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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 182340, 21 pages
http://dx.doi.org/10.1155/2010/182340
Research Article

Asymptotic Properties of a Hepatitis B Virus Infection Model with Time Delay

Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, No. 97 Heping West Road, Shijiazhuang 050003, China

Received 14 February 2010; Accepted 21 August 2010

Academic Editor: Juan J. Nieto

Copyright © 2010 Xiaohong Tian and Rui Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. Introduction

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., [115]). In [1], Anderson and May used a simple mathematical model to illustrate the effects of carriers on the transmission of HBV. In an effort to model HBV infection dynamics and its treatment with the reverse transcriptase inhibitor lamivudine, Nowak and Bangham [7] and Bonhoeffer et al. [2] proposed the following basic HBV infection model: where and are numbers of uninfected cells, infected cells, and free-virus cells, respectively. Uninfected cells are assumed to be produced at a constant rate , die at rate , and become infected at rate in which is the mass action rate constant describing the infection process. Infected cells are killed by immune cells at rate and produce free virus at rate , here is the so-called burst constant. Free-virus cells are cleared at rate . It is assumed that parameters are positive constants. In [4], by constructing novel Lyapunov functions, it was proven that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable, and if the basic reproduction ratio is greater than unity, then the infected equilibrium is globally asymptotically stable. In [9], Thornley et al. used a hepatitis B mathematical model developed by Medley et al. [5] to develop a strategy for eliminating HBV in New Zealand. In [13], Zhao et al. proposed an age-structured model to predict the dynamics of HBV transmission and evaluate the long-term effectiveness of the vaccination programme in China. In [11], Xu and Ma investigated a hepatitis B virus model with spatial diffusion and saturation response of the infection rate. In [14], Zou et al. also proposed a mathematical model to understand the transmission dynamics and prevalence of HBV in mainland China. In [12], Yu et al. considered an HBV infection model with a nonlinear infection rate. It was shown that the model has a degenerate singular infection equilibrium, and bifurcation of cusp type with codimension two (i.e., Bogdanov-Takens bifurcation) occurs under appropriate conditions. As a result, the rich dynamical behaviors indicate that the model can display an Allee effect and fluctuation effect, which are important for making strategies for controlling the invasion of virus. In [8], Pang et al. developed a mathematical model to explore the impact of vaccination and other controlling measures of HBV infection. It was shown that the vaccination is a very effective measure to control the infection, and some useful comments were given on controlling the transmission of HBV.

Usually, the rate of infection in most HBV virus models is assumed to be bilinear in the virus and the uninfected cells . Under this assumption, the basic infection reproductive number is proportional to the number of total cells of the liver, which implies that an individual with a smaller liver may be more resistant to the virus infection than an individual with a larger one. Clearly, this is not true. A typical chronically infected HBV patient has a total serum daily production rate of about to virions, and an average human liver consists of billions of liver cells. These large numbers suggest that it is reasonable to assume that the infection rate is given by the standard incidence function [3]. Based on the idea above, in [6], Min et al. proposed the following basic HBV virus model: For system (1.2), it was shown in [6] that if the basic infection reproductive number is less than unity, then every positive solution converges to the infection-free steady state. At the same time, it was also assumed that cells upon infection instantly begin producing virus. In [10], Wang et al. introduced an improved HBV model with standard incidence function and cytokine-mediated “cure” based on empirical evidences. By using the geometrical approach of Li and Muldowney [16] to global stability problems in , the global stability of the virus-infected equilibrium was established. However, in reality, there is a time delay between viral infection of a cell and the time the cell begins releasing virus. In [17], Nelson et al. considered a model that allows for less than perfect drug effects and includes a delay in the initiation of virus production. Compared with the outcomes of models without time delay, modelling on virus infection by suitable delay terms looks to be biologically reasonable [1820].

Motivated by the work of Min et al. [6] and Nelson et al. [17], in this paper, we study the following hepatitis B virus infection model with a time delay: The initial conditions for system (1.3) take the form where the Banach space of continuous functions mapping the interval into , here .

It is easy to show that all solutions of system (1.3) with initial condition (1.4) are defined on and remain positive for all .

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 3, by analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of system (1.3) is discussed. In Section 4, by using an iteration technique, we study the global stability of the infection-free equilibrium of system (1.3). By comparison arguments we discuss the global stability of the virus-infected equilibrium of system (1.3). Numerical simulations are carried out in Section 5 to illustrate the main theoretical results.

2. Preliminaries

In this section, based on the work developed by Xu and Ma [21], we introduce some notations and state several results which will be useful in the next section.

Let be the cone of nonnegative vectors in . If , we write if for . Let denote the standard basis in . Suppose that , and let be the Banach space of continuous functions mapping the interval into with supremum norm. If , we write when the indicated inequality holds at each point of . Let and let denote the inclusion by . Denote the space of functions of bounded variation on by . If and , then for any , we let be defined by

We now consider

We assume throughout this section that is continuous; is continuously differentiable in ; for all , and some . Then by [22], there exists a unique solution of (2.1) through for . This solution will be denoted by if we consider the solution in or by if we work in the space . Again by [22], is continuously differentiable in . In the following, the notation will be used as the condition of the initial data of (2.1), by which we mean that we consider the solution of (2.1) which satisfies .

To proceed further, we need the following results. Let and define

We write for a generic point of . Let Due to the ecological applications, we choose as the state space of (2.1) in the following discussions.

Fix arbitrarily. Then we set where denotes the Frechet derivation of with respect to . It is convenient to have the standard representation of as in which satisfies where is continuous from the left in .

We make the following assumptions for (2.1).(h0)If and for some , then . (h1)For all with for . (h2)The matrix defined by

is irreducible for each . (h3)For each , for which , there exists such that for all and for positive constant sufficiently small, (h4)If , then for all .

The following result was established by Wang et al. [23].

Lemma 2.1. Let (h1)–(h4) hold. Then hypothesis (h0) is valid and (i)if and are distinct elements of with and with is the intersection of the maximal intervals of existence of and , then (ii)if and is defined on with , then

This lemma shows that if (h1)–(h4) hold, then the positivity of solutions of (2.1) follows.

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

Definition 2.2 (2.1). Let be an matrix, and let be distinct points of the complex plane. For each nonzero element of , connect to with a directed line . The resulting figure in the complex plane is a directed graph for . One says that a directed graph is strongly connected if, for each pair of nodes with , there is a directed path connecting and . Here, the path consists of directed lines.

Lemma 2.3 (see [24]). A square matrix is irreducible if and only if its directed graph is strongly connected.

Lemma 2.4 (see [25]). If (2.1) is cooperative and irreducible in , where is an open subset of , and the solutions with positive initial data are bounded, then the trajectory of (2.1) tends to some single equilibrium.

We now consider the following delay differential system: with initial conditions

System (2.9) always has a trivial equilibrium . If , then system (2.9) has a unique positive equilibrium , where The characteristic equation of system (2.9) at the equilibrium takes the form where Noting that if , then the equilibrium is locally stable when ; if , then is unstable when .

It is easy to show that If , then . By Theorem in the work of Kuang [26], we see that the equilibrium is locally asymptotically stable for all . If , then is unstable for all .

The characteristic equation of system (2.9) at the positive equilibrium is of the form where note that Hence, if the positive equilibrium is locally stable when ; if is unstable when .

It is easy to show that If , then by Theorem in the work of Kuang [26], we see that the positive equilibrium is locally asymptotically stable for all . If , then is unstable for all .

Lemma 2.5. For system (2.9), one has the following. (i)If , then the positive equilibrium is globally stable.(ii)If , then the equilibrium is globally stable.

Proof. We represent the right-hand side of (2.9) by and set By a direct calculation we have We now claim that hypotheses (h1)–(h4) hold for system (2.9). It is easily seen that (h1) and (h4) hold for system (2.9). We need only to verify that (h2) and (h3) hold.
The matrix takes the form Clearly, the matrix is irreducible for each .
From the definition of and , it is readily seen that , for , for , and where is a positive Borel measure on Therefore, . Thus, for each , there is such that for all and for sufficiently small, . Hence, (h3) holds.
Thus, the conditions of Lemma 2.1 are satisfied. Therefore, the positivity of solutions of system (2.9) follows. It is easy to see that system (2.9) is cooperative. By Lemma 2.3, we see that any solution starting from converges to some single equilibrium. However, system (2.9) has only two equilibria: and . Note that if , then the positive equilibrium is locally stable and the equilibrium is unstable. Hence, any solution starting from converges to if . Using a similar argument one can show the global stability of the equilibrium when . This completes the proof.

3. Local Stability

In this section, we discuss the local stability of each of the equilibria of system (1.3) by analyzing the corresponding characteristic equations.

System (1.3) always has an infection-free equilibrium .

Let is called the basic reproduction ratio of system (1.3). It is easy to show that if system (1.3) has a virus-infected equilibrium , where

The characteristic equation of system (1.3) at the infection-free equilibrium is of the form where Obviously, (3.3) always has a negative real root . All other roots of (3.3) are determined by the following equation: It is easy to show that . If , then the infection-free equilibrium of system (1.3) is locally asymptotically stable when .

If is a solution of (3.5), by calculating, we have Note that If , then . Therefore, (3.6) has no positive roots. Accordingly, if , the infection-free equilibrium of system (1.3) is locally asymptotically stable; if , (3.6) has at least a positive real root. Accordingly, is unstable.

The characteristic equation of system (1.3) at the virus-infected equilibrium takes the form where When , (3.8) becomes Clearly, By a direct calculation we have By the Hurwitz criteria, all roots of (3.10) have only negative real parts.

If is a solution of (3.8), separating real and imaginary parts, it follows that Squaring and adding the two equations of (3.12), we derive that where Hence, (3.13) has no positive roots. Accordingly, by the general theory of characteristic equations of delay differential equations in the work of Kuang [26] (Theorem 4.1), if , the virus-infected equilibrium of system (1.3) exists and is locally asymptotically stable.

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

Theorem 3.1. For system (1.3), one has the following. (i)If , the infection-free equilibrium is locally asymptotically stable. If , then is unstable.(ii)If , the virus-infected equilibrium is locally asymptotically stable.

4. Global Stability

In this section, we discuss the global stability of the infection-free equilibrium and the virus-infected equilibrium of system (1.3), respectively. The technique of proofs is to use a comparison argument and an iteration scheme (see, e.g., [27]).

Theorem 4.1. Let . If (H1), then the virus-infected equilibrium of system (1.3) is globally asymptotically stable.

Proof. Let be any positive solution of system (1.3) with initial condition (1.4). Let Now we claim that .
It follows from the first equation of system (1.3) that By comparison we derive that Hence, for sufficiently small there exists a such that if , . We therefore derive from the second and the third equations of system (1.3) that, for , Consider the following auxiliary equations: Since , by Lemma 2.5 it follows from (4.5) that By comparison, we obtain that Since these inequalities are true for arbitrary , it follows that , where Hence, for sufficiently small, there is a such that if ,
For sufficiently small, we derive from the first equation of system (1.3) that, for , A comparison argument shows that Since this is true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that if ,
For sufficiently small, we derive from the second and the third equations of system (1.3) that, for , Consider the following auxiliary equations: Since (H1) holds, by Lemma 2.5, it follows from (4.13) that By comparison we derive that Since these two inequalities hold for arbitrary sufficiently small, we conclude that , where Therefore, for sufficiently small, there is a such that if ,
For sufficiently small, it follows from the first equation of system (1.3) that, for , A comparison argument yields Since this is true for arbitrary , it follows that , where Hence, for sufficiently small there is a such that if , . It therefore follows from the second and the third equations of system (1.3) that, for , By Lemma 2.5 and a comparison argument we derive from (4.20) that Since these inequalities are true for arbitrary , it follows that , where Hence, for sufficiently small, there exists a such that if ,
Again, for sufficiently small, we derive from the first equation of system (1.3) that, for , A comparison argument shows that Since this is true for arbitrary , we derive that , where Hence, for sufficiently small, there is a such that if ,
For sufficiently small, it follows from the second and the third equations of system (1.3) that, for , Since (H1) holds, by Lemma 2.5 and a comparison argument, it follows from (4.26) that Since these two inequalities hold for arbitrary sufficiently small, we conclude that , where Therefore, for sufficiently small, there exists a such that if , ,
Continuing this process, we derive six sequences ,,,,, and such that, for , It is readily seen that It is easy to show that the sequences , , and are nonincreasing and the sequences , , and are nondecreasing. Hence, the limit of each sequence in , , , , , and exists. Denote We therefore obtain from (4.29) and (4.31) that By having (4.32) minus (4.33), Noting that (H1) holds and , it follows that which, together with (4.34), yields . We therefore derive from (4.31) that . Noting that if (H1) holds, by Theorem 3.1, the virus-infected equilibrium is locally stable, we conclude that is globally stable. The proof is complete.

Theorem 4.2. If holds, the infection-free equilibrium of system (1.3) is globally asymptotically stable.

Proof. Let be any positive solution of system (1.3) with initial condition (1.4). It follows from the first equation of system (1.3) that A standard comparison argument shows that Hence, for sufficiently small, there is a such that if We derive from the second and the third equations of system (1.3) that for , Consider the following auxiliary equation: If , then by Lemma 2.5 it follows from (4.37) and (4.39) that By comparison, we obtain that Therefore, for sufficiently small, there is a such that if ,
It follows from the first equation of system (1.3) that for , By comparison, we derive that Letting , it follows that This together with (4.37) yields This completes the proof.

5. Numerical Example

In this section, we give one example to illustrate the main result in Section 4.

In [28], one group of HBeAg-Positive chronic hepatitis B patients received 100 mg of lamivudine once daily. The study comprised 48 weeks of treatment and a 24-week treatment-free followup. While the onset of therapy and viral levels decline rapidly, the virus returns as soon as the drug is withdrawn (see Table 1).

tab1
Table 1: 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.

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 Clearly, if , then system (5.1) becomes system (1.3), which means that the patients are assumed to return to the stable state before the drug therapy.

Example 5.1. In system (5.1), based on the work of [6] and clinical data, we let and

Before the therapy, that is, , by a direct calculation, we have the basic reproduction ratio and system (1.3) has a virus-infected equilibrium . Clearly, (H1) holds. By Theorem 4.1, we see that the virus-infected equilibrium of system (1.3) is globally asymptotically stable. Numerical simulation illustrates the previous result (see Figure 1).

fig1
Figure 1: The numerical solution of system (5.1). (a) Uninfected cells ; (b) infected cells ; (c) virus declines in response to drug treatment and virus resurges as soon as the drug is withdrawn in which the clinical data are marked by dots.

Biologically, as can be seen from Figure 1(c), based on system (5.1), during the 48 weeks of treatment, the viral levels decline rapidly. As soon as the drug is withdrawn, by Theorem 4.1, virus level returns rapidly and tends to the virus-infected equilibrium. Figure 1(c) indicates that the simulation of model (5.1) agrees well with the clinical data reported. Furthermore, compared with the work of Min et al. [6], it is easy to show that the simulation results are similar. However, for system (1.3), numerical simulation shows that the slopes of the curves generated with different delays differ, and notice the slopes of the decay with and without a delay are parallel. When the time delay increases, it is easy to see that the viral load reduces; numerical simulation illustrates that the change in the slope is affected by the delay (see Figure 2).

182340.fig.002
Figure 2: The numerical solution for viral decay of system (1.3) when and days.

Acknowledgments

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).

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