A hepatitis B virus infection model with time delay is discussed. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the model is studied. By using comparison arguments, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable. If the basic reproduction ratio is greater than unity, by means of an iteration technique, sufficient conditions are derived for the global asymptotic stability of the virus-infected equilibrium. Numerical simulations are carried out to illustrate the theoretical results.

Hepatitis B is a potentially life-threatening liver infection caused by the hepatitis B virus. It is a major global health problem and the most serious type of viral hepatitis. It can cause chronic liver disease and puts people at high risk of death from cirrhosis of the liver and liver cancer. Worldwide, estimated two billion people have been infected with the hepatitis B virus (HBV), and more than 350 million have chronic (long-term) liver infections. In the past decade, therapy for HBV has been revolutionized by the advent of drugs that directly block replication of the HBV genome. All these drugs (to date) are nucleoside or nucleotide analogues that selectively target the viral reverse transcriptase. The first successful drug, lamivudine, emerged from screening for inhibitors of the HBV reverse transcriptase and was introduced into clinical practice for the management of HBV infection.

Recently, mathematical models have been used frequently to study the transmission dynamics of HBV (see, e.g., [

Usually, the rate of infection in most HBV virus models is assumed to be bilinear in the virus

Motivated by the work of Min et al. [

It is easy to show that all solutions of system (

The organization of this paper is as follows. In the next section, we introduce some notations and state several lemmas which will be essential to our proofs. In Section

In this section, based on the work developed by Xu and Ma [

Let

We now consider

We assume throughout this section that

To proceed further, we need the following results. Let

We write

Fix

We make the following assumptions for (

If

For all

The matrix

is irreducible for each

For each

If

The following result was established by Wang et al. [

Let (h1)–(h4) hold. Then hypothesis (h0) is valid and

if

if

This lemma shows that if (h1)–(h4) hold, then the positivity of solutions of (

The following definition and results are useful in proving our main result.

Let

A square matrix is irreducible if and only if its directed graph is strongly connected.

If (

We now consider the following delay differential system:

System (

It is easy to show that

The characteristic equation of system (

It is easy to show that

For system (

If

If

We represent the right-hand side of (

The matrix

From the definition of

Thus, the conditions of Lemma

In this section, we discuss the local stability of each of the equilibria of system (

System (

Let

The characteristic equation of system (

If

The characteristic equation of system (

If

Based on the discussions above, we have the following result.

For system (

If

If

In this section, we discuss the global stability of the infection-free equilibrium and the virus-infected equilibrium of system (

Let

Let

It follows from the first equation of system (

For

For

For

Again, for

For

Continuing this process, we derive six sequences

If

Let

It follows from the first equation of system (

In this section, we give one example to illustrate the main result in Section

In [

Rapid deline in plasma virus: mean HBV DNA levels (log copies/ml) in response to the therapy, and the virus level returning rapidly after the treatment was stopped.

Week | 0 | 1 | 2 | 4 | 6 | 8 | 12 | 18 |

Patient Nos. | 272 | 272 | 272 | 267 | 267 | 267 | 267 | 267 |

Virus load | 9.8 | 7.8 | 6.6 | 5.6 | 5.1 | 4.8 | 4.4 | 4.3 |

Week | 24 | 30 | 36 | 42 | 48 | 52 | 60 | 72 |

Patient Nos. | 263 | 263 | 259 | 260 | 249 | 248 | 228 | 241 |

Virus load | 4.2 | 4.0 | 4.15 | 4.2 | 4.5 | 7.0 | 8.0 | 8.20 |

In the following, we will use the set of clinical data to formulate a hepatitis B virus infection therapy model. Assume that, during the lamivudine drug treatment, the dynamic model of the patient with the mean load HBV DNA is of the form

In system (

Before the therapy, that is,

The numerical solution of system (

Biologically, as can be seen from Figure

The numerical solution for viral decay of system (

The authors wish to thank the reviewers and the editor for their valuable comments and suggestions that greatly improved the presentation of this paper.

This work was supported by the National Natural Science Foundation of China (nos. 11071254, 10671209) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and the Science Research Foundation of JCB (no. JCB 1005).