0}^{t}\int_{H^{-}\cup
H^{0}\cup\\{I_{0}=0\\}}\left(\beta\frac{SI}{S+I}-\gamma
I\right)dxds=\int_{H^{-}\cup H^{0}\cup\\{I_{0}=0\\}}I(x,t)dx-\int_{H^{-}\cup
H^{0}\cup\\{I_{0}=0\\}}I_{0}dx.$

Note that

$\beta\frac{SI}{S+I}-\gamma I=\frac{((\beta-\gamma)S-I)I}{S+I}\leq 0,\ \
\forall x\in H^{-}\cup H^{0}\cup\\{I_{0}=0\\}.$

We have

$0\leq\int_{0}^{\infty}\int_{H^{-}\cup
H^{0}\cup\\{I_{0}=0\\}}\left(-\beta\frac{SI}{S+I}+\gamma I\right)dxds<\infty,$

which together with (4.24) yields that

$0\leq\int_{0}^{\infty}\int_{H^{-}\cup
H^{0}\cup\\{I_{0}=0\\}}S\left(-{\beta}\frac{SI}{S+I}+\gamma
I\right)dxdt<\infty.$

Hence integrating (4.22) over $(0,\infty)$, we obtain that

$\int_{0}^{\infty}\int_{\Omega}|\triangledown S|^{2}dxdt<\infty$ (4.33)

and

$\int_{0}^{\infty}\int_{H^{+}\cap\\{I_{0}>0\\}}(\beta-\gamma)_{+}I\frac{(S-\kappa
I)^{2}}{S+I}dxdt<\infty.$ (4.34)

By (4.24), we have

$\sup_{t\geq 0}\|\frac{\beta SI}{S+I}-\gamma I\|_{L^{\infty}(\Omega)}<\infty.$

So by the regularity theory for parabolic equations and Sobolev embedding
theorem, the mappings $\nabla S(x,t)$ and $S$ are Hölder continuous on
$\overline{\Omega}\times[1,\infty)$, and $\\{S(\cdot,t)\\}_{t\geq 1}$ is
precompact in $C^{1+\alpha}(\Omega)$, $0<\alpha<1$. Then by (4.33) and Lemma
2.2, $\int_{\Omega}|\nabla S|^{2}dx\to 0$ as $t\to\infty$. Hence the set
$w_{S}:=\cap_{t\geq 1}\overline{\cup_{s\geq t}\\{S(\cdot,s)\\}}$ consists of
positive constant functions. Furthermore since $\sup_{t\geq
0}\|\partial_{t}I(\cdot,t)\|_{\infty}<\infty$ and $S$ is Hölder continuous on
$\overline{\Omega}\times[1,\infty)$, the mapping

$t\mapsto\int_{H^{+}\cap\\{I_{0}>0\\}}(\beta-\gamma)_{+}\frac{(S-\kappa
I)^{2}}{S+I}Idx$

is Hölder continuous on $[1,\infty)$. Therefore by Lemma 2.2 and (4.34), we
have

$\lim_{t\to\infty}\int_{H^{+}\cap\\{I_{0}>0\\}}(\beta-\gamma)_{+}\frac{(S-\kappa
I)^{2}}{S+I}Idx=0.$ (4.35)

Now, let $S^{*}\in w_{S}$. Then there is a sequence $t_{k}\to\infty$ such that
$S(\cdot,t_{k})\to S^{*}$ uniformly on $\bar{\Omega}$ as $k\to\infty$. Since
(4.35) holds, after passing to a subsequence if necessary, we may suppose that

$\lim_{n\to\infty}I(x,t_{k})\frac{(S^{*}-\kappa(x)I(x,t_{k}))^{2}}{S^{*}+I(x,t_{k})}=0\quad\text{
a.e. on }\ H^{+}\cap\\{I_{0}>0\\}.$ (4.36)

However, when $x\in H^{+}\cap\\{I_{0}>0\\}$, we have from (4.26) that
$\liminf_{t\to\infty}I(x,t)\geq\underline{S}(\beta(x)-\gamma(x))>0$.
Therefore, we conclude from (4.36) that

$\lim_{t\to\infty}I(x,t_{k})=\frac{S^{*}}{\kappa(x)}=\frac{(\beta(x)-\gamma(x))S^{*}}{\gamma(x)}\quad\text{
a.e. on }\ H^{+}\cap\\{I_{0}>0\\}.$

Therefore by (4.26) again, $\lim_{t\to\infty}I(x,t)=0$ almost everywhere for
$x\in\Omega\setminus(H^{+}\cap\\{I_{0}>0\\})$. So by the dominated convergence
theorem, we have

$\displaystyle N$ $\displaystyle=$
$\displaystyle\lim_{k\to\infty}\int_{\Omega}(S(\cdot,t_{k})+I(\cdot,t_{k}))dx$
$\displaystyle=$
$\displaystyle|\Omega|S^{*}+S^{*}\int_{H^{+}\cap\\{I_{0}>0\\}}\frac{1}{\kappa}dx=\Big{(}|\Omega|+\int_{H^{+}\cap\\{I_{0}>0\\}}\frac{\beta-\gamma}{\gamma}dx\Big{)}S^{*}.$

This yields that

$S^{*}=\frac{N}{|\Omega|+\int_{H^{+}\cap\\{I_{0}>0\\}}\frac{\beta-\gamma}{\gamma}dx}.$

Since $S^{*}$ is independent of the chosen subsequence, we have

$w_{S}=\Big{\\{}\frac{N}{|\Omega|+\int_{H^{+}\cap\\{I_{0}>0\\}}\frac{\beta-\gamma}{\gamma}dx}\Big{\\}},$

and therefore (4.29) holds. Since $S(\cdot,t)\to S^{*}$ uniformly on $\Omega$
as $t\to\infty$, we obtain from (4.21) that
$I(\cdot,t)\to\frac{S^{*}(\beta-\gamma)^{+}}{\gamma}\chi_{H^{+}\cap\\{I_{0}>0\\}}$
uniformly on $\bar{\Omega}$ as $t\to\infty$. ∎

##  5 Simulations

In this section, we run numerical simulations to illustrate the results. Let
$\Omega=[0,1]$, $S_{0}=2+\cos(\pi x)$ and $I_{0}=1.5+\cos(\pi x)$. Then the
total population is $N=\int_{\Omega}(S_{0}+I_{0})dx=3.5$.

###  5.1 Mass action mechanism

We first simulate the models with the mass action mechanism.

Simulation 1: control the movement of susceptible people. Let $d_{S}=0$,
$d_{I}=1$, and $\gamma=4-\pi\sin(\pi x)$. Firstly, choose $\beta=0.5$, and so
$N<\int_{\Omega}\gamma/\beta dx=4$, i.e. the total population is small. By
Theorem 3.3, we have $I(\cdot,t)\to 0$ as $t\to\infty$ and the infected
population will be eliminated, which is confirmed by Figure 1(a). Then, choose
$\beta=2$, and so $N>\int_{\Omega}\gamma/\beta dx=1$, i.e. the total
population is large. Now Corollary 3.5 predicts that
$S(\cdot,t)\to\gamma/\beta$ and $I\to
I^{*}={(N-\int_{\Omega}rdx)}/{|\Omega|}=2.5$, which is confirmed by Figure
1(b). From the simulations, we can see that the disease can be eliminated only
when the total population is small by controlling the movement of susceptible
people.

(a)

(b)

Figure 1: Simulations of the model with mass action mechanism and $d_{S}=0$.
Parameters: $d_{I}=1$, $\gamma=4-\pi\sin(\pi x)$. Left figure: $\beta=0.5$ and
$N<\int_{\Omega}\gamma/\beta dx$; right figure: $\beta=2$ and
$N>\int_{\Omega}\gamma/\beta dx$.

Simulation 2: control the movement of infected people. Let $d_{S}=1$ and
$d_{I}=0$. Firstly, choose $\beta=0.2$ and $\gamma=4-\pi\sin(\pi x)$ such that
$H^{+}=\emptyset$. By Theorem 3.11, $S(\cdot,t)$ converges to $N/|\Omega|=3.5$
and $I(\cdot,t)$ converges to 0, which is confirmed by Figure 2(a). Then,
choose $\beta=1$ and $\gamma=4-\pi\sin(\pi x)$ such that $H^{+}\neq\emptyset$
and the minimum of $\gamma/\beta$ is attached at $x=0.5$. By Theorem 3.11, the
infected people will concentrate at $x=0.5$, which is confirmed by Figure
2(b). Finally we choose $\beta=2$ and $\gamma=14-4\pi\sin(4\pi x)$ such that
$H^{+}\neq\emptyset$. The minimum of $\gamma/\beta$ is attached at
$x=1/8,5/8$, and the infected people concentrate at these two points as shown
in Figure 2(c). From the simulations, we can see that the infected people may
not be eliminated by controlling the movement of infected people if
$H^{+}\neq\emptyset$. Instead, the infected people will concentrate at certain
points that are of the highest risk.

(a)

(b)

(c)

Figure 2: Simulations of the model with mass action mechanism and
$d_{S}=1,d_{I}=0$. Left figure: $\beta=0.2$, $\gamma=4-\pi\sin(\pi x)$, and
$H^{+}=\emptyset$; Middle figure: $\beta=1$, $\gamma=4-\pi\sin(\pi x)$,
$H^{+}\neq\emptyset$, and the minimum of $\gamma/\beta$ is attached at
$x=0.5$; right figure: $\beta=2$, $\gamma=14-4\pi\sin(4\pi x)$,
$H^{+}\neq\emptyset$, and the minimum of $\gamma/\beta$ is attached at
$x=1/8,5/8$.

###  5.2 Standard incidence mechanism

We then simulate the models with standard incidence mechanism.

Simulation 3: control the movement of susceptible people. Let $d_{S}=0$ and
$d_{I}=1$. Firstly, choose $\beta=1+\sin(\pi x)$ and $\gamma=1.5$ such that
$\int_{\Omega}(\beta-\gamma)dx={2}/{\pi}-0.5>0$, the high-risk sites are
$H^{+}=(1/6,5/6)$, the low-risk sites are $H^{-}=[0,1/6)\cup(5/6,1]$ and
moderate-risk sites are $H^{0}=\\{1/6,5/6\\}$. Since $H^{-}\neq\emptyset$,
Theorem 4.2-(i) predicts that infected population will be eliminated and
susceptible people occupy $H^{-}\cup H^{0}$. Moreover since
$\int_{\Omega}(\beta-\gamma)>0$, then $\mathcal{R}_{0}>1$ and Remark 4.3
suggests that the local size of the susceptible population maybe significantly
low on some portion of the high-risk area, which is confirmed by Figure 3(a).
Then choose $\beta=2.5+\sin(\pi x)$ and $\gamma=1.5+\sin(\pi x)$ such that
$\beta>\gamma$ and $I^{*}=N/\int_{\Omega}(\beta/(\beta-\gamma))dx\approx
1.1159$. Theorem 4.2-(ii) predicts that $I(\cdot,t)\to I^{*}$ as $t\to\infty$,
which is confirmed by Figure 3(b). Finally, choose $\beta=2-\sin(\pi x)$ and
$\gamma=1$ such that $\beta\geq\gamma$, $H^{0}=\\{0.5\\}$, and
$\int_{\Omega}1/(\beta-\gamma)dx=\infty$. As shown in Figure 4, infected
people are eliminated which agrees with Theorem 4.4. Moreover, susceptible
people seem to concentrate near $H^{0}$, which we cannot prove. Our
theoretical results and simulations show that the disease may be controlled by
limiting the movement of susceptible people if there exist low-risk or
moderate-risk sites.

(a)

(b)

Figure 3: Simulations of the model with standard incidence mechanism and
$d_{S}=0,d_{I}=1$. Left figure: $\beta=1+\sin(\pi x)$ and $\gamma=1.5$ such
that $H^{+}=(1/6,5/6)$, $H^{-}=[0,1/6)\cup(5/6,1]$ and $H^{0}=\\{1/6,5/6\\}$;
right figure: $\beta=2.5+\sin(\pi x)$ and $\gamma=1.5+\sin(\pi x)$ such that
$\beta>\gamma$ and $I^{*}=N/\int_{\Omega}(\beta/(\beta-\gamma))dx\approx
1.1159$.

(a)

(b)

(c)

Figure 4: Simulations of the model with standard incidence mechanism and
$d_{S}=0,d_{I}=1$. Parameters: $\beta=2-\sin(\pi x)$ and $\gamma=1$ such that
$\int_{\Omega}1/(\beta-\gamma)dx=\infty$.

Simulation 4: control the movement of infected people. Let $d_{S}=1$ and
$d_{I}=0$. Firstly, choose $\beta=2-|x-0.5|^{0.5}$ and $\gamma=1.5$ such that
the high-risk sites are $H^{+}=(0.25,0.75)$ and
$\int_{\Omega}1/|\beta-\gamma|dx<\infty$. As shown in Figure 5(a),
$S(\cdot,t)$ converges to a positive constant and $I(x,t)$ is positive when
$x\in H^{+}$. Then we choose $\beta=2-\sin(\pi x)$ and $\gamma=1.5$ such that
the high-risk sites are $H^{+}=(0,1/6)\cup(5/6,1)$ and
$\int_{\Omega}1/|\beta-\gamma|dx=\infty$. The infected people will live in
$H^{+}$ as shown in Figure 5(b). From the simulations, we can see that the
infected people may be eliminated exactly at the low-risk sites by controlling
the movement of infected people, which is predicted by Theorem 4.6.

(a)

(b)

Figure 5: Simulations of the model with standard incidence mechanism and
$d_{S}=1,d_{I}=0$. Left figure: $\beta=2-|x-0.5|^{0.5}$ and $\gamma=1.5$ such
that the high-risk sites are $H^{+}=(0.25,0.75)$ and
$\int_{\Omega}1/|\beta-\gamma|dx<\infty$.; right figure: $\beta=2-\sin(\pi x)$
and $\gamma=1.5$ such that the high-risk sites are $H^{+}=(0,1/6)\cup(5/6,1)$
and $\int_{\Omega}1/|\beta-\gamma|dx=\infty$.

##  6 Conclusions

We studied the impact of limiting population movement on disease outbreak by
examining the large time behavior of classical solutions of a class of
epidemic models with mass action or standard incidence transmission mechanism.
To this end, we set the diffusion rate of the population subgroups
(susceptible or infected) to zero separately and presented detailed
mathematical analysis of the corresponding degenerate epidemic models. First,
we established the existence and uniqueness of global classical solutions (see
Propositions 3.1, 3.7, 4.1 and 4.5). Next, we discussed the global dynamics of
the solutions (Theorems 3.3, 3.11, 4.4, and 4.6). Finally, we conducted some
numerical simulations to complement and illustrate our theoretical results
(see Figures 1-5). Our results revealed the intricate effects of restricting
population movement on disease dynamics and how predictions may dependent on
the choices of transmission mechanisms and spatially heterogeneous parameters.

First, we considered the global dynamics of the model with mass action
transmission mechanism. We defined a risk function $r:={\gamma}/{\beta}$ as
the ratio of the recovery and transmission rates. When the susceptible
population movement is restricted, Theorem 3.3 indicates that the average
population size, ${N}/{|\Omega|}$, compared to that of the risk function,
$\int_{\Omega}rdx/|\Omega|$, largely determines the disease outcome. In
particular, regardless of the magnitude of the movement rate of the infected
population, the disease may be eradicated only if the average population size
is smaller than the average risk function. Moreover if the disease persists,
the infected population would eventually be uniformly distributed across the
whole habitat. On the other hand, when the movement of the infected population
is restricted, Theorem 3.11 suggests that the disease could be eradicated only
if the habitat does not have a high-risk area. Moreover, whether the magnitude
of the movement rate of the susceptible population affects the results depends
on whether the average size of the population is larger than the risk function
in at least one location. Furthermore when the disease persists, the infected
population would concentrate on the highest-risk area of the habitat. Our
simulations in Figures 1-2 illustrated the above theoretical results.

Next, we studied the global dynamics of the model with standard incidence
mechanism. Theorems 4.2 and 4.4 indicate that restricting the susceptible
population’s movement could completely eradicate the disease only if the
habitat accommodates either a low-risk or moderate-risk area. These
requirements are achieved if the local distribution of the risk function is
less than or equal to one. However when the habitat consists of only high-risk
areas, the disease would persist with the infected population being uniformly
distributed across the whole habitat. On the other hand, when the movement of
the infected population is restricted, Theorem 4.6 indicates that the disease
may persist if the habitat has a nonempty high-risk area. Moreover, the
infected population precisely occupy the high-risk sites, and the magnitude of
the movement of the susceptible subgroup does not influence the disease
persistence. Our simulations in Figures 3-5 illustrated these theoretical
results.

Our above theoretical results suggest that the transmission mechanisms play an
important role when the population movement of one subgroup is restricted.
Indeed, when mass action mechanism is used, the persistence of the disease
depends on the average population size compared to either the average of the
risk function (when $d_{S}=0$) or its local distribution (when $d_{I}=0$).
However, when standard incidence mechanism is adopted, the disease persistence
prediction depends on whether the habitat has only high-risk areas (when
$d_{S}=0$) or has a nonempty high-risk area (when $d_{I}=0$). Interestingly,
our results suggest that for either mass action or standard incidence
mechanism disease control strategies which focus on limiting the movement of
the susceptible population should be preferred to those focusing on
restricting the movement of the infected population.

Finally, we point out that most of the previous works (e.g., [2, 10, 14, 46,
63]) considered the asymptotic profiles of the EE solutions to understand the
effects of limiting population movements on the dynamics of infectious
diseases, under the assumptions that the evolution of the disease happens on a
fast scale compared to control strategies and populations ultimately stabilize
at the EE solutions. In this work, we study the effects of limiting population
movement by focusing on the global dynamics of the degenerate systems with
either $d_{S}=0$ or $d_{I}=0$, assuming instead that the control strategies
happen on much faster scale than the evolution of the disease. We remark that
the theoretical implications from these two approaches seem to align well.

## Appendix

######  Lemma 6.1.

[9, Theorem 2.3-(iii)] Let $\beta$ and $\gamma$ be positive Hölder continuous
functions on $\bar{\Omega}$ and set $r={\gamma}/{\beta}$. Let $d_{I}>0$ and
$\mathcal{R}_{0}$ be defined by (4.4). Suppose that
$\int_{\Omega}rdx<\tilde{N}<{|\Omega|}/{\mathcal{R}_{0}}$ for some positive
number $\tilde{N}$. Then there is a positive function $\tilde{S}^{*}\in
C(\overline{\Omega})$ satisfying $\int_{\Omega}\tilde{S}^{*}(x)dx=\tilde{N}$
such that $\sigma(d_{I},\beta(\tilde{S}^{*}-r))=0$.

###### Proof.

Take $\tilde{S}^{*}:=S^{*}_{\rm low}$ in [9, Equation (2.9)] and note by [9,
System (2.10)] that $\sigma(d_{I},\beta(\tilde{S}^{*}-r))=0$. ∎

######  Proposition 6.2.

Let $\beta$ and $\gamma$ be positive Hölder continuous functions on
$\bar{\Omega}$ and set $r={\gamma}/{\beta}$. Let $d_{I}>0$ and
$\mathcal{R}_{0}$ be defined by (4.4). Suppose that
$\int_{\Omega}rdx<\tilde{N}<{|\Omega|}/{\mathcal{R}_{0}}$ for some positive
number $\tilde{N}$. Then there is a positive function $S_{0}\in
C(\overline{\Omega})$ such that

$\max\Big{\\{}\int_{\Omega}rdx,\int_{\Omega}S_{0}dx\Big{\\}}<\tilde{N}\leq
N^{*}_{S_{0},r},$ (6.1)

where $N^{*}_{S_{0},r}$ is defined as in (3.4).

###### Proof.

Let $\tilde{S}^{*}$ be given by Lemma 6.1. We construct functions of the form
$S_{0}(x)=\tilde{S}^{*}(x)+\varepsilon(x)(\tilde{S}^{*}(x)-r(x))$, where the
positive function $\varepsilon\in C(\overline{\Omega})$ will be specified
later. First, since $\tilde{S}^{*}>0$ and
$\int_{\Omega}\tilde{S}^{*}dx>\int_{\Omega}rdx$, then $S_{0}>0$ and
$\int_{\Omega}S_{0}dx>\int_{\Omega}rdx$ for any positive function
$\varepsilon\in C(\overline{\Omega})$ satisfying

$\|\varepsilon\|_{L^{\infty}(\Omega)}<\eta_{0}:=\min\Big{\\{}\frac{\tilde{S}^{*}}{\|\tilde{S}^{*}-r\|_{L^{\infty}(\Omega)}},\frac{\tilde{N}-\int_{\Omega}r(x)dx}{\|\tilde{S}^{*}-r\|_{L^{1}(\Omega)}}\Big{\\}}.$

Next, since $\int_{\Omega}\tilde{S}^{*}dx=\tilde{N}>\int_{\Omega}rdx$ and
$\sigma(d_{I},\beta(\tilde{S}^{*}-r))=0$, then $\tilde{S}^{*}-r$ must change
sign on $\Omega$. Then, we can choose a positive and continuous function
$\varepsilon$ satisfying $\|\varepsilon\|_{L^{\infty}(\Omega)}<\eta_{0}$ and
$\int_{\Omega}\varepsilon(\tilde{S}^{*}-r)dx<0$. With this choice of
$\varepsilon$, we have that

$\int_{\Omega}rdx<\int_{\Omega}S_{0}dx=\int_{\Omega}\tilde{S}^{*}dx+\int_{\Omega}\varepsilon(\tilde{S}^{*}-r)dx=\tilde{N}+\int_{\Omega}\varepsilon(\tilde{S}^{*}-r)dx<\tilde{N},$
$\tilde{S}^{*}=\frac{1}{1+\varepsilon}S_{0}+\frac{\varepsilon}{1+\varepsilon}r:=\lambda^{*}S_{0}+(1-\lambda^{*})r,$

and

$\sigma(d_{I},\beta\lambda^{*}(S_{0}-r))=\sigma(d_{I},\beta(\tilde{S}^{*}-r))=0.$

Therefore by the definition of $N^{*}_{S_{0},r}$, we have that

$N^{*}_{S_{0},r}\geq\int_{\Omega}\tilde{S}^{*}dx=\tilde{N}>\int_{\Omega}S_{0}dx=\max\Big{\\{}\int_{\Omega}rdx,\int_{\Omega}S_{0}dx\Big{\\}}.$

and the result follows. ∎

Declarations

Conflict of interest The authors declare that they have no conflict of
interest.

## References

  * [1] N. D. Alikakos.  An application of the invariance principle to reaction-diffusion equations.  Journal of Differential Equations, 33(2):201–225, 1979.
  * [2] L. J. S. Allen, B. M. Bolker, Y. Lou, and A. L. Nevai.  Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model.  Discrete Contin. Dyn. Syst., 21(1):1–20, 2008.
  * [3] H. Amann.  Dual semigroups and second order linear elliptic boundary value problem.  Israel J. Math., 45:225–254, 1983.
  * [4] R. M. Anderson and R. M. May.  Infectious diseases of humans: dynamics and control.  Oxford University Press, 1991.
  * [5] Y. Apostolopoulos and Sevil F. Sönmez.  Population mobility and infectious disease.  Springer, 2007.
  * [6] F. Brauer, C. Castillo-Chavez, Z. Feng, et al.  Mathematical models in epidemiology, volume 32.  Springer, 2019.
  * [7] F. Brauer, P. van den Driessche, and J. Wu, editors.  Mathematical epidemiology, volume 1945 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 2008.  Mathematical Biosciences Subseries.
  * [8] R. S. Cantrell and C. Cosner.  Spatial ecology via reaction-diffusion equations.  Series in Mathematical and Computational Biology, 2003.
  * [9] K. Castellano and R. B. Salako.  Multiplicity of endemic equilibria for a diffusive SIS epidemic model with mass-action transmission mechanism.  arXiv:2307.15155.
  * [10] K. Castellano and R. B. Salako.  On the effect of lowering population’s movement to control the spread of an infectious disease.  Journal of Differential Equations, 316:1–27, 2022.
  * [11] S. Chen and J. Shi.  Asymptotic profiles of basic reproduction number for epidemic spreading in heterogeneous environment.  SIAM J. Appl. Math., 80(3):1247–1271, 2020.
  * [12] R. Cui, K.-Y. Lam, and Y. Lou.  Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments.  J. Differential Equations, 263(4):2343–2373, 2017.
  * [13] R. Cui and Y. Lou.  A spatial SIS model in advective heterogeneous environments.  J. Differential Equations, 261(6):3305–3343, 2016.
  * [14] K. Deng and Y. Wu.  Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model.  Proc. Roy. Soc. Edinburgh Sect. A, 146(5):929–946, 2016.
  * [15] O. Diekmann and J. A. P. Heesterbeek.  Mathematical epidemiology of infectious diseases. Model building, analysis and interpretation.  Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2000.
  * [16] J. T. Doumatè and R. B. Salako.  Asymptotic behavior of solutions of an ode-pde hybrid competition system.  Journal of Differential Equations, 334:216–255, 2022.
  * [17] M. A. Efendiev, M. Otani, and H. J. Eberl.  Approaching the ideal free distribution in two-species competition models with fitness-dependent dispersal.  Journal of Differential Equations, 339:602–636, 2022.
  * [18] W. E. Fitzgibbon and M. Langlais.  Simple models for the transmission of microparasites between host populations living on noncoincident spatial domains.  In Structured population models in biology and epidemiology, volume 1936 of Lecture Notes in Math., pages 115–164. Springer, Berlin, 2008.
  * [19] W. E. Fitzgibbon, J. J. Morgan, G. F. Webb, and Y. Wu.  A diffusive seir model for community transmission of covid-19 epidemics: application to brazil.  Mathematics in Applied Sciences and Engineering, 2(4):290–309, 2021\.
  * [20] T. J. Hagenaars, C. A. Donnelly, and N. M. Ferguson.  Spatial heterogeneity and the persistence of infectious diseases.  Journal of Theoretical Biology, 229(3):349–359, 2004.
  * [21] D. Henry.  Geometric theory of semilinear parabolic equations.  1981\.
  * [22] J. Húska.  Harnack inequality and exponential separation for oblique derivative problems on lipschitz domains.  Journal of Differential Equations, 226(2):541–557, 2006.
  * [23] D. Jiang, Z. Wang, and L. Zhang.  A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment.  Discrete Contin. Dyn. Syst. Ser. B, 23(10):4557–4578, 2018.
  * [24] A. Källén, P. Arcuri, and J. D. Murray.  A simple model for the spatial spread and control of rabies.  Journal of theoretical biology, 116(3):377–393, 1985.
  * [25] W. O. Kermack and A. G. McKendrick.  A contribution to the mathematical theory of epidemics.  Proceedings of the royal society of london. Series A, Containing papers of a mathematical and physical character, 115(772):700–721, 1927.
  * [26] P. G. Kevrekidis, J. Cuevas-Maraver, Y. Drossinos, Z. Rapti, and G. A. Kevrekidis.  Reaction-diffusion spatial modeling of covid-19: Greece and andalusia as case examples.  Physical Review E, 104(2):024412, 2021.
  * [27] K. Kuto, H. Matsuzawa, and R. Peng.  Concentration profile of endemic equilibrium of a reaction-diffusion-advection SIS epidemic model.  Calc. Var. Partial Differential Equations, 56(4):112, 2017.
  * [28] M. Lewis, J. Rencławowicz, and P van den Driessche.  Traveling waves and spread rates for a west nile virus model.  Bulletin of mathematical biology, 68:3–23, 2006.
  * [29] H. Li and R. Peng.  Dynamics and asymptotic profiles of endemic equilibrium for SIS epidemic patch models.  J. Math. Biol., 79(4):1279–1317, 2019.
  * [30] H. Li, R. Peng, and Z. Wang.  On a diffusive susceptible-infected-susceptible epidemic model with mass action mechanism and birth-death effect: analysis, simulations, and comparison with other mechanisms.  SIAM J. Appl. Math., 78(4):2129–2153, 2018.
  * [31] H. Li, R. Peng, and T. Xiang.  Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion.  European Journal of Applied Mathematics, 31(1):26–56, 2020.
  * [32] Z. Lin and H. Zhu.  Spatial spreading model and dynamics of west nile virus in birds and mosquitoes with free boundary.  Journal of Mathematical Biology, 75:1381–1409, 2017.
  * [33] A. L. Lloyd and R. M. May.  Spatial heterogeneity in epidemic models.  Journal of Theoretical Biology, 179(1):1–11, 1996.
  * [34] Y. Lou and R. B. Salako.  Dynamics of a parabolic-ode competition system in heterogeneous environments.  Proc. Amer. Math. Soc, 148(7):025–3038, 2020.
  * [35] Y. Lou and R. B. Salako.  Control strategy for multi-strains epidemic model.  Bulletin of Mathematical Biology, 84(10):1–47, 2022.
  * [36] Y. Lou, R. B. Salako, and P. Song.  Human mobility and disease prevalence.  Journal of Mathematical Biology, 87(1):1–32, 2023.
  * [37] Y. Lou, Y. Tao, and M. Winkler.  Approaching the ideal free distribution in two-species competition models with fitness-dependent dispersal.  SIAM Journal on Mathematical Analysis, 42(2):1228–1262, 2014.
  * [38] Y. Lou and M. Winkler.  Advantage and disadvantage of dispersal in two-species competition models.  CSIAM Trans. Appl. Math, 1(1):86–103, 2020.
  * [39] Y. Lou and X.-Q. Zhao.  The periodic ross–macdonald model with diffusion and advection.  Applicable Analysis, 89(7):1067–1089, 2010.
  * [40] Y. Lou and X.-Q. Zhao.  A reaction–diffusion malaria model with incubation period in the vector population.  Journal of mathematical biology, 62(4):543–568, 2011.
  * [41] P. Magal, G. F. Webb, and Y. Wu.  Spatial spread of epidemic diseases in geographical settings: Seasonal influenza epidemics in puerto rico.  Discrete & Continuous Dynamical Systems-Series B, 25(6), 2020.
  * [42] M. Martcheva.  An introduction to mathematical epidemiology, volume 61.  Springer, 2015.
  * [43] J. D. Murray, E. A. Stanley, and D. L. Brown.  On the spatial spread of rabies among foxes.  Proceedings of the Royal society of London. Series B. Biological sciences, 229(1255):111–150, 1986.
  * [44] M. A. Onyido and R. B. Salako.  Asymptotic dynamics of a three-species hybrid competition system.  Journal of Dynamics and Differential Equations, 2023.
  * [45] A. Pazy.  Semigroups of linear operators and applications to partial differential equations, volume 44.  Springer Science & Business Media, 2012.
  * [46] R. Peng.  Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. I.  J. Differential Equations, 247(4):1096–1119, 2009.
  * [47] R. Peng and S. Liu.  Global stability of the steady states of an SIS epidemic reaction-diffusion model.  Nonlinear Anal., 71(1-2):239–247, 2009.
  * [48] R. Peng, Z.-A. Wang, G. Zhang, and M. Zhou.  Novel spatial profiles of population distribution of two diffusive SIS epidemic models with mass action infection mechanism and small movement rate for the infected individuals.  arXiv preprint arXiv:2301.01012, 2023.
  * [49] R. Peng and Y. Wu.  Global $l^{\infty}$-bounds and long-time behavior of a diffusive epidemic system in a heterogeneous environment.  SIAM Journal on Mathematical Analysis, 53(3):2776–2810, 2021.
  * [50] R. Peng and F. Yi.  Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: effects of epidemic risk and population movement.  Phys. D, 259:8–25, 2013.
  * [51] R. Peng and X.-Q. Zhao.  A reaction-diffusion SIS epidemic model in a time-periodic environment.  Nonlinearity, 25(5):1451–1471, 2012.
  * [52] L. Setti, F. Passarini, G. De Gennaro, et al.  Potential role of particulate matter in the spreading of covid-19 in northern italy: first observational study based on initial epidemic diffusion.  BMJ open, 10(9):e039338, 2020.
  * [53] S. T. Stoddard, A. C. Morrison, G. M. Vazquez-Prokopec, and _et al._ The role of human movement in the transmission of vector-borne pathogens.  PLoS Neglected Tropical Diseases, 3(7):e481, 2009.
  * [54] L. T. Takahashi, N. A. Maidana, W. C. Ferreira, P. Pulino, and H. M. Yang.  Mathematical models for the aedes aegypti dispersal dynamics: travelling waves by wing and wind.  Bulletin of mathematical Biology, 67:509–528, 2005.
  * [55] Y. Tao and M. Winkler.  Analysis of a chemotaxis-SIS epidemic model with unbounded infection force.  Nonlinear Analysis: Real World Applications, 71:103820, 2023.
  * [56] A. J. Tatem, D. J. Rogers, and S. I. Hay.  Global transport networks and infectious disease spread.  Advances in Parasitology, 62:293–343, 2006.
  * [57] N. Tuncer and M. Martcheva.  Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion.  J. Biol. Dyn., 6(2):406–439, 2012.
  * [58] A. Viguerie, A. Veneziani, G. Lorenzo, D. Baroli, N. Aretz-Nellesen, A. Patton, T. E. Yankeelov, A. Reali, T. JR Hughes, and F. Auricchio.  Diffusion–reaction compartmental models formulated in a continuum mechanics framework: application to covid-19, mathematical analysis, and numerical study.  Computational Mechanics, 66:1131–1152, 2020.
  * [59] W. Wang and X.-Q. Zhao.  Basic reproduction numbers for reaction-diffusion epidemic models.  SIAM J. Appl. Dyn. Syst., 11(4):1652–1673, 2012.
  * [60] G. F. Webb.  A reaction-diffusion model for a deterministic diffusive epidemic.  Journal of Mathematical Analysis and Applications, 84(1):150–161, 1981.
  * [61] X. Wen, J. Ji, and B. Li.  Asymptotic profiles of the endemic equilibrium to a diffusive SIS epidemic model with mass action infection mechanism.  Journal of Mathematical Analysis and Applications, 458(1):715–729, 2018.
  * [62] M. Winkler.  Aggregation vs. global diffusive behavior in the higher-dimensional keller–segel model.  Journal of Differential Equations, 248(12):2889–2905, 2010.
  * [63] Y. Wu and X. Zou.  Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism.  J. Differential Equations, 261(8):4424–4447, 2016.
  * [64] Yixiang Wu and Xingfu Zou.  Dynamics and profiles of a diffusive host–pathogen system with distinct dispersal rates.  Journal of Differential Equations, 264(8):4989–5024, 2018.