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

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.


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., 1-15 . 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 where x, y, and v are numbers of uninfected cells, infected cells, and free-virus cells, respectively. Uninfected cells are assumed to be produced at a constant rate λ, die at rate dx, and become infected at rate βxv in which β is the mass action rate constant describing the infection process. Infected cells are killed by immune cells at rate ay and produce free virus at rate ky, here k is the so-called burst constant. Free-virus cells are cleared at rate uv. It is assumed that parameters a, d, k, u, λ, and β 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  Usually, the rate of infection in most HBV virus models is assumed to be bilinear in the virus v and the uninfected cells x. 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 2 × 10 11 to 3 × 10 12 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: v t ky t − uv t .

1.2
Discrete Dynamics in Nature and Society v t ky t − uv t .

1.3
The initial conditions for system 1.3 take the form . It is easy to show that all solutions of system 1.3 with initial condition 1.4 are defined on 0, ∞ and remain positive for all t ≥ 0.
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.

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 R n be the cone of nonnegative vectors in R n . If x, y ∈ R n , we write x ≤ y x < y if x i ≤ y i x i < y i for 1 ≤ i ≤ n. Let {e 1 , e 2 , . . . , e n } denote the standard basis in R n . Suppose that r ≥ 0, and let C C −r, 0 , R n be the Banach space of continuous functions mapping the interval −r, 0 into R n with supremum norm. If φ, ψ ∈ C, we write φ ≤ ψ φ < ψ when the indicated inequality holds at each point of −r, 0 . Let C {φ ∈ C : φ ≥ 0}, and let ∧ denote the inclusion R n → C −r, 0 , R n by x → x, x θ x, θ ∈ −r, 0 . Denote the space of functions of bounded variation on −r, 0 by BV −r, 0 .
We assume throughout this section that f : In the following, the notation x t 0 φ will be used as the condition of the initial data of 2.1 , by which we mean that we consider the solution x t of 2.1 which satisfies x t 0 θ φ θ , θ ∈ −r, 0 .
To proceed further, we need the following results. Let r r 1 , r 2 , . . . , r n ∈ R n , |r| max i {r i }, and define We write φ φ 1 , φ 2 , . . . , φ n for a generic point of C r . Let C r {φ ∈ C r : φ ≥ 0}. Due to the ecological applications, we choose C r as the state space of 2.1 in the following discussions.
Fix φ 0 ∈ C r arbitrarily. Then we set L t, · D φ 0 f t, φ 0 , where D φ 0 f t, φ 0 denotes the Frechet derivation of f with respect to φ 0 . It is convenient to have the standard representation of L L 1 , L 2 , . . . , L n as where η ij ·, t is continuous from the left in −r j , 0 .
Discrete Dynamics in Nature and Society 5 We make the following assumptions for 2.1 .
h0 If φ, ψ ∈ C , φ ≤ ψ and φ i 0 h3 For each j, for which r j > 0, there exists i such that for all t ∈ R and for positive constant ε sufficiently small, η ij −r j ε, t > 0. h4 The following result was established by Wang et al. 23 .
The following definition and results are useful in proving our main result.
Let A a ij n×n be an n × n matrix, and let P 1 , . . . , P n be distinct points of the complex plane. For each nonzero element a ij of A, connect P i to P j with a directed line P i P j . The resulting figure in the complex plane is a directed graph for A. One says that a directed graph is strongly connected if, for each pair of nodes P i , P j with i / j, there is a directed path connecting P i and P j . Here, the path consists of r directed lines.

Lemma 2.3 see 24 .
A square matrix is irreducible if and only if its directed graph is strongly connected. We now consider the following delay differential system: with initial conditions System 2.9 always has a trivial equilibrium A 0 0, 0 . If kβe −mτ > au, then system 2.9 has a unique positive equilibrium The characteristic equation of system 2.9 at the equilibrium A 0 takes the form Noting that Kuang 26 , we see that the equilibrium A 0 is locally asymptotically stable for all τ > 0. If kβe −mτ > au, then A 0 is unstable for all τ > 0.
The characteristic equation of system 2.9 at the positive equilibrium A * is of the form Discrete Dynamics in Nature and Society 7 note that Hence, if kβe −mτ > au, the positive equilibrium A * is locally stable when τ 0; if kβe −mτ < au, A * is unstable when τ 0. It is easy to show that

2.18
If kβe −mτ > au, then by Theorem 3.4.1 in the work of Kuang 26 , we see that the positive equilibrium A * is locally asymptotically stable for all τ > 0. If kβe −mτ < au, then A * is unstable for all τ > 0.
Lemma 2.5. For system 2.9 , one has the following.
Proof. We represent the right-hand side of 2.9 by f t, x t f 1 t, x t , f 2 t, x t and set L t, · D φ f t, φ .

2.19
By a direct calculation we have

2.20
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 A t takes the form Clearly, the matrix A t is irreducible for each t ∈ R.
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 D C τ converges to some single equilibrium. However, system 2.9 has only two equilibria: A 0 and A * . Note that if kβe −mτ > au, then the positive equilibrium A * is locally stable and the equilibrium A 0 is unstable. Hence, any solution starting from D converges to A * u * 1 , u * 2 if kβe −mτ > au. Using a similar argument one can show the global stability of the equilibrium A 0 when kβe −mτ < au. This completes the proof.

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
R 0 is called the basic reproduction ratio of system 1.3 . It is easy to show that if R 0 > 1, system 1.3 has a virus-infected equilibrium E * x * , y * , v * , where The characteristic equation It is easy to show that p 1 > 0, p 0 q 0 au−kβe −mτ . If R 0 < 1, then the infection-free equilibrium E 0 of system 1.3 is locally asymptotically stable when τ 0. If iω ω > 0 is a solution of 3.5 , by calculating, we have Note that If R 0 < 1, then p 2 0 − q 2 0 > 0. Therefore, 3.6 has no positive roots. Accordingly, if R 0 < 1, the infection-free equilibrium E 0 of system 1.3 is locally asymptotically stable; if R 0 > 1, 3.6 has at least a positive real root. Accordingly, E 0 is unstable.
The characteristic equation of system 1.3 at the virus-infected equilibrium E * x * , y * , v * takes the form

3.11
By the Hurwitz criteria, all roots of 3.10 have only negative real parts. If iω ω > 0 is a solution of 3.8 , separating real and imaginary parts, it follows that

3.12
Squaring and adding the two equations of 3.12 , we derive that 3.14 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 R 0 > 1, the virus-infected equilibrium E * of system 1.3 exists and is locally asymptotically stable. Based on the discussions above, we have the following result. i If R 0 < 1, the infection-free equilibrium E 0 λ/d, 0, 0 is locally asymptotically stable. If ii If R 0 > 1, the virus-infected equilibrium E * x * , y * , v * is locally asymptotically stable.

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 .
then the virus-infected equilibrium E * x * , y * , v * of system 1.3 is globally asymptotically stable.
Proof. Let x t , y t , v t be any positive solution of system 1.3 with initial condition 1.4 . Let

4.1
Now we claim that U 1 V 1 x * , U 2 V 2 y * , and U 3 V 3 v * . It follows from the first equation of system 1.3 thaṫ By comparison we derive that Hence, for ε > 0 sufficiently small there exists a T 1 > 0 such that if t > T 1 , x t ≤ M x 1 ε. We therefore derive from the second and the third equations of system 1.3 that, for t > T 1 τ, v t ky t − uv t .

12
Discrete Dynamics in Nature and Society Consider the following auxiliary equations:

4.5
Since R 0 > 1, by Lemma 2.5 it follows from 4.5 that

4.6
By comparison, we obtain that

4.7
Since these inequalities are true for arbitrary ε > 0, it follows that Hence, for ε > 0 sufficiently small, there is a For ε > 0 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 ε > 0 sufficiently small, we conclude that Hence, for ε > 0 sufficiently small, there is a Discrete Dynamics in Nature and Society

13
For ε > 0 sufficiently small, we derive from the second and the third equations of system 1.3 that, for t > T 3 τ, v t ky t − uv t .

4.12
Consider the following auxiliary equations:

4.13
Since H1 holds, by Lemma 2.5, it follows from 4.13 that

4.14
By comparison we derive that

4.15
Since these two inequalities hold for arbitrary ε > 0 sufficiently small, we conclude that

4.16
Therefore, for ε > 0 sufficiently small, there is a For ε > 0 sufficiently small, it follows from the first equation of system 1.3 that, for t > T 4 ,ẋ

4.17
A comparison argument shows that

4.24
Since this is true for arbitrary ε > 0, we derive that

4.25
Hence, for ε > 0 sufficiently small, there is a T 7 ≥ T 6 such that if t > T 7 , x t ≥ N x 2 − ε. For ε > 0 sufficiently small, it follows from the second and the third equations of system 1.3 that, for t > T 7 τ,

4.26
Since H1 holds, by Lemma 2.5 and a comparison argument, it follows from 4.26 that

4.27
Since these two inequalities hold for arbitrary ε > 0 sufficiently small, we conclude that

4.28
Therefore, for ε > 0 sufficiently small, there exists a T 8 ≥ T 7 τ such that if t > T 8 ,

4.29
It is readily seen that Hence, for ε > 0 sufficiently small, there is a T 1 > 0 such that if t > T 1 , x t ≤ λ/d ε. We derive from the second and the third equations of system 1.3 that for t > T 1 τ,

4.38
Consider the following auxiliary equation:

4.39
If R 0 < 1, then by Lemma 2.5 it follows from 4.37 and 4.39 that By comparison, we obtain that Therefore, for ε > 0 sufficiently small, there is a T 2 > T 1 τ such that if t > T 2 , y t < ε, v t < ε. It follows from the first equation of system 1.3 that for t > T 2 , By comparison, we derive that Letting ε → 0, it follows that This together with 4.37 yields This completes the proof.

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 treatmentfree followup. While the onset of therapy and viral levels decline rapidly, the virus returns as soon as the drug is withdrawn see Table 1 . 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 forṁ v t 1 − n 2 ky t − uv t .

5.1
Clearly, if n 1 n 2 0, 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.  Before the therapy, that is, n 1 n 2 0, by a direct calculation, we have the basic reproduction ratio R 0 ≈ 1.33, and system 1.3 has a virus-infected equilibrium E * . Clearly, H1 holds. By Theorem 4.1, we see that the virus-infected equilibrium E * of system 1.3 is globally asymptotically stable. Numerical simulation illustrates the previous result see Figure 1 .
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 .