\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 94, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2013/94\hfil Global stability]
{Global stability of a delay differential equation of hepatitis
B virus infection with immune response}
\author[J. Wang, X. Tian \hfil EJDE-2013/94\hfilneg]
{Jinliang Wang, Xinxin Tian}
\address{Jinliang Wang \newline
School of Mathematical Science, Heilongjiang University,
Harbin 150080, China}
\email{jinliangwang@yahoo.cn}
\address{Xinxin Tian \newline
School of Mathematical Science, Heilongjiang University,
Harbin 150080, China}
\email{xinxintian@yahoo.cn}
\thanks{Submitted March 4, 2013. Published April 11, 2013.}
\subjclass[2000]{34K20, 92D25}
\keywords{HBV infection model; delay; CTLs; global stability}
\begin{abstract}
The global stability for a delayed HBV infection model with CTL immune
response is investigated. We show that the global dynamics is
determined by two sharp thresholds, basic reproduction number $\Re_0$
and CTL immune-response reproduction number $\Re_1$. When $\Re_0 \leq 1$,
the infection-free equilibrium is globally asymptotically stable, which
means that the viruses are cleared and immune is not active;
when $\Re_1 \leq 1 < \Re_0$, the CTL-inactivated infection equilibrium
exists and is globally asymptotically stable, which means that CTLs immune
response would not be activated and viral infection becomes chronic;
and when $\Re_1 > 1$, the CTL-activated infection equilibrium exists and
is globally asymptotically stable, in this case the infection causes
a persistent CTLs immune response. Our model is formulated by incorporating
a Cytotoxic T lymphocytes (CTLs) immune response to recent work
[Gourley, Kuang, Nagy, J. Bio. Dyn., 2(2008), 140-153]
to model the role in antiviral by attacking virus infected cells.
Our analysis provides a quantitative understandings of HBV replication
dynamics in vivo and has implications for the optimal timing of drug
treatment and immunotherapy in chronic HBV infection.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks
\section{Introduction}
Approximately more than 350 million people worldwide live with chronic
hepatitis B virus (HBV) infection\cite{WHO}, and 25-40 percent of these
chronic infection carrier will at risk of developing chronic liver disease,
cirrhosis and hepatocellular carcinoma \cite{Nowak2}.
HBV infection therefore represents a significant global public health problem.
A basic within-host viral infection model introduced by Nowak et al
\cite{Nowak1, Nowak2} and Perelson et al \cite{Perelson3} have been
widely used in the studies of HBV and HIV infection dynamics and its
treatment with the reverse transcriptase inhibitor lamivudine.
After then several mathematical models have been modified to study of
anti-HBV infection treatment and its dynamics. Most of these
models focus on cell-free viral spread in a compartment such as the
bloodstream, see, for example, In \cite{Song}, saturated mass action
incidence rates $\beta xv/(1 +\alpha v)$ was proposed under the assumption
that a less than linear response in $v$ could occur due to saturation at
high virus concentration.
Min et al \cite{Min} and Zheng et al \cite{Zheng} employed a standard
incidence function instead of the mass action incidence to
describe the hepatitis B virus infection as follows
\begin{equation}\label{M1}
\begin{gathered}
\dot{x}(t)= \lambda-dx(t)-\frac{\beta x(t)v(t)}{x(t)+y(t)}, \\
\dot{y}(t)= \frac{\beta x(t)v(t)}{x(t)+y(t)}-ay(t), \\
\dot{v}(t)= ky(t)-\mu y(t),
\end{gathered}
\end{equation}
where $x$, $y$ and $v$ are numbers of uninfected (susceptible) liver cells,
infected liver cells and free virions, respectively. Uninfected liver
cells are assumed to be produced at a constant rate, $\lambda$, to
maintain tissue homeostasis in the face of hepatocyte turnover,
described by the linear term $dx$, where $d$ is the per-capita death rate.
Infected liver cells are killed by immune cells at rate $ay$.
Free virions are generated from infected cells at the rate of $ky$
and decay by lymphatic and other mechanisms at the rate of $\mu v$,
where $k$ is the so-called ``burst'' constant.
Upon infection with HIV-1, there is a short intracellular ``eclipse phase''
(often referred to as ``latency'' in the literature),
during which the cell is infected but has not yet begun producing virus.
There are two methods to model this eclipse phase, by a
time delay or by an explicit class of latently infected cells.
Recently, Gourley et al \cite{Gourley} proposed the following model
(As an extension of this model \eqref{M1}) under some biologically
motivated modifications:
\begin{equation}\label{M2}
\begin{gathered}
\dot{x}(t)= \lambda-dx(t)-\frac{\beta kx(t)y(t)}{\mu(x(t)+y(t))}, \\
\dot{e}(t)= -de(t)+\frac{\beta kx(t)y(t)}{\mu(x(t)+y(t))}
-\frac{\beta ke^{-d\tau}x(t-\tau)y(t-\tau)}{\mu(x(t-\tau)+y(t-\tau))}, \\
\dot{y}(t)= \frac{\beta e^{-d\tau} kx(t-\tau)y(t-\tau)}
{\mu(x(t-\tau)+y(t-\tau))}-ay(t),
\end{gathered}
\end{equation}
where $e(t)$ represents the number of exposed cells (i.e., cells that have
acquired the virus but are not yet producing new virions). Exposed cells begin
shedding virions after $\tau$ units of time, representing the time required
to construct, transcribe and translate the episomal viral genome,
construct and then release mature virions. Other parameters
are the same as in the basic virus model \eqref{M1}.
Model \eqref{M2} is obtained from the following three observations:
(1) A typical chronically infected HBV patient has a total serum load
of about $2\times 10^{11}$ to $3\times 10^{12}$
virions \cite{Nowak1}. The average human liver has about an equal number
of cells (assuming a liver mass of about 1.5 kg). These large numbers
suggest that a more plausible HBV model should
employ a standard incidence function, instead of the mass action incidence.
On the other hand, the time delay associated with virus production is on
the order of a day or two \cite{Murray}, much shorter than the life
expectancy of a typical hepatocyte, which is 6-12 months or more
\cite{Seeger}. This makes $e$ much smaller than $x$ and $y$.
Hence, $e$ can be omitted from the denominators of the infection term.
(2) The HBV incubation period, which varies from 45 to 180 days, and
the delay in viral shedding mentioned previously both suggest that viral
production delay may significantly impact infection dynamics and, hence,
should be explicitly modeled.
(3) Variable $v$ (virus particles) evolves on much faster time scale
than the liver cells do, so a quasi-steady state assumption can be
applied to $v$; i.e., to a good approximation, $v = ky/\mu$.
In fact, it has been reported (Dimitrov et al. \cite{Dimitrov}
and Sato et al. \cite{Sato}) that cell-to-cell spread of virus is favored over
infections with cell-free virus inocula. For example, HTLV-I infection in
vivo is achieved through cell-to-cell contact among healthy and infected CD4+
T cells \cite{Bangham}. It is evidently that cell-to-cell infection is the
predominant route of viral spread since viral replication in a system with
rapid cell turnover kinetics depends on cell-to-cell
transfer of virus(see e.g. Gummuluru et al. \cite{Gummuluru},
Haase et al \cite{Haase1, Haase2}, Philips et al \cite{Philips}).
Then the above HBV infection model cab be termed as cell to cell infection model.
Note that in above simpler model \eqref{M2}, the $x$ and $y$ equations
do not involve variable $e$ and form a closed subsystem
of two equations. Guo and Cai \cite{GuoCai} resolved the global stability
of infection equalibrium of model \eqref{M2}, without other additional
conditions, which is left as an open problem in \cite{Gourley}.
They showed that the infected equilibrium of system \eqref{M2} is always
globally asymptotically stable as long as it exists by constructing
suitable Lyapunov functional and LaSalle invariance principle.
In most viral infections,
cytotoxic T lymphocyte cells (CTLs), which attack infected cells,
and antibody cells, which attack viruses, play a key role in antiviral
defense. Chronic HBV infection is often the result of
exposure early in life, leading to viral persistence in the
absence of strong antibody or cellular immune responses \cite{Ferrari}.
Therapy of HBV carriers can aim to either inhibit viral
replication or enhance immunological responses against the
virus, or both \cite{Regenstein}. It is currently widely accepted that HBV
infection is non-cytopathic. Infected hepatocytes are
killed not by the virus but by HBV-specific cytotoxic T lymphocytes
(CTLs) \cite{Guidotti, Murray}. This mortality
is somehow amplified by inflammation responses within the liver, but CTLs
appear to be the major mediator of hepatitis B pathogenesis \cite{Iannacone}.
Therefore, one of the dynamics of viral infection model with CTLs response
have recently drawn much attention of researchers in the related areas and
the interaction between infected cells and CTLs response in vivo has been
studied by ordinary differential equations (ODEs) or delay differential
equations (DDEs) (see e.g.\cite{Arnaout, Culshaw, WWL}).
In this article, letting $z(t)$ be the density of CTLs, we propose
the model
\begin{equation}\label{M3}
\begin{gathered}
\dot{x}(t)= \lambda-d_1x(t)-\frac{\beta x(t)y(t)}{x(t)+y(t)}, \\
\dot{y}(t)= \frac{\beta e^{-d_1\tau} x(t-\tau)y(t-\tau)}{x(t-\tau)
+y(t-\tau)}-d_2y(t)-ay(t)z(t), \\
\dot{z}(t)= py(t)z(t)-d_3z(t),
\end{gathered}
\end{equation}
where the infected cells $y(t)$ are removed at a rate $ayz$ by the CTL immune
response and the virus-specific CTL cells proliferate at a rate $pyz$ by
contact with the infected cells, and die at a rate $d_3z$. The aim of
the present paper is to carry out a complete mathematical analysis of
model \eqref{M3}, we will show that the global properties of model \eqref{M3}
for $\Re_1\leq 1<\Re_0$ and $\Re_1>1$ without any further conditions on the
parameters. More precisely, we show that if $\Re_1\leq 1<\Re_0$,
CTL-inactivated infection equilibrium $E_1$ is globally asymptotically stable;
if $\Re_1>1$, CTL-activated infection equilibrium $E_2$ is globally
asymptotically stable. The global stabilities of these models are established
by constructing Lyapunov functionals and Lyapunov-LaSalle invariance
principle (see e.g. \cite{Haddock}). Similar methods and techniques had been
engaged by motivated by the works by Huang et al \cite{Huang},
Korobeinikov \cite{Korobeinikov}, McCluskey \cite{McCluskey} and
Wang et al \cite{Wang1, Wang2}.
The organization of this paper is as follows.
In Section 2, we give the preliminaries of model \eqref{M3} including
basic reproduction number, CTL immune-response reproduction number
and equilibria. In Section 3, The global stability results is
proved by Lyapunov functionals. A
brief discussion is given in Section 4 to conclude this work.
\section{Preliminaries}
We denote by $\mathcal{C}$ the Banach space of continuous real-valued
functions $\mathcal{C} = \mathcal{C}([-\tau, 0], \mathbb{R}^3)$ with the
sup-norm
\begin{equation}
\|\varphi\|= \max\Big\{\sup_{-\tau\leq\theta\leq 0}|\varphi_1(\theta)|,
\sup_{-\tau\leq\theta\leq 0}|\varphi_2(\theta)|,
\sup_{-\tau\leq\theta\leq 0}|\varphi_3(\theta)|\Big\}
\end{equation}
for $\varphi=(\varphi_1, \varphi_2, \varphi_3)\in \mathcal{C}$.
Further, the nonnegative cone of $\mathcal{C}$ is defined as
$\mathcal{C}^+ = \mathcal{C}([-\tau, 0], \mathbb{R}^3_+)$.
The initial conditions of system \eqref{M3} at $t=0$ are given as
$x(\theta) =\varphi_1(\theta)$,
$y(\theta) = \varphi_2(\theta)$,
$z(\theta) = \varphi_3(\theta)$, $\theta\in [-\tau, 0]$.
where
\begin{equation}\label{Ic}
\varphi=(\varphi_1, \varphi_2, \varphi_3)\in \mathcal{C}^+, \quad \varphi(0)>0.
\end{equation}
The following theorem establishes the positivity and boundedness
of solutions for system \eqref{M3} with initial conditions \eqref{Ic}.
\begin{theorem}
Under the preceding initial conditions \eqref{Ic}, then $x(t), y(t)$ and
$z(t)$ are all nonnegative and bounded for all $t$ at which the solution exists.
\end{theorem}
\begin{proof}
By the existence and uniqueness theorem \cite[Theorem 2.1]{Kuang}
of delay differential equations, there exists a $t_0 > 0$ such that
there exists a solution $(x(t), y(t), z(t))$ of system
\eqref{M3} for $0 0$ be the first time such
that $x(t_1) = 0$. If $x(t)$ ere to lose its non-negativity,
there would have to be $x' (t_1)\leq 0$, by the first
equation of system \eqref{M1}, this is clearly impossible given the
equation for $x(t)$ in system \eqref{M3}.
It follows that $x(t) > 0$ for $t > 0$ as long as $x(t)$ exists.
By the second equation of system \eqref{M3}, we have
\begin{align*}\label{E11}
y(t)&=\ y(0)\exp\Big(-d_2t-a\int_0^tz(\theta)d\theta\Big)\\
&\quad +\int_0^t\frac{\beta e^{-d\tau}x(\theta-\tau)y(\theta-\tau)}
{x(\theta-\tau)+y(\theta-\tau)}e^{d_2(\theta-t)}
\exp\Big(-a\int_{\theta}^tz(\sigma)d\sigma\Big)d\theta.
\end{align*}
It follows that $y(t) > 0$ for $t > 0$.
From the third equation of system \eqref{M3}, we have
$z(t)=z(0)\exp[(py-d_3)t]$.
This shows that $z(t)\geq 0$ for $ 0 \leq t < t_1$.
Next we show that positive solutions of \eqref{M3} are
ultimately uniformly bounded for $t \geq 0$. Let
$$
G(t)=e^{-d_1\tau}x(t)+y(t+\tau)+\frac{a}{p}z(t).
$$
Adding all the equations of \eqref{M3} we obtain
\begin{align*}
G'(t)&= \lambda e^{-d_1\tau}-d_1e^{-d_1\tau}x(t)
-d_2y(t+\tau)-\frac{d_3}{a}z(t) \\
&\leq \lambda e^{-d_1\tau}-dG(t),
\end{align*}
where $d=\min\{d_1, d_2, d_3\}$. Then $G(t)\leq M_1$ for some $M_1 > 0$
for sufficiently large $t$. For example, we can take as
$M_1 = \frac{2\lambda e^{-d\tau}}{d}$, which implies that $G(t)$
is ultimately bounded, and so are $x(t), y(t)$ and $z(t)$.
This proof is complete.
\end{proof}
System \eqref{M3} always exists an infection-free equilibrium
$E_0=(x_0,0,0)$, where $x_0=\frac{\lambda}{d_1}$,
which represents the state that the viruses are absent.
The basic reproduction number of system \eqref{M3} is given by
$$
\Re_0=\frac{\beta e^{-d\tau}}{d_2}.
$$
If $\Re_0\leq 1$, an infection-free equilibrium $E_0$ is the unique
equilibrium, corresponding to the extinction of free viruses.
If $\Re_0 > 1$, in addition to $E_0$, there exists an CTL-inactivated
infection equilibrium $E_1(x_1, y_1,0)$, where
$$
x_1=\frac{\lambda}{d_1+d_2e^{d_1\tau}(\Re_0-1)},\quad
y_1=\frac{\lambda (\Re_0-1)}{d_1+d_2e^{d_1\tau}(\Re_0-1)},
$$
which represents the state that the viruses are present whereas the CTLs
are absent. We introduce a CTL immune-response reproduction number
$$
\Re_1=\frac{d_1+d_2 e^{d_1\tau}(\Re_0-1)}{p\lambda}(py_1-d_3)+1.
$$
Given $\Re_1>1$, then system \eqref{M3} has an CTL-activated infection
equilibrium $E_2(x_2, y_2, z_2)$, where
\begin{gather*}
x_2=\frac{(\lambda p-d_1d_3-\beta d_3)
+\sqrt{(\lambda p-d_1d_3-\beta d_3)^2+4d_1d_3\lambda p}}{2d_1p},
\\
y_2=\frac{d_3}{p},\quad
z_2=\frac{\beta pe^{-d_1\tau}x_2}{ax_2p+d_3}-\frac{d_2}{a}.
\end{gather*}
Clearly, the endemic equilibrium represents the state that
both the viruses and CTL response are present.
\section{Main results}
Throughout the article, we let $g(x)=x-1-\ln x$, to simplify many
of the expressions which follow.
Note that $g: \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ has strict
global minimum $g(1)=0$.
\begin{theorem}\label{T1}
If $\Re_0 \leq 1$, then the disease free
equilibrium $E_0$ is globally asymptotically stable.
\end{theorem}
\begin{proof}
Define a Lyapunov functional
\begin{equation}
L_0=e^{-d_1\tau}\Big[x-x_0-\int_{x_0}^{x}
\frac{x_0(\theta+y)}{\theta(x_0+y)}d\theta\Big]+y
+\frac{a}{p}z+\beta e^{-d_1\tau}\int_{t-\tau}^t\frac{x(\theta)
y(\theta)}{x(\theta)+y(\theta)}d\theta.
\end{equation}
Calculating the time derivative of $L_0$ along the solution of \eqref{M3},
it follows that
\begin{align*}
&\frac{dL_0}{dt}\Big|_{\eqref{M3}}\\
&= e^{-d_1\tau}\big[1-\frac{x_0(x+y)}{x(x_0+y)}\big]\dot{x}+\dot{y}+\frac{a}{p}\dot{z}+\frac{\beta e^{-d_1\tau}xy}{x+y}-\frac{\beta e^{-d_1\tau}x(t-\tau)y(t-\tau)}{x(t-\tau)+y(t-\tau)}\\
&= e^{-d_1\tau}\big[1-\frac{x_0(x+y)}{x(x_0+y)}\big]
\big[\lambda-d_1x-\frac{\beta xy}{x+y}\big]+\frac{\beta e^{-d_1\tau} x(t-\tau)y(t-\tau)}{x(t-\tau)+y(t-\tau)}\\
&\quad -d_2y-ayz+\frac{a}{p}(pyz-d_3z)
+\frac{\beta e^{-d_1\tau}xy}{x+y}
-\frac{\beta e^{-d_1\tau}x(t-\tau)y(t-\tau)}{x(t-\tau)+y(t-\tau)}\\
&= e^{-d_1\tau}d_1(x_0-x)\big[1-\frac{x_0(x+y)}{x(x_0+y)}\big]
+\frac{\beta e^{-d_1\tau}x_0y}{x_0+y}-d_2y-\frac{ad_3z}{p}\\
&= e^{-d_1\tau}d_1(x_0-x)\big[1-\frac{x_0(x+y)}{x(x_0+y)}\big]
+\frac{d_2x_0y(\Re_0-1)}{x_0+y}-\frac{d_2y^2}{x_0+y}-\frac{ad_3z}{p},
\end{align*}
where
$$
e^{-d_1\tau}d_1(x_0-x)\big[1-\frac{x_0(x+y)}{x(x_0+y)}\big]
=-\frac{e^{-d_1\tau}d_1y(x_0-x)^2}{x(x_0+y)}\leq 0.
$$
Therefore, $\Re_0\leq 1$ ensures that $dL_0/dt \leq 0$ for all $x >0$,
$y \geq 0$, $z \geq 0$, and $dL_0/dt = 0$ holds if
and only if $x = x_0, y = 0$, and $z(t) = 0$ for $\Re_0 \leq 1$.
If follows that the largest invariant set in
$\{(x_t, y_t, v_t, z_t)|dL_0/dt = 0\}$ is $E_0$.
The classical Lyapunov-LaSalle invariance principle
\cite[Theorem 2.5.3]{Kuang} shows that
$E_0$ is globally asymptotically stable when $\Re_0\leq 1$.
\end{proof}
\begin{theorem} \label{T2}
If $\Re_1\leq 1<\Re_0$, then CTL-inactivated infection equilibrium $E_1$
is globally asymptotically stable.
\end{theorem}
\begin{proof}
Define a Lyapunov functional
\begin{align*}
L_1&= x-x_1-\int_{x_1}^{x}\frac{x_1(\theta+y_1)}{\theta(x_1+y_1)}d\theta
+e^{d_1\tau}y_1g\big(\frac{y}{y_1}\big)+\frac{ae^{d_1\tau}}{p}z\\
&\quad + e^{d_1\tau}d_2y_1 \int_{t-\tau}^tg
\Big(\frac{\beta x(\theta)y(\theta)}{e^{d_1\tau}d_2y_1(x(\theta)+y(\theta))}\Big)
d\theta.
\end{align*}
Calculating the time derivative of $L_1$ along the solution of \eqref{M3},
it follows that
\begin{equation} \label{E1}
\begin{aligned}
\frac{dL_1}{dt}\Big|_{\eqref{M1}}
&= \big[1-\frac{x_1(x+y_1)}{x(x_1+y_1)}\big]\dot{x}
+e^{d_1\tau}\big(1-\frac{y_1}{y}\big)\dot{y}+\frac{ae^{d_1\tau}}{p}\dot{z} \\
&\quad +\frac{\beta xy}{x+y}-\frac{\beta x(t-\tau)y(t-\tau)}{x(t-\tau)
+y(t-\tau)}-e^{d_1\tau}d_2y_1\ln\frac{\beta xy}{e^{d_1\tau}d_2y_1(x+y)} \\
&\quad + e^{d_1\tau}d_2y_1\ln\frac{\beta x(t-\tau)y(t-\tau)}{e^{d_1\tau}
d_2y_1(x(t-\tau)+y(t-\tau))} \\
&= \big[1-\frac{x_1(x+y_1)}{x(x_1+y_1)}\big]
\Big[d_1(x_1-x)+\Big(\frac{\beta x_1y_1}{x_1+y_1}-\frac{\beta xy}{x+y}\Big)\Big] \\
&\quad + \big[\frac{\beta x(t-\tau)y(t-\tau)}{x(t-\tau)+y(t-\tau)}
-e^{d_1\tau} (d_2y(t)-ay(t)z(t))\big]\big(1-\frac{y_1}{y}\big) \\
&\quad +\frac{ae^{d_1\tau}}{p}(py-d_3)z +\frac{\beta xy}{x+y}
-\frac{\beta x(t-\tau)y(t-\tau)}{x(t-\tau)+y(t-\tau)} \\
&\quad -e^{d_1\tau}d_2y_1\ln\frac{\beta xy}{e^{d_1\tau}d_2y_1(x+y)} \\
&\quad + e^{d_1\tau}d_2y_1\ln\frac{\beta x(t-\tau)y(t-\tau)}{e^{d_1
\tau}d_2y_1(x(t-\tau)+y(t-\tau))}.
\end{aligned}
\end{equation}
Here we used that
\begin{equation}\label{E2}
\lambda=d_1x_1+\frac{\beta x_1y_1}{x_1+y_1},\quad
d_2y_1=\frac{\beta e^{-d_1\tau x_1y_1}}{x_1+y_1}.
\end{equation}
Combining the \eqref{E1} and \eqref{E2} we obtain
\begin{align*}
&\frac{dL_1}{dt}\Big|_{\eqref{M1}}\\
&= d_1(x_1-x)\big[1-\frac{x_1(x+y_1)}{x(x_1+y_1)}\big]
+\frac{\beta x_1y_1}{x_1+y_1}\big[1-\frac{x_1(x+y_1)}{x(x_1+y_1)}
+\frac{y(x+y_1)}{y_1(x+y)}-\frac{y}{y_1}\big]\\
&\quad -\frac{y_1}{y}\frac{\beta x(t-\tau)y(t-\tau)}{x(t-\tau)+y(t-\tau)}
+e^{d_1\tau}ay_1z-\frac{e^{d_1\tau} ad_3z}{p}+\frac{\beta x_1y_1}{x_1+y_1}\\
&\quad -e^{d_1\tau}d_2y_1\ln\frac{\beta xy}{e^{d_1\tau}d_2y_1(x+y)}+
e^{d_1\tau}d_2y_1\ln\frac{\beta x(t-\tau)y(t-\tau)}
{e^{d_1\tau}d_2y_1(x(t-\tau)+y(t-\tau))}\\
&= d_1(x_1-x)\big[1-\frac{x_1(x+y_1)}{x(x_1+y_1)}\big]
+\frac{\beta x_1y_1}{x_1+y_1}\Big[\Big(1-\frac{y(x+y_1)}{y_1(x+y)}\Big)
\Big(\frac{x+y}{x+y_1}-1\Big) \\
&\quad -\Big(\frac{x+y}{x+y_1}-1-\ln\frac{x+y}{x+y_1}\Big)
-\Big(\frac{x_1(x+y_1)}{x(x_1+y_1)}-1-\ln\frac{x_1(x+y_1)}{x(x_1+y_1)}\Big)\\
&\quad -\ln\frac{x+y}{x+y_1}-\ln\frac{x_1(x+y_1)}{x(x_1+y_1)}\Big]
+\frac{ae^{d_1\tau}}{p}(py_1-d_3)
\\
&\quad -\frac{\beta x_1y_1}{x_1+y_1}
\Big[\frac{(x_1+y_1)x(t-\tau)y(t-\tau)}{x_1y(x(t-\tau)+y(t-\tau))}-1
-\ln\frac{(x_1+y_1)x(t-\tau)y(t-\tau)}{x_1y(x(t-\tau)+y(t-\tau))}\Big]
\\
&\quad -\frac{\beta x_1y_1}{x_1+y_1}
\Big[\ln\frac{(x_1+y_1)x(t-\tau)y(t-\tau)}{x_1y(x(t-\tau)+y(t-\tau))}
+\ln\frac{\beta xy}{e^{d_1\tau}d_2y_1(x+y)} \\
&\quad -\ln\frac{\beta x(t-\tau)y(t-\tau)}{e^{d_1\tau}d_1y_1(x(t-\tau)
+y(t-\tau))}\Big]\\
&= d_1(x_1-x)\Big[1-\frac{x_1(x+y_1)}{x(x_1+y_1)}\Big]
+\frac{\beta x_1y_1}{x_1+y_1}\Big(1-\frac{y(x+y_1)}{y_1(x+y)}\Big)
\Big(\frac{x+y}{x+y_1}-1\Big)\\
&\quad -\frac{\beta x_1y_1}{x_1+y_1}g\Big(\frac{x+y}{x+y_1}\Big)
-\frac{\beta x_1y_1}{x_1+y_1}g\Big(\frac{x_1(x+y_1)}{x(x_1+y_1)}\Big)\\
&\quad -\frac{\beta x_1y_1}{x_1+y_1}g
\Big(\frac{(x_1+y_1)x(t-\tau)y(t-\tau)}{x_1y(x(t-\tau)+y(t-\tau))}\Big)
+\frac{ae^{d_1\tau}}{p}(py_1-d_3),
\end{align*}
where
$$
d_1(x_1-x)\big[1-\frac{x_1(x+y_1)}{x(x_1+y_1)}\big]
=-\frac{d_1y(x_1-x)^2}{x(x_1+y_1)}\leq 0.
$$
Note that
\[
\Re_1=\frac{d_1+d_2 e^{d_1\tau}(\Re_0-1)}{p\lambda}(py_1-d_3)+1,
\]
which implies that $\Re_1 \leq 1$ is equivalent to $py_1/d_3 \leq 1$. The
latter $py_1/d_3$ is seen to be the immune reproductive number,
which expresses the average number of activated CTLs generated from
one CTL during its life time $1/d_3$ through the stimulation of
the infected cells $y_1$. It is reasonable that immune is activated
in the case where $\Re_1 > 1$. Hence $\frac{dL_1}{dt}$ is always
non-positive under the condition $\Re_1\leq 1<\Re_0$, and it can be
verified that $\frac{dL_1}{dt}=0$ if and only if $x=x_1$ and
$\frac{x+y}{x+y_1}=\frac{x_1(x+y_1)}{x(x_1+y_1)}
=\frac{(x_1+y_1)x(t-\tau)y(t-\tau)}{x_1y(x(t-\tau)+y(t-\tau))}=1$.
Using the first two equations of system \eqref{M3}, we have
\begin{gather*}
0= \dot{x}(t)= \lambda-d_1x_1-\frac{\beta x_1y(t)}{x_1+y(t)}, \\
0= \dot{y}(t)= \frac{\beta e^{-d_1\tau} x_1y(t-\tau)}{x_1+y(t-\tau)}
-d_2y(t)-ay(t)z(t).
\end{gather*}
This gives $y=y_1$, $z=0$. So, the global asymptotic stability of $E_1$
follows from the LaSalle's invariant principle.
\end{proof}
\begin{theorem} \label{T3}
If $\Re_1>1$, then CTL-activated infection equilibrium $E_2$
is globally asymptotically stable; i.e., $E_2$
is globally asymptotically stable whenever it exists.
\end{theorem}
\begin{proof}
Define a Lyapunov functional
\begin{align*}
L_2&= x(t)-x_2-\int_{x_2}^{x(t)}\frac{x_2(\theta+y_2)}{\theta(x_2+y_2)}d\theta
+e^{d_1\tau}y_2g\Big(\frac{y(t)}{y_2}\Big)
+\frac{ae^{d_1\tau}}{p}z_2g\left(\frac{z(t)}{z_2}\right)\\
&\quad + e^{d_1\tau}(d_2y_2+ay_2z_2) \int_{t-\tau}^tg
\Big(\frac{\beta x(\theta)y(\theta)}{e^{d_1\tau}(d_2y_2+ay_2z_2)(x(\theta)
+y(\theta))}\Big)d\theta.
\end{align*}
Calculating the time derivative of $L_2$ along the solution of \eqref{M1},
we obtain
\begin{equation} \label{E3}
\begin{aligned}
\frac{dL_2}{dt}\Big|_{\eqref{M1}}
&= \Big[1-\frac{x_2(x+y_2)}{x(x_2+y_2)}\Big]
\Big[d_1(x_2-x)+\Big(\frac{\beta x_2y_2}{x_2+y_2}-\frac{\beta xy}{x+y}\Big)\Big] \\
&\quad + \Big[\frac{\beta x(t-\tau)y(t-\tau)}{x(t-\tau)+y(t-\tau)}
-e^{d_1\tau} (d_2y(t)-ay(t)z(t))\Big]\big(1-\frac{y_2}{y}\big) \\
&\quad +\frac{ae^{d_1\tau}}{p}(pyz-d_3z)\big(1-\frac{z_2}{z}\big)
+\frac{\beta xy}{x+y}-\frac{\beta x(t-\tau)y(t-\tau)}{x(t-\tau)+y(t-\tau)} \\
&\quad -\frac{\beta x_2y_2}{x_2+y_2}\ln\frac{\beta xy}{\frac{\beta x_2y_2}{x_2+y_2}(x+y)}+
\frac{\beta x_2y_2}{x_2+y_2}\ln\frac{\beta x(t-\tau)y(t-\tau)}
{\frac{\beta x_2y_2}{x_2+y_2}(x(t-\tau)+y(t-\tau))}.
\end{aligned}
\end{equation}
Here we used that
\begin{equation}\label{E4}
\lambda=d_1x_2+\frac{\beta x_2y_2}{x_2+y_2},\quad
d_2y_2+ay_2z_2=\frac{\beta e^{-d_1\tau}x_2y_2}{x_2+y_2}, \quad
py_2=d_3.
\end{equation}
Combining \eqref{E3} and \eqref{E4} we obtain
\begin{align*}
&\frac{dL_2}{dt}\Big|_{\eqref{M1}}\\
&= d_1(x_2-x)\Big[1-\frac{x_2(x+y_2)}{x(x_2+y_2)}\Big]
+\frac{\beta x_2y_2}{x_2+y_2}
\Big[1-\frac{x_2(x+y_2)}{x(x_2+y_2)}+\frac{y(x+y_2)}{y_2(x+y)}-\frac{y}{y_2}
\Big]\\
&\quad +\frac{\beta x_2y_2}{x_2+y_2}-\frac{y_2}{y}
\frac{x(t-\tau)y(t-\tau)}{x(t-\tau)+y(t-\tau)}\\
&\quad -\frac{\beta x_2y_2}{x_2+y_2}\ln\frac{\beta xy}{\frac{\beta x_2y_2}
{x_2+y_2}(x+y)}+
\frac{\beta x_2y_2}{x_2+y_2}\ln\frac{\beta x(t-\tau)y(t-\tau)}
{\frac{\beta x_2y_2}{x_2+y_2}(x(t-\tau)+y(t-\tau))} \\
&= d_1(x_2-x)\Big[1-\frac{x_2(x+y_2)}{x(x_2+y_2)}\Big]
+\frac{\beta x_2y_2}{x_2+y_2}\Big[\Big(1-\frac{y(x+y_2)}{y_2(x+y)}\Big)
\Big(\frac{x+y}{x+y_2}-1\Big)\\
&\quad -\Big(\frac{x+y}{x+y_2}-1-\ln\frac{x+y}{x+y_2}\Big)
-\Big(\frac{x_2(x+y_2)}{x(x_2+y_2)}-1-\ln\frac{x_2(x+y_2)}{x(x_2+y_2)}\Big)\\
&\quad -\ln\frac{x+y}{x+y_2}-\ln\frac{x_2(x+y_2)}{x(x_2+y_2)}\Big]\\
&\quad -\frac{\beta x_2y_2}{x_2+y_2}
\Big[\frac{(x_2+y_2)x(t-\tau)y(t-\tau)}{x_2y(x(t-\tau)
+y(t-\tau)}-1-\ln\frac{(x_2+y_2)x(t-\tau)y(t-\tau)}{x_2y(x(t-\tau)
+y(t-\tau)}\Big]\\
&\quad -\frac{\beta x_2y_2}{x_2+y_2}
\Big[\ln\frac{(x_2+y_2)x(t-\tau)y(t-\tau)}{x_2y(x(t-\tau)+y(t-\tau)}
+\ln\frac{\beta xy}{\frac{\beta x_2y_2}{x_2+y_2}(x+y)} \\
&\quad -\ln\frac{\beta x(t-\tau)y(t-\tau)}
{\frac{\beta x_2y_2}{x_2+y_2}(x(t-\tau)+y(t-\tau))}\Big]\\
&= d_1(x_2-x)\Big[1-\frac{x_2(x+y_2)}{x(x_2+y_2)}\Big]
+\frac{\beta x_2y_2}{x_2+y_2}\Big(1-\frac{y(x+y_2)}{y_2(x+y)}\Big)
\Big(\frac{x+y}{x+y_2}-1\Big)\\
&\quad -\frac{\beta x_2y_2}{x_2+y_2}g\Big(\frac{x+y}{x+y_2}\Big)
-\frac{\beta x_2y_2}{x_2+y_2}g\Big(\frac{x_2(x+y_2)}{x(x_2+y_2)}\Big)\\
&\quad -\frac{\beta x_2y_2}{x_2+y_2}g
\Big(\frac{(x_2+y_2)x(t-\tau)y(t-\tau)}{x_2y(x(t-\tau)+y(t-\tau))}\Big).
\end{align*}
Similar to the proof of Theorem \ref{T2}, the terms of $dL_2/dt$
always are non-positive. Hence
$dL_2/dt$ for all $x > 0, y > 0 $and $z > 0$, and $dL_2/dt=0$ if
and only if $x = x_2$ and $y=y_2, z=z_2$. The largest invariant set
in $\{(x_t, y_t,z_t)\mid dL_2/dt=0\}$ is $E_2$. From the Lyapunov-LaSalle invariance principle, it shows that equilibrium $E_2(x_1, y_2, z_2)$ is
globally asymptotically stable.
\end{proof}
\section{Summary and Discussion}
In this article, we have modified the delay differential equation model
for cell-to-cell infection of HBV in tissue
cultures proposed by Gourley et al \cite{Gourley} by incorporating a Cytotoxic
T lymphocytes (CTLs) immune response to model the role in antiviral by attacking
virus infected cells. Since immune response after viral infection
is universal and necessary to eliminate or control the disease.
Our analysis provides a quantitative understandings of HBV replication
dynamics in vivo and has implications for the optimal timing
of drug treatment and immunotherapy in chronic HBV infection.
By constructing Lyapunov functionals, we obtain the global stability of the
equilibria of \eqref{M3} that depends only on the basic reproductive
number $\Re_0$ and the basic immune reproductive number $\Re_1$.
For delay differential equations model \eqref{M3}, the basic reproductive
number is given by $\Re_0=\frac{\beta e^{-d\tau}}{d_2}$ and it is a
decreasing function on intracellular delay $\tau$ such that $\Re_0(\infty) = 0$.
Theorems \ref{T1}-\ref{T3} show that when $\Re_0 \leq 1$, the infection-free
equilibrium is globally asymptotically stable, which means that the
viruses are cleared and immune is not active; when $\Re_1 \leq 1 < \Re_0$,
the CTL-inactivated infection equilibrium exists and is globally asymptotically
stable, which means that CTLs immune response would not be activated and
viral infection becomes chronic but with a low level of proviral load;
and when $\Re_1 > 1$, the CTL-activated infection equilibrium
exists and is globally asymptotically stable, in this case the infection
causes a persistent CTLs immune response and is chronic with a high level
of proviral load. We can see that under the condition $\Re_1(\tau) > 1$,
as delay $\tau$ increases, the number of CTLs immune response does not
change in this situation. However, when the delay $\tau$ is sufficiently
large, and brings $\Re_1(\tau)$) to a level lower than unity,
the CTL-inactivated infection equilibrium $E_1$ becomes globally asymptotically
stable.
It has been repointed in Nowak \cite{Nowak1} that \lq\lq
Treatment of chronic HBV infections with lamivudine leads
to a rapid and sustained decline of plasma virus levels, but
clinical benefit with a reduced risk of cirrhosis and development
of liver cancer will greatly depend on the decline of
infected cells. Immunotherapy without antiviral
treatment could be problematic because of the very large
number of infected liver cells in the typical HBV carrier.
Therefore, the drugs which can prolong the latent period, and/or
decrease the needed time of immune response activation and/or
inhibit infection can slow down the virus production process. This gives us a
good guidance on the development of treatment strategies.
On the other hand, cell-to-cell models may be applicable to study the
within-host dynamics of other types of viral infections such as human
T-cell leukaemia virus type 1 (HTLV-1), hepatitis C, etc. We leave
the modeling and study of the cell-to-cell HTLV-1 infection
for future consideration. Other realistic modifications can be made.
For example, we can modify target-cell dynamics to be
a mitosis component given by a logistic term and the loss/gain term
as nonlinear incidence function. Another possible modification would be to
incorporate diffusion term into the delayed model to more accurately
reflect the realistic situation in tissue cultures.
\subsection*{Acknowledgments}
The authors want to thank the anonymous referees and the editor
for their valuable suggestions and comments.
J. Wang is supported by National Natural Science Foundation of
China (TianYuan No.11226255).
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\end{document}