\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 170, pp. 1--21.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2018/170\hfil Epidemic reaction-diffusion systems]
{Epidemic reaction-diffusion systems with two types
 of boundary conditions}

\author[K. Li, J. Li, W. Wang \hfil EJDE-2018/170\hfilneg]
{Kehua Li, Jiemei Li, Wei Wang}

\address{Kehua Li  \newline
 School of Applied Mathematics,
 Xiamen University of Technology,
 Xiamen, Fujian 361024, China}
\email{khli@xmut.edu.cn}

\address{Jiemei Li \newline
School of Mathematics and Physics,
Lanzhou Jiaotong University,
Lanzhou, Gansu 730070, China}
\email{lijiemei81@126.com}

\address{Wei Wang (corresponding author) \newline
Department of Mathematics, 
College of Mathematics and Systems Science, 
Shandong University of Science and Technology,
Qingdao 266590, China.\newline
 School of Mathematics and Physics,
 University of Science and Technology Beijing,
 Beijing 100083, China}
\email{weiw10437@gmail.com}

\thanks{Submitted November 12, 2017. Published October 11, 2018.}
\subjclass[2010]{58F15, 58F17, 53C35}
\keywords{SIRS model; reaction-diffusion system; global dynamics;
\hfill\break\indent Neumann boundary condition; free boundary condition}

\begin{abstract}
 We investigate an epidemic reaction-diffusion system with  two different
 types of boundary conditions. For the problem with the Neumann boundary
 condition, the global dynamics is fully determined by the basic
 reproduction number $\mathcal{R}_0$.
 For the problem with the free boundary condition,
 the disease will vanish if the basic reproduction number
 $\mathcal{R}_0<1$ or the initial infected radius $g_0$ is sufficiently
 small. Furthermore, it is shown that the disease will spread to the
 whole domain if $\mathcal{R}_0>1$ and the initial infected radius
 $g_0$ is suitably large. Main results reveal that besides the basic
 reproduction number, the size of initial epidemic
 region and the diffusion rates of the disease also
 have an important influence to the disease transmission.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks


\section{Introduction and model derivation}

Mathematical modeling has been shown to be an effective  approach to
study the spread of infectious diseases as they can capture the main
factors underlying the transmission mechanisms and provide feasible
control strategies for health agencies. One of the simplest
epidemic models is the Kermack-McKendrick model, which can be divided
the population into susceptible ($S$), infectious ($I$) and recovered
individuals ($R$) \cite{Kermack1927}. In recent years, mathematical
analyses for epidemic models have received wide attentions
(see, e.g., \cite{Chenxingfu2000,Duyihong2010,liguo2017ejde,lihufengli2017,
Liting2017,Meng2016,Uggenti2018, Wang1, wangmalai, Zhang1}).


In the classical  SIR models, it is assumed that recovered individuals
have gotten permanent immunity. However, the acquired immunity may
disappear and recovered individuals will
become susceptible after a period of time  \cite{Mena-Lorca1992}.
Moreover, for some bacterial agent diseases, infected individuals may
recover after some treatments and go back directly to the susceptible
class because of transient antibody \cite{Mena-Lorca1992}.
Li et al \cite{Liting2017}  proposed the following SIRS epidemic system with
nonlinear response function $Sf(I)$ and transfer from the infected class
to the susceptible class, which is governed by a set of ordinary
differential equations
\begin{equation} \label{e1.1}
\begin{gathered}
 \frac{dS}{ dt} =\Lambda-\mu S-Sf(I)+\gamma_1I+\delta R,\\
 \frac{d I}{d t} =Sf(I)-(\mu+\gamma_1+\gamma_2+\alpha)I,\\
 \frac{d R}{d t} =\gamma_2I-(\mu+\delta)R,
 \end{gathered}
\end{equation}
where $\Lambda>0$ is the recruitment rate of susceptible individuals,
$\gamma_1\geq 0$ denotes the transfer rate from the infected class to
the susceptible class, $\gamma_2\geq 0$ represents
the transfer rate from the infected class to the recovered class,
$\alpha \geq 0$ stands for the disease-induced death rate,
$\delta \geq 0$ is the immunity loss rate, and $\mu>0$ is the natural
 death rate.

Li et al \cite{Liting2017}  obtained
the global dynamics of system  \eqref{e1.1}, which is determined by
the basic reproduction number
$$
\mathcal{R}_0=\frac{\Lambda\beta}{\mu(\mu+\gamma_1+\gamma_2+\alpha)},
$$
with  LaSalle's invariance principle and the Lyapunov direct method.

Most of epidemic systems are governed by a set of ordinary differential
equations, which only reflect the
epidemiological process as the time changes. To closely match
the reality, we consider a SIRS epidemic reaction-diffusion system as follows
\begin{equation} \label{e1.2}
\begin{gathered}
\begin{aligned}
 \frac{\partial S(x,t)}{\partial t}
&=D\Delta S(x,t)  +\Lambda-\mu S(x,t)-S(x,t)f(I(x,t))\\
&\quad +\gamma_1I(x,t)+\delta R(x,t),\quad x\in\Omega,\; t>0,
\end{aligned}\\
 \frac{\partial I(x,t)}{\partial t} = D\Delta I(x,t)
+S(x,t)f(I(x,t))-(\mu+\gamma_1+\gamma_2+\alpha)I(x,t),\\\
 x\in\Omega,\; t>0,\\
\frac{\partial R(x,t)}{\partial t} =D\Delta R(x,t)
+\gamma_2I(x,t)-(\mu+\delta)R(x,t),\quad x\in\Omega,\; t>0,\\
\frac{\partial S}{\partial \nu}=\frac{\partial I}{\partial \nu}
=\frac{\partial R}{\partial \nu}=0,\quad x\in\partial\Omega,\; t>0,\\
  S(x,0)=S_0(x)>0,\quad I(x,0)=I_0(x)>0,\quad
 R(x,0)=R_0(x)>0,\quad  x\in\overline{\Omega},
 \end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary
$\partial\Omega$. $\nu$ is the outward normal to $\partial\Omega$.
$D>0$ stands for the diffusion rate. To  continue our study,
we make the same hypotheses on $f$ as in \cite{Liting2017}.
 Namely,  $f$ is a real locally Lipschitz function on
$\mathbb{R}_+=[0,+\infty)$ satisfying the following assumptions
\begin{itemize}
\item[(A1)] $ f(0)=0$, $f(I)>0$, and $f'(I)\geq 0 $ for $I>0$.

\item[(A2)] $\frac{f(I)}{I}$ is continuous and  nonincreasing  for $I>0$,
and  $\lim_{I \to 0^+ }\frac{f(I)}{I}$
exists, denoted by $\beta>0$.

\item[(A3)] $f''(I)\leq 0$ for $I>0$.
\end{itemize}

In recent years, the free boundary problems have received tremendous
attentions (see, e.g., \cite{Crank1984,Chenxingfu2000,Duyihong2010,
DuLin2014,Ghidouche2001,GuoWu2012,Hilhorst2003,Kaneko2014,
Linzhigui2007,LinZhaoZhou2013,Rubinstein1971,Wangmingxin2015,ZhaoWang2014}).
To make a better understanding for  the dynamics of spatial transmission
of the disease, the free  boundary condition is introduced to epidemic systems.
Kim et al \cite{Kim2013} investigated a reaction-diffusion SIR epidemic
 system with the free boundary condition and derived some sufficient
conditions for the disease vanishing or spreading.
Huang and Wang \cite{Huangmwangmingxin2015} studied a diffusive SIR system
with the free boundary condition. The dynamical behavior of the susceptible
population is obtained. A SIS reaction-diffusion-advection system with
the free boundary condition was proposed  to discuss the persistence and
eradication of infectious disease \cite{Gejing2015}.
Cao et al \cite{Cao2017} explored a free boundary problem of a diffusive
SIRS system with nonlinear incidence.
The estimate of the expanding speed was discussed.

Motivated by the works mentioned above, we make further investigation
for a SIRS epidemic system with nonlinear incidence and the free boundary
condition. For the sake of simplicity, we assume that the environment
is radially symmetric. We study the behavior of the positive solution
$(S(z,t), I(z,t),\, R(z,t); g(t))$ with $z= |x|$ and $x\in \mathbb{R}^n$ for the following
problem
\begin{equation} \label{e1.3}
\begin{gathered}
\begin{aligned}
 \frac{\partial S(z,t)}{\partial t}
&=D\Delta S(z,t) +\Lambda-\mu S(z,t)-S(z,t)f(I(z,t)) \\
&\quad +\gamma_1I(z,t)+\delta R(z,t),\quad z>0,\; t>0,
\end{aligned}\\
\begin{aligned}
 \frac{\partial I(z,t)}{\partial t}
&= D\Delta I(z,t) +S(z,t)f(I(z,t))-(\mu+\gamma_1 \\
&\quad +\gamma_2+\alpha)I(z,t),\quad  0<z<g(t),\; t>0,
\end{aligned}\\
\frac{\partial R(z,t)}{\partial t}
 =D\Delta R(z,t)+\gamma_2I(z,t)-(\mu+\delta)R(z,t),\quad 0<z<g(t),\; t>0,\\
S_{z}(0,t)=I_{z}(0,t)=R_{z}(0,t)=0,\quad t>0,\\
I(z,t)=R(z,t)=0,\quad z\geq g(t),\; t>0,\\
g'(t)=-\mu_1 I_z(g(t),t),\quad  g(0)=g_0>0,\quad t>0,\\
S(z,0)=S_0(z),\quad  I(z,0)=I_0(z),\quad R(z,0)=R_0(z),\quad z\geq 0,
\end{gathered}
\end{equation}
where $g_0$, $D$ and $\mu_1$ are positive constants.
From the biological perspective,  the Neumann boundary condition at
$x=0$ indicates that the left boundary is fixed, with the population confined
to its right.  Beyond the free boundary $z= g(t)$, there only exist susceptible
individuals. The equation  $g'(t)=-\mu_1 I_z(g(t),t)$ is a special case of
the well-known Stefan condition, which has been proposed in \cite{Linzhigui2007}.  $[0, g_0]$ is the initial epidemic region where infective individuals $I$ and  removed individuals $R$ exist. The constant $\mu_1$ denotes the ratio of
expanding speed of the free boundary.
The initial functions $S_0$, $I_0$ and $R_0$ are nonnegative and satisfy
\begin{equation} \label{e1.4}
\begin{gathered}
 S_0\in C^{2}([0,+\infty)),\quad  I_0,R_0\in C^{2}([0,g_0]),\\
 I_0(z)=R_0(z)=0,\quad z\in [g_0,+\infty), \quad I_0(z)>0,\quad z\in[0,g_0).
 \end{gathered}
\end{equation}
The organization of this article is  as follows.
In Section 2, we study the Neumann boundary problem in a bounded domain.
We first show that the solution of system \eqref{e1.2} is positive and bounded,
then study the global dynamics of  steady states for system \eqref{e1.2}.
Main results reveal that if $\mathcal{R}_0 < 1$, then the disease-free
steady state is globally asymptotically stable; while
if $\mathcal{R}_0 > 1$,  the endemic steady state is globally asymptotically
stable.
In Section 3,  we discuss the free boundary problem.
 We firstly investigate the existence and uniqueness
of the solution to system \eqref{e1.3}. We derive some sufficient conditions
for the disease vanishing or spreading. In Section 4, we perform some
numerical simulations to illustrate theoretical results.
At last, we give discussions and conclusions in Section 5.

\section{Fixed domain}

In this section, we aim to study system \eqref{e1.2} with
the Neumann boundary condition in a bounded domain.
The well-posedness of the solutions for system \eqref{e1.2} is discussed
in Theorem \ref{thm2.1}.
Furthermore, the global asymptotic stabilities of steady states of
system \eqref{e1.2}  are explored in Theorems \ref{thm2.2} and \ref{thm2.3}.

\subsection{Well-posedness of solutions}
We denote the positive cone in $\mathbb{R}^{3}$ by
$$
\mathbb{R}^{3}_{+}=\{\phi=(S,I, R)^{T}\in \mathbb{R}^{3}:
 S\geq0, I\geq0,\; R\geq0\}.
$$

Take $p>3$ so that the space $W^{1,p}(\Omega, \mathbb{R}^{3})$
is continuously embedded in the continuous function space
$C(\Omega, \mathbb{R}^{3})$ \cite{Adams75}.
We consider the well-posedness of the solutions in the  phase space
$$
\mathcal{X}_{+}=\{\phi\in W^{1,p}(\Omega,\mathbb{R}^{3}):
 \phi(\overline{\Omega})\subset \mathbb{R}^{3}_{+} \text { and }
 \partial \phi/\partial \nu=0\text{ on } \partial\Omega\}.
$$


We rewrite system \eqref{e1.2} as
\begin{gather*}
 \phi_{t}+\mathcal {S}(\phi)\phi=\mathcal {F}(x,\phi),\quad x\in \phi,\; t>0,\\
 B\phi=0,\quad x\in\partial\Omega,\; t>0,\\
\end{gather*}
where
$\mathcal {S}(e)\phi=-\sum_{i,k}\partial_{i}(a_{i,k}(e)\partial_{k}\phi)$,
$B\phi=\frac{\partial\phi}{\partial \nu}$,
$a_{i,k}=a(e)\delta_{i,k}$, $1\leq i,k\leq3$, and
\begin{equation*}
a(e)=\begin{pmatrix}
D&0&0\\
0&D&0\\
0&0&D
\end{pmatrix},
\end{equation*}
for $e(e_1, e_2, e_3)\in \mathbb{R}^{3}_{+}$.
Here $\delta_{i,k}$ is the Kronecker delta function, and
$$
\mathcal {F}(x,\phi)=\Big(\Lambda-\mu S-Sf(I)+\gamma_1I+\delta R,
 Sf(I)-(\mu+\gamma_1+\gamma_2+\alpha)I, \gamma_2I-(\mu+\delta)R\Big)^{T},
$$
for $\phi=(S,I,R)$.  Clearly, $a(e)\in C^{2}(\mathbb{R}^{3}_{+}, L)
(\mathbb{R}^{3}_{+}))$, where we identified $ L(\mathbb{R}^{3}_{+})$ with
the space of $3\times3$ real matrices.

\begin{theorem} \label{thm2.1}
 For every initial value $(S_0,I_0,R_0)$,  system \eqref{e1.2} admits a
unique nonnegative solution defined on $[0,+\infty)\times \overline{\Omega}$,
such that
 $$
(S,I, R)\in C\left([0,+\infty), \mathcal{X}_{+}\right)
\cap C^{2,1}\left([0,+\infty)\times \overline{\Omega}, \mathbb{R}^{3}\right).
$$
\end{theorem}

\begin{proof}
 In view of  \cite[Theorem 1]{Amann89} or
\cite[Theorems 14.4 and 14.6]{Amann93},  system
\eqref{e1.2} admits a unique nonnegative classical solution $(S,I, R)$
defined on $[0,\varrho_0)\times\Omega$ such that
$$
(S,I, R)\in C\left([0,\varrho_0), \mathcal{X}_{+}\right)
\cap C^{2,1}\left([0,\varrho_0)\times \overline{\Omega},
 \mathbb{R}^{3}\right),
$$
where $\varrho_0>0$ is the maximal interval of existence of the solution
for system \eqref{e1.2}. According to \cite[Theorem 15.1 ]{Amann93},
the solution of system \eqref{e1.2} is nonnegative.
Motivated by the idea developed in  \cite[Theorem 5.2]{Amann89},
we need to show that any nonnegative solution $(S(x,t), I(x,t), R(x,t))$
of system \eqref{e1.2} is bounded.

Denote $N=S+I+R$, from system \eqref{e1.2}, we get that
\begin{equation*}
\frac{\partial N}{\partial t}
\leq D\Delta N+\Lambda-\mu N.
\end{equation*}
By  \cite[Lemma 1]{Lou11},  $\frac{\Lambda}{\mu}$ is the globally
attractive steady state for the  reaction-diffusion equations
\begin{gather*}
\frac{\partial N(x,t)}{\partial t}= D\Delta N+\Lambda-\mu N,\quad
 x\in \Omega,\; t>0,\\
\frac{\partial N}{\partial \nu}=0,\quad x\in \partial\Omega,\; t>0.
\end{gather*}

In view of the parabolic comparison theorem (\cite[Theorem 7.3.4]{Smith95}),
$S+I+R$ is bounded. Since $S$, $I$, and $R$ are nonnegative, $S(x,t)$, $I(x,t)$,
 and $R(x,t)$ of system \eqref{e1.2} are bounded.
That is, $\varrho_0=+\infty.$ By \cite[Theorem 5.2]{Amann89},
the global existence of the solution can be obtained.
The proof is complete.
\end{proof}

\subsection{Global dynamics for system \eqref{e1.2}}
In this subsection, we investigate the global dynamics of steady states
of system \eqref{e1.2} by constructing  suitable Lyapunov functions.
Firstly, we obtain that the state space $\varPi$ is positively invariant
for system \eqref{e1.2}
$$
\varPi :=\big\{(S,I, R)^{T}: S(x,\cdot)+I(x,\cdot)+R(x,\cdot)
\leq \frac{\Lambda}{\mu}, \text{ for }  x\in\overline{\Omega}\big\}.
$$
Obviously, system \eqref{e1.2} always has the disease-free steady state
$E_0(\frac{\Lambda}{\mu},0,0)$.
If $\mathcal{R}_0>1$, from \cite{Liting2017},  system \eqref{e1.2}
has a unique endemic steady state $E^*=(S^*,I^*, R^*)$, where
$$
S^*=\frac{(\mu+\gamma_1+\gamma_2+\alpha)I^*}{f(I^*)},\quad
 R^*=\frac{\gamma_2I^*}{\mu+\delta}.
$$
Here $I^*$ is a unique positive zero of $\mathcal{H}$ defined by
$$
\mathcal{H}(I)=\mu(\mu+\gamma_1+\gamma_2+\alpha)\frac{I}{f(I)}
+\big(\mu+\alpha+\frac{\mu\gamma_2}{\mu+\delta}
\big)I-\Lambda.
$$

\begin{theorem} \label{thm2.2}
 If $\mathcal{R}_0<1$,  the disease-free steady state $E_0$ of
system \eqref{e1.2} is globally asymptotically stable in $\varPi$.
\end{theorem}

\begin{proof}
 We define the Lyapunov function
$$
V_0=\int_{\Omega } {I(x,t)}dx.
$$
From (A2), we obtain that
$f(I)\leq \beta I,\ \text{for}\ I\in \mathbb{R}^+$.
By the divergence theorem and the Neumann boundary condition,
we obtain
$$
D\int_{\Omega } {\Delta I}dx=0.
$$
The derivative of $V_0$ along solutions of system \eqref{e1.2} is
\begin{align*}
\frac{\partial V_0}{\partial t}
&=D\int_{\Omega } {\Delta I}dx+\int_{\Omega } {[S(x,t)f(I(x,t))
-(\mu+\gamma_1+\gamma_2+\alpha)I(x,t)]}dx\\
&\leq \int_{\Omega } {[\beta S(x,t)I(x,t)-(\mu+\gamma_1+\gamma_2
 +\alpha)I(x,t)]}dx\\
&\leq \int_{\Omega } {\big(\frac{\Lambda \beta}{\mu}
 -(\mu+\gamma_1+\gamma_2+\alpha)\big)I(x,t)}dx\\
&=(\mu+\gamma_1+\gamma_2+\alpha) \int_{\Omega } {(R_0-1)I(x,t)}dx.\\
\end{align*}
We have  $\frac{\partial V_0}{\partial t}\leq 0$, and the equality holds
if and only if $I\equiv 0$.
By LaSalle's invariance principle, the disease-free steady state $E_0$
is globally asymptotically stable if $\mathcal{R}_0<1$.
The proof is complete.
\end{proof}

Next we study the global asymptotic stability of the endemic steady
 state $E^*$. We study
the following equivalent system constituted by $I$, $R$,  and $N=S+I+R$,
\begin{equation} \label{e2.1}
\begin{gathered}
\frac{\partial I(x,t)}{\partial t}
= D\Delta I+(N-I-R)f(I)-(\mu+\gamma_1+\gamma_2+\alpha)I,\quad x\in\Omega,\;
 t>0,\\
\frac{\partial R(x,t)}{\partial t} =D\Delta R+\gamma_2I-(\mu+\delta)R,\quad
 x\in\Omega,\; t>0,\\
\frac{\partial N(x,t)}{\partial t} =D\Delta N
 +\Lambda-\mu N-\alpha I,\quad x\in\Omega,\; t>0,\\
\frac{\partial I}{\partial \nu}=\frac{\partial R}{\partial \nu}
=\frac{\partial N}{\partial \nu}=0,\quad  x\in\partial\Omega,\; t>0,\\
 I(x,0)=I_0(x)>0,\quad R(x,0)=R_0(x)>0,\\
 N(x,0)=N_0(x)>0,\quad  x\in\overline{\Omega}.
 \end{gathered}
\end{equation}
If $\mathcal{R}_0>1$, this has a unique endemic steady state
$\overline{E}^{*}=(I^*, R^*,N^*)$. Hence,
\begin{align*}
\frac{\partial I(x,t)}{\partial t}
&= D\Delta I(x,t)+f(I)\{ N-I-R-(\mu+\gamma_1+\gamma_2
+\alpha)\frac{I}{f(I)}\} \\
&\quad -f(I)\{[ N^*-I^*-R^*-(\mu+\gamma_1+\gamma_2+\alpha)
\frac{I^*}{f(I^*)}] \} .
\end{align*}
We rewrite system \eqref{e2.1} as
\begin{equation} \label{e2.2}
\begin{gathered}
\begin{aligned}
\frac{\partial I(x,t)}{\partial t}
&= D\Delta I(x,t)+f(I)\left\{(N-N^*)-(I-I^*)-(R-R^*)\right\}\\
&\quad -f(I)(\mu+\gamma_1+\gamma_2+\alpha)
 [\frac{I}{f(I)}-\frac{I^*}{f(I^*)}],\quad  x\in\Omega,\; t>0,
\end{aligned} \\
\frac{\partial R(x,t)}{\partial t}
=D\Delta R(x,t)+\gamma_2I(x,t)-(\mu+\delta)R(x,t),\quad x\in\Omega,\; t>0,\\
\frac{\partial N(x,t)}{\partial t} =D\Delta N(x,t)
 +\Lambda-\mu N(x,t)-\alpha I(x,t),\quad x\in\Omega,\; t>0,\\
\frac{\partial I}{\partial \nu}=\frac{\partial R}{\partial \nu}
=\frac{\partial N}{\partial \nu}=0,\quad  x\in\partial\Omega,\; t>0,\\
I(x,0)=I_0(x)>0,\quad R(x,0)=R_0(x)>0,\\
 N(x,0)=N_0(x)>0,\quad  x\in\overline{\Omega}.
 \end{gathered}
\end{equation}

\begin{theorem} \label{thm2.3}
 If $\mathcal{R}_0>1$, then the endemic steady state $\overline{E}^*$
of system \eqref{e2.2} is globally asymptotically stable in $\varPi$.
\end{theorem}

\begin{proof}
We define the Lyapunov function
$$
V_1=\int_{\Omega } {\int_{I^*}^{I}
{\frac{u-I^*}{f(u)}}du}dx+\frac{1}{2\gamma_2}\int_{\Omega } {(R-R^*)^2}dx
+\frac{1}{2\alpha}\int_{\Omega } {(N-N^*)^2}dx.
$$
Then we have
\begin{gather*}
D\int_{\Omega } {\frac{I-I^*}{f(I)}\Delta I}dx
=-D\int_{\Omega } {\frac{[f(I)-If'(I)]\|\nabla I\|^{2}}{f^2(I)}}dx
-DI^*\int_{\Omega } {\frac{f'(I)\|\nabla I\|^{2}}{f^2(I)}}dx, \\
\frac{D}{\gamma_2}\int_{\Omega } {(R-R^*)\Delta R}dx
=-\frac{D}{\gamma_2}\int_{\Omega } {\| \nabla R\|^2}dx, \\
\frac{D}{\alpha}\int_{\Omega } {(N-N^*)\Delta N}dx
=-\frac{D}{\alpha}\int_{\Omega } {\| \nabla N\|^2}dx.
\end{gather*}
The derivative of $V_1$ along solutions of system \eqref{e2.2} is
\begin{align*}
\frac{\partial V_1}{\partial t}
&=\int_{\Omega } {\frac{I-I^*}{f(I)}\frac{\partial I}{\partial t}}dx
 +\frac{1}{\gamma_2}\int_{\Omega } {(R-R^*)\frac{\partial R}{\partial t}}dx
 +\frac{1}{\alpha}\int_{\Omega } {(N-N^*)\frac{\partial N}{\partial t}}dx\\
&=-D\int_{\Omega } {\frac{[f(I)-If'(I)]\|\nabla I\|^{2}}{f^2(I)}}dx
 -DI^*\int_{\Omega } {\frac{f'(I)\|\nabla I\|^{2}}{f^2(I)}}dx\\
&\quad -\frac{D}{\gamma_2}\int_{\Omega } {\| \nabla R\|^2}dx
 -\frac{D}{\alpha}\int_{\Omega } {\| \nabla N\|^2}dx\\
&\quad +\int_{\Omega } {\frac{I-I^*}{f(I)}f(I)
 \{(N-N^*)-(I-I^*)-(R-R^*)\}}dx\\
&\quad-\int_{\Omega } {\frac{I-I^*}{f(I)}f(I)
 \big\{(\mu+\gamma_1+\gamma_2+\alpha)[\frac{I}{f(I)}-\frac{I^*}{f(I^*)}]\big\}}
 dx\\
&\quad  +\int_{\Omega } {\frac{R-R^*}{\gamma_2}[\gamma_2(I-I^*)
 -(\mu+\delta)(R-R^*)]}dx\\
&\quad  +\int_{\Omega } {\frac{N-N^*}{\alpha}[-\mu(N-N^*)-\alpha(I-I^*)]}dx\\
&=-D\int_{\Omega } {\frac{[f(I)-If'(I)]\|\nabla I\|^{2}}{f^2(I)}}dx
 -DI^*\int_{\Omega } {\frac{f'(I)\|\nabla I\|^{2}}{f^2(I)}}dx\\
&\quad -\frac{D}{\gamma_2}\int_{\Omega } {\| \nabla R\|^2}dx
 -\frac{D}{\alpha}\int_{\Omega } {\| \nabla N\|^2}dx\\
&\quad  -\int_{\Omega } {(I-I^*)^2}dx-(\mu+\gamma_1+\gamma_2+\alpha)
 \int_{\Omega } {(I-I^*)[\frac{I}{f(I)}-\frac{I^*}{f(I^*)}]}dx\\
&\quad -\frac{\mu+\delta}{\gamma_2}\int_{\Omega }
 {(R-R^*)^2}dx-\frac{\mu}{\alpha}\int_{\Omega } {(N-N^*)^2}dx.
\end{align*}
Thus, $\frac{\partial V_1}{\partial t}\leq 0$, and the equality holds if
and only if $S\equiv S^*$,  $I\equiv I^*$, and $R\equiv R^*$.
By LaSalle's invariance principle, the endemic steady state
$\overline{E}^*$ is globally attractive if $\mathcal{R}_0>1$.
The proof is complete.
\end{proof}

From Theorem \ref{thm2.3}, we immediately obtain the following corollary.

\begin{corollary} \label{coro2.4}
 If $\mathcal{R}_0>1$, then the endemic steady state $E^*$ of
system \eqref{e1.2} is globally asymptotically stable in $\varPi$.
\end{corollary}

\section{Free boundary problem}

In this section, we study the free boundary problem of system \eqref{e1.3}.
Let $g_{\infty}:=\lim_{t \to \infty }g(t)$, then
$g_{\infty}\in (0,+\infty]$. If $g_{\infty}<\infty$ and
$\lim_{t \to \infty }\| I(\cdot,t)\|_{C[0, g(t)]}=0$,
then the vanishing occurs. If $g_{\infty}=\infty$,
then the spreading occurs. In this case,  the moving domain $(0,g(t))$
becomes the whole domain $(0,+\infty)$.

\subsection{Existence and uniqueness of solutions}
We use a contraction mapping theorem.
The proof depends mainly on some existing arguments
\cite{Chenxingfu2000,Kim2013,Linzhigui2007}, with some
modifications. We sketch the details here for completeness.

\begin{theorem} \label{thm3.1}
 For any given $(S_0,I_0, R_0)$ satisfying \eqref{e1.4} and any $\iota\in (0,\,1)$,
there exists a $T>0$  such that  system \eqref{e1.3} admits a unique bounded solution
 $$
 (S,I, R;g)\in C^{1+\iota,\frac{1+\iota}{2}}(Z_T^{\infty})
\times \big[C^{1+\iota,\frac{1+\iota}{2}}(Z_T)\big]^{2}
\times C^{1+\frac{\iota}{2}}([0,T]);
 $$
 Furthermore,
 $$
 \| S\|_{C^{1+\iota,\frac{1+\iota}{2}}(Z_T^{\infty})}
+\| I\|_{C^{1+\iota,\frac{1+\iota}{2}}(Z_T)}
+\| R\|_{C^{1+\iota,\frac{1+\iota}{2}}(Z_T)}
+\| g\|_{ C^{1+\frac{\iota}{2}}([0, T])}\leq \mathcal{K},
 $$
 where
\begin{gather*}
 Z_T^{\infty}=\{(z,t)\in \mathbb{R}^2:z\in[0,+\infty),t\in[0,T]\}, \\
 Z_T=\{(z,t)\in \mathbb{R}^2:z\in[0, g(t)],t\in[0,T]\}.
\end{gather*}
 Here $\mathcal{K}$ and $T$ only depend on $g_0$, $\iota$,
 $\| S_0\|_{C^2([0,+\infty))}$, $\| I_0\|_{C^2([0,g_0])}$, and
$\| R_0\|_{C^2([0,g_0])}$.
\end{theorem}

\begin{proof}
Let $\kappa(s)$ be a function in $C^3[0,+\infty)$ satisfying
$$
\kappa(s)=\begin{cases}
1, & \text{if }| s-g_0|<\frac{g_0}{8},\\
0, & \text{if }|s-g_0|>\frac{g_0}{2},
\end{cases} \quad \text{and }  |\kappa'(s)|<\frac{5}{g_0}\quad
\text{for all } s.
$$
We considering the transformation
$$
(y,t)\to (x,t),\quad \text{where }
 x=y+\kappa(|y|)\big(g(t)-\frac{g_0y}{|y|}\big),\quad y\in \mathbb{R}^n.
$$
Then
$$
(s,t)\to (z,t),\quad \text{where } z=s+\kappa(s)(g(t)-g_0),\; 0\leq s<\infty.
$$
By adopting the method similar to \cite{Chenxingfu2000},
the free boundary $z = g(t)$ can be changed to the line $s = g_0$.
Direct calculations yield that
\begin{gather*}
\frac{\partial s}{\partial z}=\frac{1}{1+\kappa'(s)(g(t)-g_0)}
:=\mathcal{A}(g(t),s), \\
\frac{\partial^2 s}{\partial z^2}
=-\frac{\kappa''(s)(g(t)-g_0)}{[1+\kappa'(s)(g(t)-g_0)]^3}
:=\mathcal{B}(g(t),s), \\
-\frac{1}{g(t)}\frac{\partial s}{\partial t}
=\frac{\kappa(s)}{1+\kappa'(s)(g(t)-g_0)}:=\mathcal{C}(g(t),s).
\end{gather*}
We set
\begin{gather*}
S(z,t)=S(s+\kappa(s)(g(t)-g_0),t):=m(s,t), \\
I(z,t)=I(s+\kappa(s)(g(t)-g_0),t):=n(s,t),\\
R(z,t)=R(s+\kappa(s)(g(t)-g_0),t):=j(s,t).
\end{gather*}
We rewrite system \eqref{e1.3} as
\begin{gather*}
 m_t-\mathcal{A}D\Delta_sm-\left(\mathcal{B}D+g'\mathcal{C}\right)m_s
=\Lambda-\mu m-mf(n)-\gamma_1n+\delta j,\quad s>0,\; t>0,\\
n_t-\mathcal{A}D\Delta_sn-(\mathcal{B}D+g'\mathcal{C})n_s
=mf(n)-(\mu+\gamma_1+\gamma_2+\alpha)n,\quad 0<s<g_0,\; t>0,\\
j_t-\mathcal{A}D\Delta_sj-(\mathcal{B}D+g'\mathcal{C})j_s
=\gamma_2n-(\mu+\delta)j,\quad  0<s<g_0,\;  t>0,\\
 m_{s}(0,t)=n_{s}(0,t)=j_{s}(0,t)=0,\quad t>0,\\
 n(s,t)=j(s,t)=0,\quad s\geq g_0,\; t>0,\\
 g'(t)=-\mu_1 n_s(g_0,t),\quad g(0)=g_0>0,\; t>0,\\
 m(s,0)=m_0(s),\quad n(s,0)=n_0(s),\quad j(s,0)=j_0(s),\; s\geq 0,
 \end{gather*}
where  $m_0=S_0$, $n_0=I_0$, and $j_0=R_0$.

We denote $g^*=-\mu_1 n_0'(g_0)$, and for $0<T\leq \frac{g_0}{8(1+g^*)}$, we set
\begin{gather*}
H_T=\big\{g\in C^1[0, T]:g(0)=g_0,\ g'(0)=g^*,\;
 \| g'-g^*\|_{C([0, T])}\leq 1\big\}, \\
M_T=\big\{m\in C([0,+\infty)\times[0, T]): m(s,0)=m_0(s),
 \| m-m_0\|_{L^{\infty}([0,+\infty)\times[0, T])}\leq 1\big\}, \\
\begin{aligned}
N_T=\Big\{&n\in C([0,+\infty)\times[0, T]): n(s,0)\equiv 0 \text{ for }
  s\geq g_0,\; 0\leq t\leq T,\\
&n(s,0)=n_0(s),  \text{ for }  0\leq s \leq g_0,\;
 \| n-n_0\|_{L^{\infty}([0,+\infty)\times[0, T])}\leq 1\Big\},
\end{aligned} \\
\begin{aligned}
J_T=\Big\{&j\in C([0,+\infty)\times[0, T]): j(s,0)\equiv 0\text{ for }
  s\geq g_0,\; 0\leq t\leq T,\\
&j(s,0)=j_0(s),  \text{ for } 0\leq s \leq g_0,\;
 \| j-j_0\|_{L^{\infty}([0,+\infty)\times[0, T])}\leq 1\Big\}.
\end{aligned}
\end{gather*}
Since $g_1,g_2\in H_T$ and $g_1(0) = g_2(0) = g_0$, one gets
$$
\| g_1-g_2\|_{C([0, T])}\leq T\| g_1'-g_2'\|_{C([0, T])}.
$$
 $\Gamma_T:=M_T\times N_T\times J_T\times H_T$  is a complete metric
space with the metric
\begin{align*}
&\mathfrak{D}\big((m_1,n_1,j_1;g_1),(m_2,n_2,j_2;g_2)\big) \\
&=\| m_1-m_2\|_{L^{\infty}([0,+\infty)\times[0, T])}
 +\| n_1-n_2\|_{L^{\infty}([0,+\infty)\times[0, T])} \\
&\quad +\| j_1-j_2\|_{L^{\infty}([0,+\infty)\times[0, T])}
 +\| g_1'-g_2'\|_{C([0, T])}.
\end{align*}
By adopting standard $L^p$ theory and the Sobolev embedding
theorem \cite{Ladyzenskaja1968},  for $(m, n, j; g)\in \Gamma_T$,
the  initial boundary value problem
\begin{gather*}
 \widetilde{m}_t-\mathcal{A}D\Delta_s\widetilde{m}
-(\mathcal{B}D+g'\mathcal{C})\widetilde{m}_s
 =\Lambda-\mu m-mf(n)-\gamma_1n+\delta j,\quad s>0,\; t>0,\\
\widetilde{n}_t-\mathcal{A}D\Delta_s\widetilde{n}
-(\mathcal{B}D+g'\mathcal{C})\widetilde{n}_s
=mf(n)-(\mu+\gamma_1+\gamma_2+\alpha)n,\quad 0<s<g_0,\; t>0,\\
\widetilde{j}_t-\mathcal{A}D\Delta_s\widetilde{j}
-(\mathcal{B}D+g'\mathcal{C})\widetilde{j}_s
 =\gamma_2n-(\mu+\delta)j,\quad 0<s<g_0,\; t>0,\\
\widetilde{m}_{s}(0,t)=\widetilde{n}_{s}(0,t)=\widetilde{j}_{s}(0,t)=0,\quad
 t>0,\\
 \widetilde{n}(s,t)=\widetilde{j}(s,t)=0,\quad s\geq g_0,\; t>0,\\
\widetilde{m}(s,0)=m_0(s),\quad \widetilde{n}(s,0)=n_0(s),\quad
 \widetilde{j}(s,0)=j_0(s),\quad  s\geq 0,
 \end{gather*}
has a unique solution
$$
(\widetilde{m},\widetilde{n},\widetilde{j})
\in \left[C^{1+\iota,\frac{1+\iota}{2}}([0,+\infty)\times[0, T])\right]^3,
$$
and it satisfies
\begin{gather*}
\|\widetilde{m}\|_{C^{1+\iota,\frac{1+\iota}{2}}([0,+\infty)\times[0, T])}
\leq \mathcal{K}_1, \\
\|\widetilde{n}\|_{C^{1+\iota,\frac{1+\iota}{2}}([0, g_0]\times[0, T])}
\leq \mathcal{K}_1, \\
\|\widetilde{j}\|_{C^{1+\iota,\frac{1+\iota}{2}}([0, g_0]\times[0, T])}\leq
\mathcal{K}_1,
\end{gather*}
where $\mathcal{K}_1$ is a constant depending on $\iota$,
$g_0$, $\| S_0\|_{C^2([0,+\infty))}$, $\| I_0\|_{C^2([0, g_0])}$,
 and $\| R_0\|_{C^2([0, g_0])}$.

We define
$$
\widetilde{g}(t)=g_0-\mu_1\int_{0}^{t}{\widetilde{n}_s(g_0,\tau)}d\tau.
$$
Then it follows that $\widetilde{g}'(t)=-\mu_1 \widetilde{n}_s(g_0,t)$,
$\widetilde{g}(0)=g_0$, and $\widetilde{g}'(0)=-\mu_1 n'_0(g_0)=g^*$.
 Thus, $\widetilde{g}'(t)\in C^{\iota/2}([0, T])$ and
$$
\| \widetilde{g}'(t) \|_{C^{\iota/2}([0, T])}
\leq \mathcal{K}_2:=\mu_1 \mathcal{K}_1.
$$
Next, we define a map
$\mathfrak{F}: \Gamma_T\to[C([0,+\infty)\times [0, T])]^{3}\times C^{1}([0, T])$
by
\[
\mathfrak{F}(m(s,t), n(s,t), j(s,t);g(t))
=(\widetilde{m}(s,t),\widetilde{n}(s,t),\widetilde{j}(s,t);\widetilde{g}(t)).
\]
 Then
$(m(s,t), n(s,t), j(s,t);g(t))\in \Gamma_T$ is  a fixed point of $\mathfrak{F}$.

From \cite{Duyihong2010},  there is a $T>0$ such that $\mathfrak{F}$
is a contraction mapping in $\Gamma_T$. In view of the contraction
 mapping theorem,  there exists a
$(m(s,t), n(s, t), j(s, t); g(t))$ in $\Gamma_T$ such that
$$
\mathfrak{F}(m(s,t), n(s,t), j(s,t);g(t))=(m(s,t), n(s,t), j(s,t);g(t)).
$$
Thus, $(S(z,t),I(z,t), R(z,t);g(t))$ is the solution of system \eqref{e1.3}.
Further, by employing the Schauder estimates,
$h(t)\in C^{1+\frac{\iota}{2}}([0, T])$,
$S\in C^{2+\iota,1+\frac{\iota}{2}}((0,+\infty)\times[0, T])$ and
$I,R\in C^{2+\iota,1+\frac{\iota}{2}}((0, g(t))\times[0, T])$.
 Hence, $(S(z,t),I(z,t), R(z,t); g(t))$ is the classical solution of
system \eqref{e1.3}. The proof is complete.
\end{proof}

To show the existence of solution for $t > 0$, we need to show the
following lemma. For mathematical considerations, we assume that
$\gamma_1=\delta=0$.

\begin{lemma} \label{lem3.2}
 Let $(S, I, R; g)$ be a bounded solution to system \eqref{e1.3} defined on
$t\in(0, T_0)$ for some $T_0 \in(0,+\infty]$. Then there
 exist positive constants $\mathcal{C}_1$ and $\mathcal{C}_2$
independent of $T_0$ such that
\begin{gather*}
0<S(z,t)\leq \mathcal{C}_1,\quad \text{for } \ 0\leq z<+\infty,\; t\in (0, T_0),\\
0<I(z,t),\quad R(z,t)\leq \mathcal{C}_2,\text{ for } \ 0\leq z<g(t),\;
 t\in (0, T_0).
\end{gather*}
\end{lemma}

\begin{proof}
By employing the strong maximum principle to system \eqref{e1.3} in
$[0, g(t)]\times [0, T_0)$,
$S(z, t), I(z,t), R(z,t) > 0$ for $0 \leq z < g(t)$, $0 < t < T_0$.
Note that $S(z,t)$ satisfies
\begin{gather*}
S_t-D\Delta S= \Lambda-\mu S-Sf(I),\quad z>0,\; t>0,\\
S(z,0)=S_0(z)\geq 0,\quad z\geq0.
 \end{gather*}
Thus, $S(z,t)\leq \mathcal{C}_1:=\max\{\|S_0(z)\|_{L^{\infty}(0,+\infty)},
 \frac{\Lambda}{\mu}\}$.
Let $H(z,t)=S(z,t)+I(z,t)+R(z,t)$.  Then
\begin{gather*}
 H_t-D\Delta H= \Lambda-\mu H-\alpha I,\quad 0<z<g(t),\; t>0,\\
  H=S\leq C_1,\quad z=g(t),\; t>0,\\
 H(z,0)=S_0(z)+I_0(z)+R_0(z),\quad 0\leq z\leq g_0.
 \end{gather*}
Hence, there exists a constant $\mathcal{C}_2>0$ such that
$$
S+I+R\leq \mathcal{C}_2,\quad \text{for }(z,t)\in [0, g(t)]\times [0, T_0).
$$
The proof is complete.
\end{proof}

Similar to the proof of  \cite[Lemma 3.2]{LinZhaoZhou2013}, we have
 the following result.

\begin{lemma} \label{lem3.3}
 There  exists a positive constant $\mathcal{C}_3$ independent of
$T_0$ such that
$ 0<g'(t)\leq \mathcal{C}_3$ for $t\in (0, T_0)$.
\end{lemma}

By adopting the similar arguments to \cite[Theorem 3.3]{LinZhaoZhou2013},
combined with Lemmas \ref{lem3.2} and \ref{lem3.3}, we obtain the following result.

\begin{theorem} \label{thm3.4}
 The solution of system \eqref{e1.3} exists and is unique for $t \in(0,\,\infty)$.
\end{theorem}

\subsection{Spreading and vanishing}

\begin{theorem} \label{thm3.5}
 If $\mathcal{R}_0<1$, then
$\lim_{t \to \infty }S(z,t)=\frac{\Lambda}{\mu}$,
$\lim_{t \to \infty }\| I(\cdot,t)\|_{C[0, g(t)]}=0$, and
$\lim_{t \to \infty }\| R(\cdot,t)\|_{C[0, g(t)]}=0$
uniformly in any bounded subset of $[0,+\infty)$.
 Moreover, $g_{\infty}<\infty$.
\end{theorem}

\begin{proof}
From the comparison principle, $S(z,t) \leq \overline{S}(t)$ for
 $z\geq 0$ and $t\in (0,+\infty)$, where
$$
\overline{S}(t):=\frac{\Lambda}{\mu}
+\big(\| S_0\|_{\infty}-\frac{\Lambda}{\mu}\big)e^{-\mu t}.
$$
$\overline{S}(t)$ is the solution of the problem
$$
\frac{d\overline{S}}{dt}=\Lambda-\mu \overline{S},\quad t>0;\;
 \overline{S}(0)=\| S_0\|_{\infty}.
$$
Since $\lim_{t \to \infty }\overline{S}(t)=\frac{\Lambda}{\mu}$, it follows that
$\limsup_{t \to \infty }S(z,t)
\leq\frac{\Lambda}{\mu}$ uniformly for $z\in [0,+\infty)$.
From $\mathcal{R}_0<1$,  there exists $T_0$ such that
 $S(z,t)\leq \frac{\Lambda}{\mu}\frac{1+\mathcal{R}_0}{2\mathcal{R}_0}$
 in $[0,+\infty)\times [T_0,+\infty)$. We find that $I(z,t)$ satisfies
\begin{gather*}
 I_t-D\Delta I \leq \big[\frac{\beta\Lambda}{\mu}
\frac{1+\mathcal{R}_0}{2\mathcal{R}_0}-(\mu+\gamma_2+\alpha)\big]I(z,t),
\quad 0<z<g(t),\; t>T_0,\\
 I(z,t)=0,\quad I_z(0,t)=0,\quad z=g(t),\; t>0,\\
 I(z, T_0)>0,\quad 0\leq z\leq g(T_0).
 \end{gather*}
Because of
$$
\frac{\beta\Lambda}{\mu(\mu+\gamma_2+\alpha)}
\frac{1+\mathcal{R}_0}{2\mathcal{R}_0}<1,
$$
we have $\lim_{t \to \infty }\| I(\cdot,t)\|_{C[0,g(t)]}=0$.
From  \eqref{e1.3}, we have
$\lim_{t \to \infty }\| R(\cdot,t)\|_{C[0, g(t)]}=0$.
 Next, we show that $g_{\infty}<+\infty$. In fact,
\begin{align*}
&\frac{d}{dt}\int_{0}^{g(t)}{z^{n-1}I(z,t)}dt \\
&=\int_{0}^{g(t)}{z^{n-1}
 I_t(z,t)}dz+g'(t)g^{n-1}(t)I(g(t),t)\\
&=\int_{0}^{g(t)}{Dz^{n-1}\Delta I}dz+\int_{0}^{g(t)}{z^{n-1}I(z,t)
 \big[\frac{Sf(I)}{I}-(\mu+\gamma_2+\alpha)\big]}dz\\
&=\int_{0}^{g(t)}{D(z^{n-1} I_z(z,t))_{z}}dz
 +\int_{0}^{g(t)}{z^{n-1}I(z,t)
 \big[\frac{Sf(I)}{I}-(\mu+\gamma_2+\alpha)\big]}dz\\
&=-\frac{D}{\mu_1}g^{n-1}g'(t)+\int_{0}^{g(t)}{z^{n-1}I(z,t)
 \big[\frac{Sf(I)}{I}-(\mu+\gamma_2+\alpha)\big]}dz.
\end{align*}
Integrating from $T_0$ to $t$ ($t>T_0$) gives
\begin{align*}
\int_{0}^{g(t)}{z^{n-1}I(z,t)}
&=\int_{0}^{g(T_0)}{z^{n-1}I_t(z, T_0)}dz+\frac{D}{n\mu_1}g^n(T_0)
 -\frac{D}{n\mu_1}g^n(t)\\
&\quad + \int_{T_0}^{t}\int_{0}^{g(s)}{z^{n-1}I(z,s)
 \big[\frac{Sf(I)}{I}-(\mu+\gamma_2+\alpha)\big]}dz\,ds.
\end{align*}
Since $0<S(z,t)\leq \frac{\Lambda}{\mu}\frac{1+\mathcal{R}_0}{2\mathcal{R}_0}$
for $z\in [0, g(t))$ and $t\geq T_0$, it follows that
\begin{equation*}
\big[\frac{Sf(I)}{I}-(\mu+\gamma_2+\alpha)\big]
\leq \beta S-(\mu+\gamma_2+\alpha)
\leq\frac{\beta\Lambda}{\mu}\frac{1+\mathcal{R}_0}{2\mathcal{R}_0}
 -(\mu+\gamma_2+\alpha)\leq 0.
\end{equation*}
For $t\geq T_0,$ it  follows that
$$
\int_{0}^{g(t)}{z^{n-1}I(z,t)}dz
\leq \int_{0}^{g(T_0)}{z^{n-1}I(z, T_0)}dz+\frac{D}{n\mu_1}g^n(T_0)
-\frac{D}{n\mu_1}g^n(t),\quad \text{for } t\geq T_0.
$$
Hence, $g_{\infty}<\infty$.
From system \eqref{e1.3},  $\lim_{t \to \infty }S(z,t)=\frac{\Lambda}{\mu}$
uniformly in any bounded subset of $[0,+\infty)$. The proof is complete.
\end{proof}

By using an argument analogous to \cite{Duyihong2010}  with some
minor modifications, we can obtain the following result.

\begin{lemma} \label{lem3.6}
Suppose that $T\in (0,+\infty)$, $\overline{g}\in C^1([0, T])$,
$\overline{S}\in C([0,+\infty)\times [0, T])\cap C^{2,1}([0,+\infty)
\times [0, T])$,
$\overline{I},\overline{R}\in C(\overline{Z}_T^{*})\cap C^{2,1}(Z_T^{*})$
with $Z_T^{*}=\{(z,t)\in \mathbb{R}^2:0<z<\overline{g}(t), 0<t\leq T\}$, and
 \begin{gather*}
 \overline{S}_t-D\Delta \overline{S} \geq
  \Lambda-\mu \overline{S},\quad z>0,\; 0<t\leq \overline{T},\\
\overline{I}_t-D\Delta \overline{I} \geq (\beta\overline{S}
-(\mu+\gamma_2+\alpha))\overline{I},\quad 0<z<\overline{g}(t),\;
 0<t\leq \overline{T},\\
\overline{R}_t-D\Delta \overline{R} \geq\gamma_2\overline{I}-\mu\overline{R},
\quad  0<z<\overline{g}(t),\; 0<t\leq \overline{T},\\
\overline{S}_{z}(0,t)\geq 0,\quad \overline{I}_{z}(0,t)\geq 0,\quad
\overline{R}_{z}(0,t)\geq 0,\quad 0<t\leq \overline{T},\\
\overline{I}(z,t)=\overline{R}(z,t)=0,\quad z\geq \overline{g}(t),\quad
 0<t\leq \overline{T},\\
\overline{g}'(t)\geq-\mu_1 \overline{I}_z(\overline{g}(t),t),\quad
\overline{g}(0)=g_0>0,\quad 0<t\leq \overline{T},\\
\overline{S}(z,0)=S_0(z),\quad \overline{I}(z,0)=I_0(z),\quad
\overline{R}(z,0)=R_0(z),\quad z\geq 0.
  \end{gather*}
\end{lemma}

Then the solution $(S,I, R;g)$ of system \eqref{e1.3} satisfies
\begin{gather*}
S(z,t)\leq \overline{S}(z,t),\quad g(t)\leq\overline{g}(t),
\quad \text{for }z\in (0,+\infty) \text{ and }  t\in (0, T], \\
I(z,t)\leq \overline{I}(z,t),\quad R(z,t)\leq \overline{R}(z,t),
\quad \text{for } z\in (0,g(t)) \text{ and } t\in (0, T].
\end{gather*}

\begin{theorem} \label{thm3.7}
 If $g_{\infty}<\infty$, then
$\lim_{t \to \infty }S(z,t)=\frac{\Lambda}{\mu}$,
$\lim_{t \to \infty }\| I(\cdot,t)\|_{C[0, g(t)]}=0$, and
 $\lim_{t \to \infty }\| R(\cdot,t)\|_{C[0, g(t)]}=0$
uniformly in any bounded subset of $[0,+\infty)$.
\end{theorem}

\begin{proof}
By contradiction, we assume that
$\limsup_{t \to \infty }\| I(\cdot,t)\|_{C[0, g(t)]}=\delta_1>0$.
There exists a sequence $(z_q,t_q)$ in $[0, g(t))\times (0,+\infty)$
such that $I(z_q,t_q)\geq \frac{\delta_1}{2}$ for $q\in N$, and
$t_q\to +\infty$. Since $0\leq z_q<g(t)<g_{\infty}<\infty$,
there exists a subsequence of $\{z_n\}$ converging to
$z_0\in [0, g_{\infty})$. We assume $z_q\to z_0$ as $q\to \infty$.

Define
\begin{gather*}
S_q(z,t)=S(z,t_q+t),\quad I_q(z,t)=I(z,t_q+t), \\
R_q(z,t)=R(z,t_q+t),\quad \text{for } (z,t)\in (0,g(t_q+t))\times
(-t_q,+\infty).
\end{gather*}
From the parabolic regularity,  $\{(S_q,I_q, R_q)\}$ has a
subsequence $\{(S_{q_{i}},I_{q_{i}}, R_{q_{i}})\}$ such that
$(S_{q_{i}},I_{q_{i}}, R_{q_{i}})\to (\widetilde{S},\widetilde{I},
\widetilde{R})$ satisfies
\begin{gather*}
\widetilde{S}_t-D\Delta \widetilde{S}
= \Lambda-\mu \widetilde{S}-\widetilde{S}f\left(\widetilde{I}
 \right),\quad 0<z<g_{\infty},\; t\in (-\infty,+\infty),\\
\widetilde{I}_t-D\Delta \widetilde{I}
= \widetilde{S}f(\widetilde{I})
-(\mu+\gamma_2+\alpha)\widetilde{I},\quad 0<z<g_{\infty},\;
 t\in (-\infty,+\infty),\\
\widetilde{R}_t-D\Delta \widetilde{R}
= \gamma_2\widetilde{I}-\mu\widetilde{R},\quad 0<z<g_{\infty},\;
 t\in (-\infty,+\infty).
 \end{gather*}
Because  $\widetilde{I}(z_0,0)\geq \delta_1/2$, we obtain
$\widetilde{I}>0$ in $[0,g_{\infty})\times(-\infty,+\infty)$. Noting that
$$
\widetilde{S}f(\widetilde{I})-(\mu+\gamma_2+\alpha)\widetilde{I}
=\Big(\widetilde{S}\frac{f(\widetilde{I})}{\widetilde{I}}
-(\mu+\gamma_2+\alpha)\Big)\widetilde{I}
$$
is bounded by $Q_1:=\beta\max\left\{\|S_0\|_{L^\infty},
\frac{\Lambda}{\mu}\right\}+\mu+\gamma_2+\alpha$.

Further,  $\widetilde{I}_z(g_{\infty},0)\leq -\sigma_0$ for some
$\sigma_0>0$. For any $0<\varpi<1$, there exists a constant
$\widetilde{C}$, which depends on $\varpi$, $g_0$,
$\| I_0\|_{C^{1+\varpi}[0, g_0]}$, and $g_{\infty}$, such that
$$
\| I_0\|_{C^{1+\varpi,\frac{1+\varpi}{2}}([0,g(t))
\times [0,+\infty))}+\| g\|_{C^{1+\frac{\varpi}{2}}( [0,+\infty))}
\leq \widetilde{C}.
$$
Define
$$
S=\frac{g_0z}{g(t)},\quad m(s,t)=S(z,t),\quad n(s,t)=I(z,t),\quad
j(s,t)=R(z,t).
$$
It then follows that
$$
I_t=n_t-\frac{g'(t)}{g(t)}sn_s,\quad I_z=\frac{g_0}{g(t)}n_s,\quad
\Delta_z I=\frac{g_0^2}{g^2(t)}\Delta_s n.
$$
Thus, $n(s,t)$ satisfies
\begin{gather*}
 n_t-D\frac{g_0^2}{g^2(t)}\Delta_s n-\frac{g'(t)}{g(t)}sn_s
=n\Big(\frac{f(n)m}{n}-(\mu+\gamma_2+\alpha)\Big),\quad 0<s<g_0,\; t>0,\\
 n_s(0,t)=n(g_0,t)=0,\quad t>0,\\
 n(s,0)=I_0(s)\geq 0,\quad 0\leq s\leq g_0.
 \end{gather*} \
Lemmas \ref{lem3.2} and \ref{lem3.3} yield
$$
\| n\big(\frac{f(n)m}{n}-(\mu+\gamma_2+\alpha)\big)\|_{L^{\infty}}\leq Q_2.
$$
By employing standard $L^P$ theory and  Sobolev embedding
theorem \cite{Ladyzenskaja1968}, one gets
$$
\| n\|_{ C^{1+\varpi, \frac{1+\varpi}{2}}([0,g_0)\times [0,+\infty))}\leq Q_3,
$$
where $Q_3$ is a positive constant depending on $\varpi$, $g_0$, $Q_1$, $Q_2$,
and $\| I_0\|_{C^2[0,g_0]}$.

Since $\| g\|_{C^{1+\frac{\varpi}{2}}([0,+\infty))}\leq \widetilde{C}$,
it follows that $g'(t)\to 0$ as $t\to \infty$, namely,
$I_z(g(t_q),t_q)\to 0$ as $t_q\to +\infty$.
Furthermore,  from 
$\| g\|_{C^{1+\varpi,\frac{1+\varpi}{2}}([0, g(t))\times[0,+\infty))}
\leq \widetilde{C}$, it follows that
$I_z(g(t_q),t_q+0)=(I_q)_z(g(t_q),0)\to \widetilde{I}_z(g_{\infty},0)$
as $q\to \infty$, which is a contradiction.
Thus, $\lim_{t \to \infty }\| I(\cdot,t)\|_{C[0, g(t)]}=0$.
Then $\lim_{t \to \infty }\| R(\cdot,t)\|_{C[0, g(t)]}=0$
and $\lim_{t \to \infty }S(z,t)=\frac{\Lambda}{\mu}$ uniformly in any
 bounded subset of $[0,+\infty)$.  The proof is complete.
\end{proof}

\begin{theorem} \label{thm3.8}
 If $\mathcal{R}_0>1$, $g_0\leq \min \big\{\sqrt{\frac{D}{16q_0}},
 \sqrt{\frac{D}{16\gamma_2}}\big\}$, and
$\mu_1\leq \frac{D}{8\mathcal{M}}$, then $g_{\infty}<\infty$,
where $q_0=\beta \mathcal{C}_1-\mu-\gamma_2-\alpha>0$ and
$\mathcal{M}=\frac{4}{3}\max\{\|I_0\|_{\infty},\|R_0\|_{\infty}\}$.
\end{theorem}

\begin{proof}
We construct suitable upper solutions for system \eqref{e1.3}.
As in \cite{Duyihong2010},  we define upper solutions as follows:
\begin{gather*} % \label{eq10}
\overline{S}(z,t)=\mathcal{C}_1, \\
\overline{I}=\overline{R}
=  \begin{cases}
 \mathcal{M}e^{-\gamma t}\mathcal{V}\big(\frac{z}{\overline{g}(t)}\big),
& 0\leq z\leq\overline{g}(t),\\
0, & z>\overline{g}(t),
 \end{cases}  \\
\overline{g}(t)=2g_0\left(2-e^{-\gamma t}\right),\quad
 t\geq 0,\quad \mathcal{V}(x)=1-x^2,\quad 0\leq x\leq 1,
\end{gather*}
where $\gamma$ and $\mathcal{M}$ are positive constants to be determined.
From $\mathcal{R}_0>1$, we get $k_0=\beta \mathcal{C}_1-\mu-\gamma_2-\alpha>0$.
A simple calculation yields
\begin{gather*}
\overline{S}_t-D\Delta \overline{S}=0 \geq \Lambda-\mu \overline{S},\\
\overline{I}_t-D\Delta \overline{I}
- \big(\beta\overline{S}-(\mu+\gamma_2+\alpha)\big)\overline{I}
\geq \mathcal{M}e^{-\gamma t}\big[\frac{D}{8g_0^2}-\gamma-k_0\big],\\
\overline{R}_t-D\Delta \overline{R}
- \left(\gamma_2\overline{I}-\mu\overline{R}\right)
\geq \mathcal{M}e^{-\gamma t}\big[\frac{D}{8g_0^2}-\gamma-\gamma_2\big],
\end{gather*}
for $0<z<\overline{g}(t)$ and $t>0$.

Direct calculations yield
 $\overline{g}'(t)=2g_0\gamma e^{-\gamma t}$ and
$-\mu_1\overline{I}_z(\overline{g}(t),t)
=2\mathcal{M}\mu_1\overline{g}^{-1}(t)e^{-\gamma t}$.
 Hence, $\overline{S}(z,0)\geq S_0(z)$,
$\overline{I}(z,0)=\mathcal{M}\big(1-\frac{z^2}{4g_0^2}\big)
\geq\frac{3}{4}\mathcal{M}$, and
$\overline{R}(z,0)=\mathcal{M}\big(1-\frac{z^2}{4g_0^2}\big)
\geq\frac{3}{4}\mathcal{M}$ for $z\in [0,g_0]$. By choosing
$\mathcal{M}=\frac{4}{3}\max\{\|I_0\|_{\infty},\|R_0\|_{\infty}\}$,
$\gamma=\frac{D}{16g_0^2}$, $\mu_1 \leq \frac{D}{8\mathcal{M}}$,
and $g_0\leq \min\{\frac{D}{16q_0},\,\frac{D}{16\gamma_2}\}$, one gets
\begin{gather*}
\overline{S}_t-D\Delta \overline{S}\geq \Lambda
-\mu \overline{S}-\overline{S}f(\underline{I}),\quad  z>0,\; t>0,\\
  \overline{I}_t-D\Delta \overline{I}
\geq \overline{S}f(\overline{I})-(\mu+\gamma_2+\alpha)\overline{I},\quad
 0<z<\overline{g}(t),\; t>0,\\
  \overline{R}_t-D\Delta \overline{R}
\geq \gamma_2\overline{I}-\mu\overline{R},\quad  0<z<\overline{g}(t),\; t>0,\\
\overline{S}_{z}(0,t)=\overline{I}_{z}(0,t)=\overline{R}_{z}(0,t)=0,\quad t>0,\\
\overline{I}(z,t)=\overline{R}(z,t)=0,\quad z\geq g(t),\; t>0,\\
 \overline{g}'(t)=-\mu_1 \overline{I}_z(\overline{g}(t),t),\quad
 \overline{g}(0)=2g_0>g_0>0,\; t>0,\\
 \overline{S}(z,0)\geq S_0(z),\quad
 \overline{I}(z,0)\geq I_0(z),\quad \overline{R}(z,0)\geq R_0(z),\ ;z\geq 0.
 \end{gather*}
From Lemma \ref{lem3.6},  $g(t)\leq \overline{g}(t)$ for $t>0$.
Hence, $g_{\infty}\leq \lim_{t \to \infty }\overline{g}(t)=4g_0<\infty$.
The proof is complete.
\end{proof}

Let $\lambda_1$ represent the principle eigenvalue of the operator $-\Delta$
with respect to the  homogeneous Dirichlet
boundary condition.
We then have the following result.

\begin{theorem} \label{thm3.9}
 If $\mathcal{R}_0>1$, then $g_{\infty}=\infty$ provided that $g_0>g_0^*$,
 where $\lambda_1(g_0^*)=\frac{\mu+\gamma_2+\alpha}{D}(\mathcal{R}_0-1)$.
\end{theorem}

\begin{proof}
By a way of contradiction, we assume that $g_{\infty}<\infty$.
From Theorem \ref{thm3.7},  $\lim_{t \to \infty }\| I(\cdot,t)\|_{C[0, g(t)]}=0$.
Further, $\lim_{t \to \infty }S(z,t)=\frac{\Lambda}{\mu}$ uniformly
in the bounded subset. Consequently, for $\varepsilon>0$,
there exists $T^{*}>0$ such that $S(z,t)\geq \frac{\Lambda}{\mu}-\varepsilon$
for $r\in [0, g(t))$, $t\geq T^*$. $I(z,t)$ satisfies
\begin{gather*}
 I_t-D\Delta I\geq I\Big(f'(\varepsilon)
\big(\frac{\Lambda}{\mu}-\varepsilon\big)-(\mu+\gamma_2+\alpha)\Big),\quad
  0<s<g_0,\; t>T^*,\\
 I_z(0,t)=I(g_0,t)=0,\quad t>T^*,\\
 I(z, T^*)> 0,\quad 0\leq z< g_0.
 \end{gather*}
 $I(z,\, t)$ has a lower solution $\underline{I}(z,t)$ satisfying
\begin{gather*}
\underline{I}_t-D\Delta \underline{I}
=\underline{I}\Big(f'(\varepsilon)\big(\frac{\Lambda}{\mu}-\varepsilon\big)
-(\mu+\gamma_2+\alpha)\Big),\quad 0<s<g_0,\; t>T^*,\\
 \underline{I}_z(0,t)=\underline{I}(g_0,t)=0,\quad t>T^*,\\
 \underline{I}(z, T^*)=I(z, T^*),\quad 0\leq z< g_0.
 \end{gather*}
From $g_0>g_0^*$, we can choose sufficiently small $\varepsilon$ satisfying
$$
f'(\varepsilon)\big(\frac{\Lambda}{\mu}-\varepsilon\big)
-(\mu+\gamma_2+\alpha)>D\lambda_1(g_0).
$$
Hence, $I$ is unbounded in $(0, g_0) \times [T^*,+\infty)$,
 which leads to a contradiction.  The proof is complete.
\end{proof}

\section{Numerical simulations}

In this section, we perform some numerical simulations to illustrate
the theoretical results.
For the homogeneous system,  we choose parameters
\begin{equation} \label{e4.1}
\Lambda=0.1,\quad d_U=0.01,\quad \beta=0.3,\quad  \gamma_1=1,\quad
 \delta=1,\quad \gamma_2=2,\quad \alpha=2.
\end{equation}
We choose the initial conditions as follows
\begin{gather*}
S_0(x)=5\Big(1+0.5\cos\big(\frac{9}{10}\pi\big)\Big),\quad
I_0(x)=5\Big(1+0.8\sin\big(\frac{9}{10}\pi\big)\Big),\\
R_0(x)=5\Big(1+0.6\sin\big(\frac{9}{10}\pi\big)\Big),\quad x\in [0,10],
\end{gather*}
and the Neumann boundary condition
\begin{equation*}
\frac{\partial S(x,t)}{\partial \nu}=\frac{\partial I(x,t)}{\partial \nu}
=\frac{\partial R(x,t)}{\partial \nu}=0,\quad  t>0,\; x\in \partial \Omega.
\end{equation*}

\begin{figure}[ht]
 \begin{center}
\includegraphics[width=6cm, height=4cm]{fig1a}  %001
\includegraphics[width=6cm, height=4cm]{fig1b}  %002
\includegraphics[width=6cm, height=4cm]{fig1c}  %003
 \end{center}
\caption{$E_0$ is globally asymptotically stable when $D=0.1$.}
\label{fig1}
\end{figure}

\begin{figure}[ht]
 \begin{center}
 \includegraphics[width=6cm, height=4cm]{fig2a} % 011.eps
 \includegraphics[width=6cm, height=4cm]{fig2b} % 012.eps
 \includegraphics[width=6cm, height=4cm]{fig2c} % 013.eps
 \end{center}
 \caption{$E^*$ is globally asymptotically stable when $D=0.1.$}
\label{fig2}
\end{figure}

\begin{figure}[ht]
 \begin{center}
 \includegraphics[width=6cm, height=4cm]{fig3a} % 111
 \includegraphics[width=6cm, height=4cm]{fig3b} % 112
 \includegraphics[width=6cm, height=4cm]{fig3c} % 113
 \end{center}
\caption{$E_0$ is globally asymptotically stable when $D=10000$.}
\label{fig3}
\end{figure}

\begin{figure}[ht]
 \begin{center}
 \includegraphics[width=6cm, height=4cm]{fig4a} % 211
 \includegraphics[width=6cm, height=4cm]{fig4b} % 212
 \includegraphics[width=6cm, height=4cm]{fig4c} % 213
\end{center}
\caption{$E^*$ is globally asymptotically stable when $D=10000$.}
\label{fig4}
\end{figure}

For the case $D=0.1$, by a simple computation, we get
 $\mathcal{R}_0<1$.  In view of Theorem \ref{thm2.1},
the disease-free steady state $E_0$ of system \eqref{e1.2} is globally
asymptotically stable (see, Figure \ref{fig1}). Further, if $\Lambda=50$ and
the other parameters are the same as \eqref{e4.1},  system \eqref{e1.2} exists
a unique endemic steady state. By Theorem \ref{thm2.1}, the endemic steady
state of system \eqref{e1.2} is globally asymptotically stable (see, Figure \ref{fig2}).
Similarly, for the case $D=10000$, from Figures \ref{fig3} and
\ref{fig4},  the disease-free steady state $E_0$ of system \eqref{e1.2}
is globally asymptotically stable if $\mathcal{R}_0<1$, while
the endemic steady state of system \eqref{e1.2} is globally asymptotically
stable if $\mathcal{R}_0>1$.

\begin{figure}[ht]
 \begin{center}
 \includegraphics[width=6cm, height=4cm]{fig5a} % 7
 \includegraphics[width=6cm, height=4cm]{fig5b} % 8
 \includegraphics[width=6cm, height=4cm]{fig5c} % 9
\end{center}
 \caption{$E_0$ is globally asymptotically stable when $D=0.1$.}
\label{fig5}
\end{figure}

\begin{figure}[ht]
 \begin{center}
 \includegraphics[width=6cm, height=4cm]{fig6a} % 1
 \includegraphics[width=6cm, height=4cm]{fig6b} % 2
 \includegraphics[width=6cm, height=4cm]{fig6c} % 3
\end{center}
\caption{The endemic steady state of system \eqref{e1.2}
converges to a positive distribution which is not a constant when $D=0.1$.}
\label{fig6}
\end{figure}

\begin{figure}[ht]
 \begin{center}
 \includegraphics[width=6cm, height=4cm]{fig7a} % 10
 \includegraphics[width=6cm, height=4cm]{fig7b} % 11
 \includegraphics[width=6cm, height=4cm]{fig7c} % 12
\end{center}
\caption{$E_0$ is globally asymptotically stable when $D=10000$.}
\label{fig7}
\end{figure}

\begin{figure}[ht]
 \begin{center}
 \includegraphics[width=6cm, height=4cm]{fig8a} % 4.eps
 \includegraphics[width=6cm, height=4cm]{fig8b} % 5.eps
 \includegraphics[width=6cm, height=4cm]{fig8c} % 6.eps
 \end{center}
\caption{The endemic steady state of system \eqref{e1.2}  converges to
a positive constant distribution which is the homogeneous constant
steady state when $D=10000$.}
\label{fig8}
\end{figure}

Next, we fix  parameters as \eqref{e4.1} and vary $\beta(x)$ with the following form
$$
\beta(x)=\overline{\beta}(1+0.8\cos \pi x),
$$
where $\overline{\beta}$ is a positive constant.
We choose the function $\beta$ to explore the difference for the
dynamical behavior between the homogeneous system and the heterogeneous
system.

Let $D=0.1$.  For $\overline{\beta}=0.3$ and the other parameters as \eqref{e4.1},
we have $\mathcal{R}_0<1$. In Figure \ref{fig5}, we observe that the
disease-free steady state $E_0$ of system \eqref{e1.2} is globally asymptotically
stable. On the other hand, for $\Lambda=50$ and the other parameters are
the same as \eqref{e4.1}, we get $\mathcal{R}_0>1$. Thus, from Theorem \ref{thm2.1},
the endemic steady state of system \eqref{e1.2}  converges to a positive
distribution which is not a constant
(see, Figure \ref{fig6}).

Let $D=10000$.  For $\overline{\beta}=0.3$ and the other parameters as \eqref{e4.1},
we get $\mathcal{R}_0<1$.  In fact, in Figure \ref{fig7},
 the disease-free steady state $E_0$ of system \eqref{e1.2} is globally
asymptotically stable.
On the other hand, for $\Lambda=50$ and the other parameters are the
same as \eqref{e4.1}, it then follows that $\mathcal{R}_0>1$. From the numerical
simulations,  the endemic steady state of system \eqref{e1.2}  converges to
a positive constant distribution which is the homogeneous constant
steady state (see, Figure \ref{fig8}).

In biology, for the homogeneous system, we observe that the final state
of the infectious disease is independent on its dispersal rate,
while for the heterogeneous system, the final state of the infectious
disease is dependent on its dispersal rate.

\section*{Discussions and conclusions}

In this paper, we have proposed a SIRS epidemic reaction-diffusion system
with  two different kinds of boundary conditions. For the problem
with the Neumann boundary condition, we have obtained the global dynamics,
 which are fully determined by the basic reproduction number $\mathcal{R}_0$.
To make a better understanding for the transmissions dynamics for the
disease, we further consider a free boundary problem of system \eqref{e1.3}.
Main results reveal that besides the basic reproduction number,
the size of initial epidemic region and the diffusion rate of the
disease also play a crucial role in the disease transmission.

\subsection*{Acknowledgments}
K. Li was supported by the Scientific and Technological Research Project
of Xiamen University of Technology (YKJ14027R).
 J. Li was supported by the Natural Science Foundation of Gansu Province, China
 (1308RJZA113); by the National Science Foundation of China (11362008),
Youth Science Foundation of Lanzhou  Jiaotong University (2012019);
 and by the Fundamental Research Funds for the Universities of Gansu
Province (212084).
 W. Wang was supported by the China Scholarship Council Award.


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\end{document}