\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 23, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/23\hfil Global stability for delay SIR and SEIR]
{Global stability for delay SIR and SEIR epidemic models with
saturated incidence rates}

\author[A. Abta A. Kaddar, H. Talibi Alaoui \hfil EJDE-2012/23\hfilneg]
{Abdelhadi Abta, Abdelilah Kaddar, Hamad Talibi Alaoui}  % in alphabetical order

\address{Abdelhadi Abta \newline
Chouaib Doukkali, Facult\'e des Sciences, 
D\'epartement de Math\'ematiques et Informatique
B.P. 20, El Jadida, Morocco}
\email{abtaabdelhadi@yahoo.fr}

\address{Abdelilah Kaddar \newline
Universit\'e Mohammed V- Souissi, Facult\'e des
 Sciences Juridiques, Economiques et Sociales - Sal\'e, Morocco}
\email{a.kaddar@yahoo.fr}

\address{Hamad Talibi Alaoui \newline
Chouaib Doukkali, Facult\'e des
Sciences, D\'epartement de Math\'ematiques et Informatique
B.P. 20, El Jadida, Morocco}
\email{talibi\_1@hotmail.fr}

\thanks{Submitted October 31, 2011. Published February 7, 2012.}
\subjclass[2000]{34D23, 37B25, 00A71}
\keywords{SIR epidemic model; SEIR epidemic model; incidence rate;
 \hfill\break\indent  delay differential equations;
 Lyapunov function; global stability}

\begin{abstract}
 In this article we propose a comparison of a delayed SIR model and
 its corresponding SEIR model in terms of global stability.
 We consider a saturated incidence rate and we determine, using
 Lyapunov functionals, conditions by which the disease-free equilibrium
 and the endemic equilibrium are globally asymptotically stable.
 Also some numerical simulations are given to compare a global behaviour
 of a delayed SIR model and its corresponding SEIR model.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks


\section{Introduction}\label{s1}

In this article, we propose the following  delay SIR epidemic model
with a saturated incidence rate (see, \cite{kaddar}):
\begin{equation}
\begin{gathered}
\frac{dS}{dt}=A-\mu S(t)-\frac{\beta S(t)I(t)}{1+\alpha_1
S(t)+\alpha_2 I(t)}, \\
\frac{dI}{dt}=\frac{\beta e^{-\mu\tau} S(t-\tau)I(t-\tau)}
 {1+\alpha_1 S(t-\tau)+\alpha_2 I(t-\tau)}
 -(\mu+\alpha+\gamma) I(t).
\end{gathered}  \label{a1}
\end{equation}
The initial condition for the above system is
\begin{equation}\label{sge3}
S(\theta)=\varphi_1(\theta), \quad I(\theta)=\varphi_2(\theta), \quad
 \theta \in [-\tau,0]
\end{equation}
 with $\varphi=(\varphi_1,\varphi_2) \in C^+\times C^+$, such that
$\varphi_i(\theta)\geq 0$  ($-\tau \leq \theta \leq 0$, $i=1,2)$.
 Here  $C$ denotes the Banach space $C([-\tau,0],\mathbb{R})$ of continuous functions
mapping the interval
$[-\tau, 0]$ into $\mathbb{R}$, equipped with the supremum norm.
The nonnegative cone of $C$ is defined as
$C^+ = C([-\tau,0],\mathbb{R}^+)$.
 where $S$ is the number of susceptible individuals, $I$ is the
number of infectious individuals,  $A$ is the recruitment rate of
the population, $\mu$ is the natural death of the population,
$\alpha$ is the death rate due to disease, $\beta$ is the
transmission rate, $\alpha_1$ and $\alpha_2$ are the parameters
that measure the inhibitory effect, $\gamma$ is the recovery rate
of the infectious individuals, and $\tau$ is the incubation period.

The corresponding SEIR model of system \eqref{a1} is described in \cite{kaddar}
as
\begin{equation}
\begin{gathered}
\frac{dS}{dt}=A-\mu S(t)-\frac{\beta S(t)I(t)}{1+\alpha_1
S(t)+\alpha_2 I(t)},\\
\frac{dE}{dt}=\frac{\beta S(t)I(t)}{1+\alpha_1
S(t)+\alpha_2 I(t)}-(\sigma +\mu )E(t),\\
\frac{dI}{dt}=\sigma E(t)-(\mu+\alpha +\gamma )I(t).
\end{gathered}  \label{a2}
\end{equation}
where $E$ is the number of exposed individuals, and $\sigma$ is the rate at
which exposed individuals become infectious. Thus $\frac{1}{\sigma}$ is the
mean latent period.

In models \eqref{a1} and \eqref{a2} the formulation of the incidence rate
$\frac{\beta S(t)I(t)}{1+\alpha_1 S(t)+\alpha_2 I(t)}$;  i.e.,
 the infection rate of susceptible individuals
through their contacts with infectious (see, for example, \cite{Gao,Yorke}),
includes the three forms: The first one is the bilinear incidence rate $\beta SI$,
\cite{Gabriela,Zhou}. The second one is the saturated incidence rate of the form
 $\frac{\beta S(t)I(t)}{1+\alpha_1 S(t)}$ \cite{Anderson,Zhang2}.
The third one is the saturated incidence rate of the form
$\frac{\beta S(t)I(t)}{1+\alpha_2 I(t)}$ \cite{Jiang,Capasso,Xu}.

In \cite{kaddar}, we  considered  a local properties of a delayed SIR
model (system \eqref{a1}) and its corresponding SEIR model
(system \eqref{a2}), and we observed that if $\mu\tau$ is close enough to $0$,
then the two above models  generate identical local asymptotic behavior.

For, the model \eqref{a1} with $\tau=0 $ and $\mu=A$, Korobeinikov \cite{koro}
proved that the endemic equilibrium is globally asymptotically stable.
 Huang and al \cite{Huang}  studied the global asymptotic stability of the
 delay SIR model
\begin{equation}
\begin{gathered}
\frac{dS}{dt}=\mu-\mu S(t)-f(S(t),I(t-\tau)), \\
\frac{dI}{dt}=f(S(t),I(t-\tau)) -(\sigma+\mu) I(t).
\end{gathered} \label{aa1}
\end{equation}
The fundamental difference  of this model with our model \eqref{a1} is the presence
of the fraction $e^{-\mu \tau}$ in  the incidence rate in the second
equation of \eqref{a1}. The Lyapunov functional proposed in \cite{Huang} in
not valid for \eqref{a1}.

For, the SEIR model, Sun and al \cite{sun,Korobeinikov2} proposed nonlinear
incidence of the form $\beta I^pS^q$ and constructed an explicit Lyapunov
function and established a global stability of this model.

In this paper, by constricting the suitable Lyapunov functionals, we determine
the global asymptotic stability of a delayed SIR model \eqref{a1} and
its corresponding SEIR model \eqref{a2}.
The rest of the paper is organized as follows.
In Section 2, global stability of the delayed SIR epidemiological model \eqref{a1}
is established. In Section 3, global stability of the SEIR epidemiological
model \eqref{a2} is determined. In Section 4, numerical simulations and
concluding remarks are provided. In the appendix, some results on the
global stability are stated.

\section{Global stability analysis of delayed SIR model}\label{s2}

In this section, we discuss the global  stability of a disease-free
equilibrium and an endemic equilibrium of system \eqref{a1}.
With the change of variables  $i(t)=I(t+\tau)$ and $s(t)=S(t)$,
the system \eqref{a1} becomes
\begin{equation}\label{ab1}
\begin{gathered}
\frac{ds(t)}{dt}=A-\mu s-\frac{\beta s(t)i(t-\tau)}{1+\alpha_1 s(t)+\alpha_2 i(t-\tau)}\\
\frac{di(t)}{dt}=\frac{e^{-\mu \tau}\beta s(t)i(t-\tau)}{1+\alpha_1 s(t)
+\alpha_2 i(t-\tau)}-(\mu+\alpha+\gamma)i(t).
\end{gathered}
\end{equation}
In the rest of this paper, we set $\mu_1=\mu+\alpha$.
Since $\frac{d}{dt}(s(t)+i(t))\leq A-\mu (s(t)+i(t))$, we have
 $\limsup (s(t)+i(t)) \leq \frac{A}{\mu}$. Hence
we discuss system \eqref{ab1} in the closed set
$$
\Omega=:\big\{(\varphi_1,\varphi_2)\in C^+\times C^+:
\|\varphi_1+\varphi_2\|\leq A/\mu  \big\} .
$$
It is easy to show that $\Omega$  is positively invariant with respect to
system \eqref{ab1}.
System \eqref{ab1} always has a disease-free equilibrium
$P_1=(A/\mu,0)$. Further, if
$$
R_{01}:=\frac{A\beta e^{-\mu\tau} }{(\alpha_1 A+\mu)(\mu_1+\gamma)}>1,
$$
system \eqref{ab1} admits a unique endemic equilibrium
$P_1^*=(S^*,I^*)$, with
\begin{gather*}
S^{\ast}=\frac{A[(\mu_1+\gamma)+\alpha_2A
e^{-\mu\tau}]}{(\mu_1 +\gamma)[\alpha_1 A(R_{01}-1)+\mu R_{01} ] +\mu\alpha_2A
e^{-\mu\tau}},
\\
I^*=\frac{A(R_{01}-1)e^{-\mu\tau}(\alpha_1 A+\mu)}{(\mu_1 +\gamma)[\alpha_1 A(R_{01}-1)+\mu R_{01} ] +\mu\alpha_2A
e^{-\mu\tau}}.
\end{gather*}
Next we consider the global asymptotic stability of the disease-free
 equilibrium $P_1$ and the endemic equilibrium $P_1^{*}$ of \eqref{ab1} by Lyapunov
functionals, respectively.

\begin{proposition} \label{prop2.1}
If $R_{01} \leq 1$, then the disease-free equilibrium $P_1$ is globally asymptotically
stable.
\end{proposition}

\begin{proof}
Define a Lyapunov functional
$V(t)=V_1(t)+i(t)+V_2(t)$,
with
\begin{gather*}
V_1(t)=e^{-\mu\tau}\int_{\frac{A}{\mu}}^{s(t)}
\Big(1-\frac{A(1+\alpha_1u)}{(\mu+\alpha_1A)u}\Big)du,\\
V_2(t)=(\mu_1+\gamma)\int_{0}^{\tau}i(t-u)du.
\end{gather*}
We will show that $\frac{dV(t)}{dt}\leq 0$ for all  $t\geq 0$.
We have
\begin{align*}
\frac{dV_1(t)}{dt}
&=e^{-\mu\tau}\Big(1-\frac{A(1+\alpha_1s(t))}{(\mu+\alpha_1A)s(t)}\Big)
\Big(A-\mu s(t)-\frac{\beta s(t)i(t-\tau)}{1+\alpha_1 s(t)+\alpha_2 i(t-\tau)}\Big)\\
&= e^{-\mu\tau}\Big(1-\frac{A(1+\alpha_1s(t))}{(\mu+\alpha_1A)s(t)}\Big)
\big(A-\mu s(t)\big)\\
&\quad -e^{-\mu\tau}\Big(1-\frac{A(1+\alpha_1s(t))}{(\mu+\alpha_1A)s(t)}\Big)
\Big(\frac{\beta s(t)i(t-\tau)}{1+\alpha_1 s(t)+\alpha_2 i(t-\tau)}\Big),
\end{align*}
and
$$
\frac{dV_2(t)}{dt}=(\mu_1+\gamma)[i(t)-i(t-\tau)]
$$
Therefore,
\begin{align*}
\frac{dV(t)}{dt}
&= e^{-\mu\tau}\Big(1-\frac{A(1+\alpha_1s(t))}{(\mu+\alpha_1A)s(t)}\Big)
 \big(A-\mu s(t)\big)\\
&\quad -e^{-\mu\tau}\Big(1-\frac{A(1+\alpha_1s(t))}{(\mu+\alpha_1A)s(t)}\Big)
 \Big(\frac{\beta s(t)i(t-\tau)}{1+\alpha_1 s(t)+\alpha_2 i(t-\tau)}\Big)\\
&\quad +\frac{e^{-\mu\tau}\beta s(t)i(t-\tau)}{1+\alpha_1 s(t)
 +\alpha_2 i(t-\tau)}-(\mu_1+\gamma)i(t)+(\mu_1+\gamma)[i(t)-i(t-\tau)]\\
&=-\frac{e^{-\mu \tau}(A-\mu s(t))^2}{(\mu+\alpha_1A)s(t)}
+(\mu_1+\gamma)\Big( \frac{R_{01} (1+\alpha_1 s(t))}{(1+\alpha_1 s(t)
+\alpha_2 i(t-\tau))}-1 \Big)i(t-\tau)
\end{align*}
Since $\frac{1+\alpha_1 s(t)}{1+\alpha_1 s(t)+\alpha_2 i(t-\tau)}\leq1$
for all $t\geq0$, it follows that
$$
\frac{dV(t)}{dt}\leq-\frac{e^{-\mu \tau}(A-\mu s(t))^2}{(\mu+\alpha_1A)s(t)}
+(\mu_1+\gamma)\left(R_{01}-1 \right)i(t-\tau).
$$
Therefore, $R_{01}\leq 1$ ensures that $\frac{dV(t)}{dt}\leq 0$ for all $t\geq 0$,
where $\frac{dV(t)}{dt}=0$ holds if $s(t)=\frac{A}{\mu}$ and $i(t)=0$ .
 Hence, it follows from system \eqref{ab1} that $\{P_1\}$ is the largest invariant
set in $\left\{(s(t),i(t))|\frac{dV(t)}{dt}=0\right\}$.
 From the Lyapunov-LaSalle asymptotic stability, we obtain that $P_1$ is
globally asymptotically stable. This completes the proof.
\end{proof}

\begin{proposition} \label{prop2.2}
If $R_{01} >1$, then the endemic equilibrium $P_1^*$ is globally asymptotically stable.
\end{proposition}

\begin{proof}
To prove global stability of the endemic equilibrium, we define a Lyapunov
functional
$V(t)=V_1(t)+V_2(t)$,
with
$$
V_1(t)={e^{-\mu\tau}\int_{s^*}^{s(t)}
\Big(1-\frac{s^*(1+\alpha_1 u+\alpha_2 i^*)}{u(1+\alpha_1 s^*+\alpha_2 i^*)}\Big)du
+i(t)-i^*-i^*\ln\big(\frac{i(t)}{i^*}\big)},
$$
and
$$
V_2(t)={\frac{e^{-\mu\tau}\beta s^* i^* }{1+\alpha_1s^*+\alpha_2 i^*}\int_{0}^{\tau}
\Big[\frac{i(t-u)}{i^*}-1-\ln\big(\frac{i(t-u)}{i^*}\big) \Big]du.}
$$
We here note that
$$\frac{\partial V_1}{\partial s}
=1-\frac{s^*(1+\alpha_1 s+\alpha_2 i^*)}{s(1+\alpha_1 s^*+\alpha_2 i^*)},\quad
\frac{\partial V_1}{\partial i}=1-\frac{i^*}{i},
$$
 which implies that the point $(s^*,i^*)$ is a stationary point of the function
 $V_1(t)$ and it is the unique stationary point and the global minimum of this function.
Using the relations
$$
{A=\mu s^*+\frac{\beta s^*i^*}{1+\alpha_1 s^*+\alpha_2 i^*}},\quad
{\mu_1+\gamma=\frac{e^{-\mu\tau}\beta s^*}{1+\alpha_1 s^*+\alpha_2 i^*}},
$$
the time derivative of the function $V_1(t)$ along the positive solution of
system \eqref{ab1} becomes
\begin{equation} \label{l1}
\begin{aligned}
\frac{dV_1(t)}{dt}
&= e^{-\mu\tau}\Big(1-\frac{s^*(1+\alpha_1 s(t)+\alpha_2 i^*)}{s(t)
 (1+\alpha_1 s^*+\alpha_2 i^*)}\Big)
 \Big(A-\mu s(t)-\frac{\beta s(t)i(t-\tau)}{1+\alpha_1 s(t)+\alpha_2i(t-\tau)}\Big)
\\
&\quad+\big(1-\frac{i^*}{i(t)}\big)
 \Big(\frac{e^{-\mu\tau}\beta s(t)i(t-\tau)}{1+\alpha_1 s(t)+\alpha_2 i(t-\tau)}
 -(\mu_1+\gamma)i(t)\Big)
\\
&=e^{-\mu\tau}\Big(1-\frac{s^*(1+\alpha_1 s(t)+\alpha_2 i^*)}{s(t)(1+\alpha_1 s^*
 +\alpha_2 i^*)}\Big)\\
&\quad\times \Big(\mu (s^*-s(t))
 +\frac{\beta s^*i^*}{1+\alpha_1 s^*+\alpha_2 i^*}
 -\frac{\beta s(t)i(t-\tau)}{1+\alpha_1 s(t)+\alpha_2i(t-\tau)}\Big)
 \\
&\quad +e^{-\mu\tau}\big(1-\frac{i^*}{i(t)}\big)
 \Big(\frac{\beta s(t)i(t-\tau)}{1+\alpha_1 s(t)+\alpha_2 i(t-\tau)}
 -\frac{\beta s^*}{1+\alpha_1 s^*+\alpha_2 i^*}i(t)\Big),
\end{aligned}
\end{equation}
and the time derivative of the function $V_2(t)$ becomes
\begin{equation}\label{l2}
\frac{dV_2(t)}{dt}=\frac{e^{-\mu\tau}\beta s^*i^*}{1+\alpha_1s^*+\alpha_2i^*}
\Big[-\frac{i(t-\tau)}{i^*}+\frac{i(t)}{i^*}+\ln\big(\frac{i(t-\tau)}{i(t)}\big)\Big].
\end{equation}
From \eqref{l1} and \eqref{l2}, we obtain
\begin{align*}
&\frac{dV(t)}{dt}\\
&= -\frac{e^{-\mu \tau}\mu(s(t)-s^*)^2(1+\alpha_2i^*)}{(1+\alpha_1s^*+\alpha_2i^*)}
  +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}\\
&\quad\times\Big( 1-\frac{s^*(1+\alpha_1s(t)+\alpha_2 i^*)}{s(t)(1+\alpha_1s^*
+\alpha_2 i^*)}\Big)
\Big(1-\frac{s(t)i(t-\tau)(1+\alpha_1s^*+\alpha_2 i^*)}{s^*i^*(1+\alpha_1s(t)
 +\alpha_2 i(t-\tau))} \Big)
\\
&\quad +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}
 \big(1-\frac{i^*}{i(t)}\big)
 \Big(\frac{s(t)i(t-\tau)(1+\alpha_1s^*+\alpha_2i^*)}{s^*i^*
 (1+\alpha_1s(t)+\alpha_2i(t-\tau))}-\frac{i(t)}{i^*}\Big)\\
&\quad +\frac{e^{-\mu \tau}\beta s^*i^*}{1+\alpha_1s^*+\alpha_2i^*}
 \Big[ -\frac{i(t-\tau)}{i^*}+\frac{i(t)}{i^*}
 +\ln\big(\frac{i(t-\tau)}{i(t)}\big)\Big]\\
&= -\frac{e^{-\mu \tau}\mu(s(t)-s^*)^2(1+\alpha_2i^*)}{(1+\alpha_1s^*+\alpha_2i^*)}
  +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}
 \Big[2-\frac{s^*(1+\alpha_1s(t)  +\alpha_2i^*)}{s(t)(1+\alpha_1s^*+\alpha_2i^*)}\\
&\quad + \frac{i(t-\tau)(1+\alpha_1s(t)+\alpha_2i^*)}{i^*(1+\alpha_1s(t)+\alpha_2i(t-\tau))}
 -\frac{s(t)i(t-\tau)(1+\alpha_1s^*+\alpha_2i^*)}{s^*i(t)(1+\alpha_1s(t)
 +\alpha_2i(t-\tau))} \\
&\quad -\frac{i(t-\tau)}{i^*}+\ln\big(\frac{i(t-\tau)}{i(t)}\big)\Big]
\\
&= -\frac{e^{-\mu \tau}\mu(s(t)-s^*)^2(1+\alpha_2i^*)}{(1+\alpha_1s^*+\alpha_2i^*)}\\
&\quad +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}
\Big[1-\frac{s^*(1+\alpha_1s(t)+\alpha_2i^*)}{s(t)(1+\alpha_1s^*+\alpha_2i^*)}
+\ln\Big(\frac{s^*(1+\alpha_1s(t)+\alpha_2i^*)}{s(t)(1+\alpha_1s^*+\alpha_2i^*)}\Big)
 \Big]\\
&\quad +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}
 \Big[1-\frac{s(t)i(t-\tau)(1+\alpha_1s^*+\alpha_2i^*)}{s^*i(t)(1+\alpha_1s(t)
 +\alpha_2i(t-\tau))}\\
&\quad +\ln\Big(\frac{s(t)i(t-\tau)(1+\alpha_1s^*+\alpha_2i^*)}{s^*i(t)
 (1+\alpha_1s(t)+\alpha_2i(t-\tau))}\Big)\Big]\\
&\quad +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}
 \Big[1-\frac{1+\alpha_1s(t)+\alpha_2 i(t-\tau)}{1+\alpha_1s(t)+\alpha_2i^*}
 +\ln\Big(\frac{1+\alpha_1s(t)+\alpha_2 i(t-\tau)}{1+\alpha_1s(t)+\alpha_2i^*}
 \Big)\Big]\\
&\quad +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}
 \Big[-1+\frac{1+\alpha_1s(t)+\alpha_2 i(t-\tau)}{1+\alpha_1s(t)+\alpha_2i^*}\\
&\quad +  \frac{i(t-\tau)(1+\alpha_1s(t)+\alpha_2i^*)}{i^*(1+\alpha_1s(t)+\alpha_2i(t-\tau))}
 -\frac{i(t-\tau)}{i^*}\Big]
 +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}
 \Big[\ln\Big( \frac{i(t-\tau)}{i(t)}\Big)\\
&\quad -\ln\Big(\frac{s^*(1+\alpha_1s(t)+\alpha_2i^*)}{s(t)(1+\alpha_1s^*+\alpha_2i^*)}\Big)
 -\ln\Big(\frac{s(t)i(t-\tau)(1+\alpha_1s^*+\alpha_2i^*)}{s^*i(t)(1+\alpha_1s(t)
 +\alpha_2i(t-\tau))}\Big) \\
&\quad -\ln\Big(\frac{1+\alpha_1s(t)+\alpha_2i(t-\tau)}{1+\alpha_1s(t)
 +\alpha_2i^*}\Big)\Big].
\end{align*}
Therefore,
\begin{align*}
&\frac{dV(t)}{dt}\\
&= -\frac{e^{-\mu \tau}\mu(s(t)-s^*)^2(1+\alpha_2i^*)}{(1+\alpha_1s^*+\alpha_2i^*)}
 +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}\\
&\quad\times \Big( 1-\frac{s^*(1+\alpha_1s(t)+\alpha_2 i^*)}{s(t)(1+\alpha_1s^*
 +\alpha_2 i^*)}\Big)
 \Big(1-\frac{s(t)i(t-\tau)(1+\alpha_1s^*+\alpha_2 i^*)}{s^*i^*(1+\alpha_1s(t)
 +\alpha_2 i(t-\tau))} \Big)
\\
&\quad +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}
 \big(1-\frac{i^*}{i(t)}\big)\Big(\frac{s(t)i(t-\tau)(1+\alpha_1s^*
 +\alpha_2i^*)}{s^*i^*(1+\alpha_1s(t)+\alpha_2i(t-\tau))}-\frac{i(t)}{i^*}\Big)
\\
&\quad +\frac{e^{-\mu \tau}\beta s^*i^*}{1+\alpha_1s^*+\alpha_2i^*}
 \Big[ -\frac{i(t-\tau)}{i^*}+\frac{i(t)}{i^*}
 +\ln\big(\frac{i(t-\tau)}{i(t)}\big)\Big]\\
&= -\frac{e^{-\mu \tau}\mu(s(t)-s^*)^2(1+\alpha_2i^*)}{(1+\alpha_1s^*+\alpha_2i^*)}
  +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}
 \Big[2-\frac{s^*(1+\alpha_1s(t)+\alpha_2i^*)}{s(t)(1+\alpha_1s^*+\alpha_2i^*)}\\
&\quad  +  \frac{i(t-\tau)(1+\alpha_1s(t)+\alpha_2i^*)}{i^*(1+\alpha_1s(t)
 +\alpha_2i(t-\tau))}-\frac{s(t)i(t-\tau)(1+\alpha_1s^*+\alpha_2i^*)}{s^*i(t)
 (1+\alpha_1s(t)+\alpha_2i(t-\tau))} \\
&\quad -\frac{i(t-\tau)}{i^*}+\ln\big(\frac{i(t-\tau)}{i(t)}\big)\Big]\\
&= -\frac{e^{-\mu \tau}\mu(s(t)-s^*)^2(1+\alpha_2i^*)}{(1+\alpha_1s^*+\alpha_2i^*)}\\
&\quad +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}
 \Big[1-\frac{s^*(1+\alpha_1s(t)+\alpha_2i^*)}{s(t)(1+\alpha_1s^*+\alpha_2i^*)}
 +\ln\Big(\frac{s^*(1+\alpha_1s(t)+\alpha_2i^*)}{s(t)(1+\alpha_1s^*+\alpha_2i^*)}
 \Big)\Big]\\
&\quad +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}
 \Big[1-\frac{s(t)i(t-\tau)(1+\alpha_1s^*+\alpha_2i^*)}{s^*i(t)(1+\alpha_1s(t)
 +\alpha_2i(t-\tau))}\\
&\quad +\ln\Big(\frac{s(t)i(t-\tau)(1+\alpha_1s^*+\alpha_2i^*)}{s^*i(t)
 (1+\alpha_1s(t)+\alpha_2i(t-\tau))}\Big)\Big]
\\
&\quad +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}
 \Big[1-\frac{1+\alpha_1s(t)+\alpha_2 i(t-\tau)}{1+\alpha_1s(t)+\alpha_2i^*}
 +\ln\Big(\frac{1+\alpha_1s(t)+\alpha_2 i(t-\tau)}{1+\alpha_1s(t)+\alpha_2i^*}\Big)\Big]
\\
&\quad +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}
 \Big[-1+\frac{1+\alpha_1s(t)+\alpha_2 i(t-\tau)}{1+\alpha_1s(t)+\alpha_2i^*}+
\frac{i(t-\tau)(1+\alpha_1s(t)+\alpha_2i^*)}{i^*(1+\alpha_1s(t)+\alpha_2i(t-\tau))}\\
&\quad -\frac{i(t-\tau)}{i^*}\Big]
 +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}
 \Big[\ln\big( \frac{i(t-\tau)}{i(t)}\big)
 -\ln\Big(\frac{s^*(1+\alpha_1s(t)+\alpha_2i^*)}{s(t)(1+\alpha_1s^*+\alpha_2i^*)}\Big)
\\
&\quad -\ln\Big(\frac{s(t)i(t-\tau)(1+\alpha_1s^*+\alpha_2i^*)}{s^*i(t)(1+\alpha_1s(t)
 +\alpha_2i(t-\tau))}\Big)
 -\ln\Big(\frac{1+\alpha_1s(t)+\alpha_2i(t-\tau)}{1+\alpha_1s(t)
 +\alpha_2i^*}\Big)\Big].
\end{align*}
Since
\begin{align*}
\ln\big( \frac{i(t-\tau)}{i(t)}\big)
&=\ln\Big(\frac{s^*(1+\alpha_1s(t)+\alpha_2i^*)}{s(t)(1+\alpha_1s^*+\alpha_2i^*)}\Big)
+\ln\Big(\frac{s(t)i(t-\tau)(1+\alpha_1s^*+\alpha_2i^*)}{s^*i(t)(1+\alpha_1s(t)
+\alpha_2i(t-\tau))}\Big) \\
&\quad +\ln\Big(\frac{1+\alpha_1s(t)+\alpha_2i(t-\tau)}{1+\alpha_1s(t)+\alpha_2i^*}\Big),
\end{align*}
and
\begin{align*}
&-1+\frac{1+\alpha_1s(t)+\alpha_2 i(t-\tau)}{1+\alpha_1s(t)+\alpha_2i^*}+
\frac{i(t-\tau)(1+\alpha_1s(t)+\alpha_2i^*)}{i^*(1+\alpha_1s(t)+\alpha_2i(t-\tau))}
-\frac{i(t-\tau)}{i^*}\\
&=-\frac{\alpha_2(1+\alpha_1s(t))(i(t-\tau)-i^*)^2}{i^*(1+\alpha_1s(t)
+\alpha_2i^*)(1+\alpha_1s(t)+\alpha_2i(t-\tau))},
\end{align*}
we have
\begin{equation}\label{sge4}
\begin{aligned}
&\frac{dV(t)}{dt}\\
&= -\frac{e^{-\mu \tau}\mu(s(t)-s^*)^2(1+\alpha_2i^*)}{(1+\alpha_1s^*+\alpha_2i^*)}
+\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}\\
&\quad\times \Big[1-\frac{s^*(1+\alpha_1s(t)+\alpha_2i^*)}{s(t)(1+\alpha_1s^*+\alpha_2i^*)}
+\ln\Big(\frac{s^*(1+\alpha_1s(t)+\alpha_2i^*)}{s(t)(1+\alpha_1s^*+\alpha_2i^*)}
 \Big)\Big]
\\
&\quad +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}
 \Big[1-\frac{s(t)i(t-\tau)(1+\alpha_1s^*+\alpha_2i^*)}{s^*i(t)(1+\alpha_1s(t)
 +\alpha_2i(t-\tau))}
\\
&\quad + \ln\Big(\frac{s(t)i(t-\tau)(1+\alpha_1s^*+\alpha_2i^*)}{s^*i(t)(1+\alpha_1s(t)
 +\alpha_2i(t-\tau))}\Big)\Big]\\
&\quad +\frac{e^{-\mu \tau}\beta s^* i^*}{1+\alpha_1s^*+\alpha_2i^*}
 \Big[1-\frac{1+\alpha_1s(t)+\alpha_2 i(t-\tau)}{1+\alpha_1s(t)+\alpha_2i^*}\\
&\quad +\ln\Big(\frac{1+\alpha_1s(t)+\alpha_2 i(t-\tau)}{1+\alpha_1s(t)+\alpha_2i^*}
 \Big)\Big]\\
&\quad -\frac{e^{-\mu \tau}\beta s^* i^* \alpha_2(1+\alpha_1s(t))
 (i(t-\tau)-i^*)^2}{i^*(1+\alpha_1s^*+\alpha_2i^*)(1+\alpha_1s(t)
 +\alpha_2i^*)(1+\alpha_1s(t)+\alpha_2i(t-\tau))}.
\end{aligned}
\end{equation}
It is easy to see that the first and the last terms in \eqref{sge4} are non-positive and
since the function $g(x) = 1 - x + \ln(x)$ is always non-positive for any $x > 0$,
and $g(x) = 0$ if and only if $x = 1$, then the second term, the third term and the
fourth term in \eqref{sge4} are non-positive. Therefore, $\frac{dV(t)}{dt}\leq 0$
for all $t\geq0$, where the equality holds only at the equilibrium point
$ (s^{*},i^{*})$.
Hence, the functional $V$ satisfies all the conditions of Theorem \ref{thm5.2}.
 This proves that $P_1^*$ is globally asymptotically stable.
\end{proof}

\section{Global Stability analysis of SEIR model}\label{s3}

In this section, we discuss the global stability of a disease-free
equilibrium and an endemic equilibrium of system \eqref{a2}.
Since $\frac{d}{dt}(S+E+I)\leq A-\mu (S+E+I)$, we have that $\limsup (S+E+I) \leq \frac{A}{\mu}$. Hence
we discuss system \eqref{a2} in the closed set:
$$
\Omega=:\big\{(S,E,I)\in (\mathbb{R}^+)^3| S+E+I\leq \frac{A}{\mu} \big\} .
$$
It is easy to show that $\Omega$  is positively invariant with respect to
 system \eqref{a2}, which
 always has a disease-free equilibrium $P_2=(\frac{A}{\mu},0,0)$.
 Further, if
$$
R_{02}:=\frac{A\beta
\sigma}{(\sigma+\mu)(\mu_1+\gamma)(\alpha_1A+\mu)}>1,
$$
then \eqref{a2} admits a unique endemic equilibrium
$P^*_2=(S^*,I^*,E^*)$, with
\begin{gather*}
S^*=\frac{A[(\sigma+\mu)(\mu_1+\gamma)+\alpha_2\sigma A]}{\alpha_2\sigma \mu A
+(\sigma+\mu)(\mu_1+\gamma)[(\alpha_1 A+\mu)(R_{02}-1)+\mu]}, \quad
E^*=\frac{\mu_1+\gamma}{\sigma}I^*, \\
 I^*=\frac{\sigma A(R_{02}- 1)(\alpha_1
A+\mu)}{\alpha_2\sigma \mu A+(\sigma+\mu)(\mu_1+\gamma)[(\alpha_1
A+\mu)(R_{02}-1)+\mu]}.
\end{gather*}
Now we consider the global asymptotic stability of the disease-free equilibrium
$P_2$ and the endemic equilibrium $P_2^{*}$ by Lyapunov functionals, respectively.

\begin{proposition} \label{prop3.1}
If $R_{02} \leq 1$, then the disease-free equilibrium $P_2$ is globally
asymptotically stable.
\end{proposition}

\begin{proof}
Define a Lyapunov functional
$V(S,E,I)=V_1(S,E,I)+V_2(S,E,I)$
with
$$
V_1(t)={\int_{\frac{A}{\mu}}^{S(t)}\Big(1-\frac{A(1+\alpha_1u))}{(\mu+\alpha_1 A)u}\Big)du}
$$
and
$$
V_2(t)={E+\frac{\sigma+\mu}{\sigma}I.}
$$
We will show that $\frac{dV(t)}{dt}\leq 0$ for all $t\geq 0$.
We have
\begin{align*}
\frac{dV_1(t)}{dt}
&= \Big(1-\frac{A(1+\alpha_1S(t))}{(\mu+\alpha_1A)S(t)}\Big)
 \Big(A-\mu S(t)-\frac{\beta S(t)I(t)}{1+\alpha_1 S(t)+\alpha_2 I(t)}\Big)\\
&= \Big(1-\frac{A(1+\alpha_1S(t))}{(\mu+\alpha_1A)S(t)}\Big)\left(A-\mu S(t)\right)\\
&\quad -\Big(1-\frac{A(1+\alpha_1S(t))}{(\mu+\alpha_1A)S(t)}\Big)
\Big(\frac{\beta S(t)I(t)}{1+\alpha_1 S(t)+\alpha_2 I(t)}\Big),
\end{align*}
and
$$
\frac{dV_2(t)}{dt}=\dot{E}+\frac{\sigma+\mu}{\sigma}\dot{I}
=\frac{\beta S(t)I(t)}{1+\alpha_1 S(t)+\alpha_2 I(t)}
-\frac{(\sigma+\mu)(\mu_1+\gamma)}{\sigma}I(t).
$$
Therefore,
\begin{align*}
\frac{dV(t)}{dt}
&= \Big(1-\frac{A(1+\alpha_1S(t))}{(\mu+\alpha_1A)S(t)}\Big)
 \left(A-\mu S(t)\right)\\
&\quad +\Big[\frac{\beta A}{(\mu+\alpha_1 A)}
 \frac{1+ \alpha_1 S(t)}{1+\alpha_1 S(t)+\alpha_2 I(t)}
 -\frac{(\sigma+\mu)(\mu_1+\gamma)}{\sigma}\Big]I(t)\\
&= \Big(1-\frac{A(1+\alpha_1S(t))}{(\mu+\alpha_1A)S(t)}\Big)
 \left(A-\mu S(t)\right)\\
&\quad +\frac{(\sigma+\mu)(\mu_1+\gamma)}{\sigma}
 \big[R_{02}\frac{1+ \alpha_1 S(t)}{1+\alpha_1 S(t)+\alpha_2 I(t)}-1\big]I(t).
\end{align*}
Hence
$$
\frac{dV(t)}{dt}\leq-\frac{(A-\mu S(t))^2}{(\mu+\alpha_1A)S(t)}
 +\frac{(\sigma+\mu)(\mu_1+\gamma)}{\sigma}\left[R_{02}-1\right]I(t).
$$
Therefore, $R_{02}\leq 1$ ensures that $\frac{dV(t)}{dt}\leq 0$ for all $t\geq 0$,
where $\frac{dV(t)}{dt}=0$ holds if $S(t)=\frac{A}{\mu}$,
$E(t)=0$ and $I(t)=0$. Hence, it follows from system \eqref{a2} that $\{P_1\}$
is the largest invariant set in $\left\{(S,E,I)|\frac{dV(t)}{dt}=0\right\}$.
From the Lyapunov-LaSalle asymptotic stability, we obtain that $P_1$ is
globally asymptotically stable. This completes the proof.
\end{proof}

\begin{proposition} \label{prop3.2}
If $R_{02} >1$, then the disease free equilibrium $P_2^*$ is globally asymptotically
 stable.
\end{proposition}

\begin{proof}
Define a Lyapunov functional $V(t)=V_1(t)+V_2(t)$
with
$$
V_1(t)=\int_{S^*}^{S(t)}
\Big(1-\frac{S^*(1+\alpha_1 u+\alpha_2 I^*)}{u(1+\alpha_1 S^*+\alpha_2 I^*)}\Big)du
$$
and
$$
V_2(t)={E(t)-E^*+E^*\ln\big(\frac{E(t)}{E^*}\big)
+\frac{\sigma+\mu}{\sigma}\big[ I(t)-I^*+I^*\ln\big(\frac{I(t)}{I^*}\big)\big].}
$$
Using the relations
$A=\mu S^*+\frac{\beta S^*I^*}{1+\alpha_1 S^*+\alpha_2 I^*},
\frac{\beta S^*I^*}{1+\alpha_1 S^*+\alpha_2 I^*}=(\sigma+\mu)E^* $ and
$\sigma E^*=(\mu_1+\gamma)I^*$, the time derivative of the function $V_1(t)$
along the positive solution of system \eqref{a2} becomes
\begin{align*}
&\frac{dV_1(t)}{dt}\\
&= \Big(1-\frac{S^*(1+\alpha_1 S(t)+\alpha_2 I^*)}{S(t)(1+\alpha_1 S^*+\alpha_2 I^*)}\Big)
 \Big(A-\mu S(t)-\frac{\beta S(t)I(t)}{1+\alpha_1 S(t)+\alpha_2I(t)}\Big)\\
&= \Big(1-\frac{S^*(1+\alpha_1 S(t)+\alpha_2 I^*)}{S(t)(1+\alpha_1 S^*+\alpha_2 I^*)}\Big)
 \Big(\mu (S^*-S(t))+\frac{\beta S^*I^*}{1+\alpha_1 S^*+\alpha_2 I^*}\\
&\quad -\frac{\beta S(t)I(t)}{1+\alpha_1 S(t)+\alpha_2I(t)}\Big)\\
&= \Big(1-\frac{S^*(1+\alpha_1 S(t)+\alpha_2 I^*)}{S(t)(1+\alpha_1 S^*+\alpha_2 I^*)}\Big)
 (\mu (S^*-S(t)))\\
&\quad +\Big(1-\frac{S^*(1+\alpha_1 S(t)+\alpha_2 I^*)}{S(t)(1+\alpha_1 S^*+\alpha_2 I^*)}
 \Big)\Big(\frac{\beta S^*I^*}{1+\alpha_1 S^*+\alpha_2 I^*}
 -\frac{\beta S(t)I(t)}{1+\alpha_1 S(t)+\alpha_2I(t)}\Big)\\
&= \Big(1-\frac{S^*(1+\alpha_1 S(t)+\alpha_2 I^*)}{S(t)(1+\alpha_1 S^*+\alpha_2 I^*)}\Big)
 (\mu (S^*-S(t)))\\
&\quad +(\sigma+\mu)E^*\Big(1-\frac{S^*(1+\alpha_1 S(t)
 +\alpha_2 I^*)}{S(t)(1+\alpha_1 S^*+\alpha_2 I^*)}\Big)
 \Big(1-\frac{S(t)I(t)(1+\alpha_1 S^*+\alpha_2 I^*)}{S^*I^*(1+\alpha_1 S(t)
 +\alpha_2 I(t))}\Big).
\end{align*}
The time derivative of the function $V_2(t)$ becomes
\begin{align*}
\frac{dV_2(t)}{dt}
&= \big(1-\frac{E^*}{E(t)} \big)\dot{E}(t)+\frac{\sigma+\mu}{\sigma}
 \big(1-\frac{I^*}{I(t)} \big)\dot{I}(t)\\
&= \big(1-\frac{E^*}{E(t)} \big)\Big(\frac{\beta S(t)I(t)}{1+\alpha_1 S(t)
 +\alpha_2 I(t)}-(\sigma+\mu)E(t)\Big)\\
&\quad +\frac{\sigma+\mu}{\sigma}
 \big(1-\frac{I^*}{I(t)} \big)(\sigma E(t)-(\mu_1+\gamma)I(t))\\
&= (\sigma+\mu)E^*\Big[\big(1-\frac{E^*}{E(t)} \big)
 \Big( \frac{S(t)I(t)(1+\alpha_1 S^*+\alpha_2 I^*)}{S^*I^*(1+\alpha_1 S(t)
 +\alpha_2 I(t))}-\frac{E(t)}{E^*} \Big)\\
&\quad + \big(1-\frac{I^*}{I(t)} \big)
 \big(\frac{E(t)}{E^*}-\frac{I(t)}{I^*}  \big) \Big]
\end{align*}
Therefore,
\begin{align*}
&\frac{dV(t)}{dt}\\
&= \Big(1-\frac{S^*(1+\alpha_1 S(t)+\alpha_2 I^*)}{S(t)(1+\alpha_1 S^*+\alpha_2 I^*)}\Big)
 (\mu (S^*-S(t)))+(\sigma+\mu)E^*\\
&\quad\times \Big[3+\frac{I(t)(1+\alpha_1 S(t)
 +\alpha_2 I^*)}{I^*(1+\alpha_1 S(t)+\alpha_2 I(t))}
 - \frac{S^*(1+\alpha_1 S(t)+\alpha_2 I^*)}{S(t)(1+\alpha_1 S^*+\alpha_2 I^*}\\
&\quad  -\frac{E^*S(t)I(t)(1+\alpha_1 S^*+\alpha_2 I^*)}{E(t)S^*I^*(1+\alpha_1 S(t)
 +\alpha_2 I(t))}-\frac{I(t)}{I^*}-\frac{I^*E(t)}{I(t)E^*}\Big]\\
&= \left(1-\frac{S^*(1+\alpha_1 S(t)+\alpha_2 I^*)}{S(t)(1+\alpha_1 S^*+\alpha_2 I^*)}\right)
 (\mu (S^*-S(t)))
\\
&\quad +(\sigma+\mu)E^*\Big[\Big(\frac{1+\alpha_1 S(t)+\alpha_2 I(t)}{1+\alpha_1 S(t)
 +\alpha_2 I^*}-\frac{I(t)}{I^*}-1+\frac{I(t)(1+\alpha_1 S(t)
 +\alpha_2 I^*)}{I^*(1+\alpha_1 S(t)+\alpha_2 I(t))}\Big)\\
&\quad +\Big(4-\frac{1+\alpha_1 S(t)+\alpha_2 I(t)}{1+\alpha_1 S(t)+\alpha_2 I^*}
 -\frac{S^*(1+\alpha_1 S(t)+\alpha_2 I^*)}{S(t)(1+\alpha_1 S^*+\alpha_2 I^*)}\\
&\quad -\frac{E^*S(t)I(t)(1+\alpha_1 S^*+\alpha_2 I^*)}{E(t)S^*I^*(1+\alpha_1 S(t)
 +\alpha_2 I(t))}-\frac{I^*E(t)}{I(t)E^*}\Big)\Big]
\\
&= \Big(1-\frac{S^*(1+\alpha_1 S(t)+\alpha_2 I^*)}{S(t)(1+\alpha_1 S^*+\alpha_2 I^*)}\Big)
 (\mu (S^*-S(t)))
\\
&\quad +(\sigma+\mu)E^*\Big[\Big(\frac{1+\alpha_1 S(t)+\alpha_2 I(t)}{1+\alpha_1 S(t)
 +\alpha_2 I^*}-1\Big)
\Big(1-\frac{I(t)(1+\alpha_1 S(t)+\alpha_2 I^*)}{I^*(1+\alpha_1 S(t)
 +\alpha_2 I(t))}\Big) 
\\
&\quad +\Big(4-\frac{1+\alpha_1 S(t)+\alpha_2 I(t)}{1+\alpha_1 S(t)+\alpha_2 I^*}
 -\frac{S^*(1+\alpha_1 S(t)+\alpha_2 I^*)}{S(t)(1+\alpha_1 S^*+\alpha_2 I^*)}\\
&\quad -\frac{E^*S(t)I(t)(1+\alpha_1 S^*+\alpha_2 I^*)}{E(t)S^*I^*(1+\alpha_1 S(t)
+\alpha_2 I(t))}-\frac{I^*E(t)}{I(t)E^*}\Big)\Big]
\\
&= -\frac{(1+\alpha_2 I^*)(S(t)-S^*)^2}{S(t)(1+\alpha_1 S^*+\alpha_2 I^*)}
 -\frac{\alpha_2(1+\alpha_1S(t))(I(t)-I^*)^2(\sigma+\mu)E^*}{I^*(1+\alpha_1 S(t)
 +\alpha_2 I(t))(1+\alpha_1 S(t)+\alpha_2 I^*)}\\
&\quad +(\sigma+\mu)E^*\Big(4-\frac{1+\alpha_1 S(t)+\alpha_2 I(t)}{1
 +\alpha_1 S(t)+\alpha_2 I^*}-\frac{S^*(1+\alpha_1 S(t)+\alpha_2 I^*)}{S(t)(1
 +\alpha_1 S^*+\alpha_2 I^*)}\\
&\quad -\frac{E^*S(t)I(t)(1+\alpha_1 S^*
 +\alpha_2 I^*)}{E(t)S^*I^*(1+\alpha_1 S(t)+\alpha_2 I(t))}-\frac{I^*E(t)}{I(t)E^*}\Big).
\end{align*} %\label{gs2}
Here $\frac{-(1+\alpha_2 I^*)(S-S^*)^2}{S(1+\alpha_1 S^*+\alpha_2 I^*)}\leq 0$ and
$\frac{-\alpha_2(1+\alpha_1S)(I-I^*)^2(\sigma+\mu)E^*}{I^*(1+\alpha_1 S+\alpha_2 I)
(1+\alpha_1 S+\alpha_2 I^*)}\leq 0$ for all $t\geq0$.
Since the arithmetic mean is greater than or equal to the geometric mean,
$$
4-\frac{1+\alpha_1 S+\alpha_2 I}{1+\alpha_1 S+\alpha_2 I^*}
-\frac{S^*(1+\alpha_1 S+\alpha_2 I^*)}{S(1+\alpha_1 S^*+\alpha_2 I^*)}
-\frac{E^*SI(1+\alpha_1 S^*+\alpha_2 I^*)}{ES^*I^*(1+\alpha_1 S+\alpha_2 I)}
-\frac{I^*E}{IE^*}\leq0
$$
for all $t\geq0$.
Therefore, $\frac{dV(t)}{dt}\leq0$ for all $t\geq0$, where the equality holds
only at the equilibrium point $(S,E,I) = (S^{*},E^{*},I^{*})$.
Thus  $\{P_2^*\}$ is the largest invariant set in
$\left\{(S,E,I)|\frac{dV(t)}{dt}=0\right\}$. Consequently, we obtain,
 by the Lyapunov-LaSalle asymptotic stability theorem, that $P_2^*$ is globally
asymptotically stable. This completes the proof.
\end{proof}

\section{Numerical Simulations and Concluding Remarks}\label{s4}

In this section, we give a numerical simulation supporting the
theoretical analysis given in section \ref{s2} and \ref{s3}.
We take the parameters of the system \eqref{a1} as follows:
\begin{gather*}
A=0.04,\quad \alpha_1=0.01,\quad
\alpha_2=0.01, \quad \mu=0.05, \\
\gamma=0.05,\quad \alpha=0.09, \quad  \beta=2.5, \quad \tau=100.
\end{gather*}
Then $R_{01}=0.07$. Therefore, by Proposition \ref{prop2.1}, the free-disease
equilibrium $P_1$ is globally asymptotically  stable; see Figure \ref{fig1}.


\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{fig1}
\end{center}
 \caption{Solutions ($S,I$) of the delay SIR model \eqref{a1}
are globally asymptotically  stable and converge to the free-disease
 equilibrium $P_1$} \label{fig1}
\end{figure}


Now we take the parameters of the system \eqref{a2} as follows:
\begin{gather*}
A=0.04,\quad \alpha_1=0.01,\quad \alpha_2=0.01, \quad \mu=0.05, \\
\gamma=0.05,\quad \alpha=0.09, \quad \beta=2.5, \quad \sigma=0.01.
\end{gather*}
Then $R_{02}=1.74$. Therefore, by Proposition \ref{prop3.2}, the endemic equilibrium
$P_2^*$ is globally asymptotically  stable; see Figure \ref{fig2}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{fig2}
\end{center}
 \caption{ Solutions ($S,E,I$) of the SEIR model \eqref{a2}
are globally asymptotically  stable and converge to the endemic equilibrium $P_2^*$}
\label{fig2}
\end{figure}

In epidemiological research literatures, a latent or incubation period can be medelled by
incorporating it as a delay effect (delayed SIR models) \cite{Cooke}, or by
introducing an exposed class (SEIR models) \cite{Hethcote}. In this paper we consider
the global stability for a delayed SIR model with saturated incidence rate
(system \eqref{a1})
 and for its corresponding SEIR model (system \eqref{a2}).

In the section 2, by modifying Lyapunov functional techniques in Huang and al \cite{Huang},
 we proved that if $R_{01}\leq1$, the disease-free equilibrium is globally asymptotically
stable, and this is the only equilibrium. On the contrary, if $R_{01} > 1$,
 then an endemic equilibrium appears which is globally asymptotically stable.
In the section 3, by proposing Lyapunov functional, we showed that if $R_{02}\leq1$,
the disease-free equilibrium is globally asymptotically stable, and this is the
only equilibrium. On the contrary, if  $R_{02} > 1$, then an endemic equilibrium appears
which is globally asymptotically stable.

Finally, numerical simulations are given to support the theoretical analysis and to
show that the delayed SIR model \eqref{a1} and its corresponding SEIR model \eqref{a2}
can generate different global asymptotic behavior, for example if the incubation
period $\tau = 100$ (thus $ \sigma =\frac{1}{\tau}= 0.01$), the system \eqref{a1} has
 only a disease free equilibrium $P_1$ which is globally asymptotically stable but
the system \eqref{a2} has a disease free equilibrium $P_2$ which is unstable and
an endemic equilibrium $P_2^{*}$
which is globally asymptotically stable (see Figure \ref{fig1} and Figure \ref{fig2}).
 In this case we ask the following question:
Which model can be adopted for modeling the incubation period in the case of human
immunodeficiency virus?

\section{Appendix: The Lyapunov-LaSalle theorem}

In the following, we present  the method of  Lyapunov functionals in the
context of a delay differential equations,
\begin{equation} \label{kad}
\frac{dx}{dt}=f(x_t),
\end{equation}
where  $f : C  \to\mathbb{R}^{n}$ is  completely  continuous  and  solutions
of  \eqref{kad} are unique  and  continuously  dependent  on  the  initial  data.
 We  denote  by $x(\phi)$ the  solution  of  \eqref{kad} through  $(0,\phi)$.
 For  a  continuous  functional $V : C  \to\mathbb{R}$, we  define
$$
\dot{V}=\limsup_{h\to0^{+}}\frac{1}{h}[V(x_h(\phi)-V(\phi)],
$$
the  derivative  of  $V$  along  a  solution  of  \eqref{kad}. To  state  the
Lyapunov-LaSalle  type  theorem  for  \eqref{kad},  we  need  the
following definition.

\begin{definition}[{\cite[p. 30]{kuang}}]   \rm
We say $V : C  \to\mathbb{R}$ is a  Lyapunov functional on a set $G$
in $C$ for \eqref{kad} if it  is continuous on $\overline{G}$
(the closure of  $G$) and $\dot{V}\leq 0$ on $G$. We also define
$E  = \{\phi \in \overline{G}   :  \dot{V}(\phi)=0\}$, and
$M$ is the largest  set  in $E $ which  is  invariant with  respect  to \eqref{kad}.
\end{definition}

The  following  result  is  the  Lyapunov-LaSalle  type  theorem  for  \eqref{kad}.

\begin{theorem}[{\cite[p. 30]{kuang}}] \label{thm5.2}
If $V$ is a Lyapunov functional on $G$ and $x_t(\phi)$  is a bounded
solution  of  \eqref{kad}  that stays  in $G$, then $\omega$-limit set
 $\omega(\phi)\subset M;$  that  is,  $x_t(\phi) \to M $ as $t \to+\infty$.
\end{theorem}


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\end{document}