Journal of University of Science and Technology Beijing
Volume 11, Number 6, December 2004, Page 572
Corresponding author: Lequan Min E-mail: lqmin@sohu.com
Communication
Research on viral dynamic models of hepatitis B virus infection
Lequan Min
1, 2)
and Xisong Dong
2)
1) Applied Science School, University of Science and Technology Beijing, Beijing 100083, China
2) Information Engineering School, University of Science and Technology Beijing, Beijing 100083, China
(Received 2004-08-28)
Abstract: A mathematical model with cytotoxic cells of hepatitis B virus (HBV) infection is set up based on a basic model of virus
dynamics without cytotoxic cells and experimental observation of anti-viral drug therapy for HBV infection patients. A quantitative
analysis of dynamic behaviors shows that the model has three kinds of equilibrium points, which represent the patient's complete re-
covery without immune ability, complete recovery with immune ability, and HBV persistent infection at the end of the treatment with
drug lamivudine, respectively. Our model may provide possible quantitative interpretations for the treatments of chronic HBV infec-
tions with the drug lamivudine, in particularly explain why the plasma virus of Nowak et al.'s patients turnover the original level after
stopping the lamivudine treatment.
Key words: mathematical model; liver; hepatitis B virus; lamivudine treatment; equilibrium point stability; treatment time; numeric
simulation
[This project is jointly supported by the National Natural Science Foundation of China (No.70271068, 60074034), and the Research
Fund for the Doctoral Program of Higher Education (No.20020008004) by the Education Ministry of China.]
1 Introduction
Hepatitis B is one of the major diseases of mankind.
It is estimated that 2 billion people worldwide have
been infected with the hepatitis B virus (HBV); more
than 350 million have chronic (lifelong) infections [1];
25%-40% of these chronic infection carriers will die
from liver cirrhosis or primary hepatocellular carci-
noma; 50 million new cases are diagnosed annually [2,
3]. Chronic HBV infection is often the result of expo-
sure early in life, leading to viral persistence in the
absence of strong antibody or cellular immune res-
ponses [4, 5].
Until very recently, there was no effective treatment
for chronic carriers of HBV. In the 1990's, some drugs
called interferon or lamivudine are used to treat pa-
tients with chronic hepatitis B [6-8]. However this
therapy will cost thousands of dollars and never be
available to most patients in developing countries [1].
In 1996, Nowak et al. proposed a mathematical model
with three variables: uninfected cells, infected cells,
and free virus [2]. This model can describe some viral
dynamics in HBV infection. The model cannot de-
scribe the phenomena reported by Nowak et al. that
the plasma virus of most of 95 chronic HBV carriers
treated with various dose of lamivudine for 28 days or
24 weeks resurges rapidly as soon as the drug is with-
drawn.
In this paper, a mathematical model consisting of
two systems is designed. One system is the model
proposed by Nowak et al., which describes the viral
dynamics in HBV infection after withdrawing the la-
mivudine therapy. The other system is a model with
four state variables: uninfected cells, infected cells,
free virus and cytotoxic cells, which represents the
viral dynamics in HBV infection during the lamivu-
dine therapy. The quantitative analysis of the dynam-
ics of our model shows that the two systems have
three kinds of equilibrium points, respectively, which
represent the patient's complete recovery without im-
mune ability, complete recovery with immune ability,
and HBV persistent infection.
The stability and unstability of the equilibrium
points are analyzed, which corresponding several
groups of system parameter inequalities. Numerical
simulations show that our model may provide possible
quantitative interpretations for the treatment of chron-
ic HBV infection with the drug lamivudine reported
by Nowak et al., in particularly explain why the plas-
ma virus of patients turnover the original level after
stopping the lamivudine therapy.