Research Article | Open Access
Stability and Bifurcation Analysis for a Delay Differential Equation of Hepatitis B Virus Infection
The stability and bifurcation analysis for a delay differential equation of hepatitis B virus infection is investigated. We show the existence of nonnegative equilibria under some appropriated conditions. The existence of the Hopf bifurcation with delay at the endemic equilibria is established by analyzing the distribution of the characteristic values. The explicit formulae which determine the direction of the bifurcations, stability, and the other properties of the bifurcating periodic solutions are given by using the normal form theory and the center manifold theorem. Numerical simulation verifies the theoretical results.
Recently, a hepatitis B virus (HBV) model with time delay that was proposed and investigated in the literature [1–4] caught the attention of a lot of mathematicians. In practice, the HBV model has suffered time delay caused by the HBV incubation period, which varies from 45 to 180 days, and the delay in viral shedding which both suggest that viral production delay may significantly impact infection dynamics . Precisely, the HBV model with time delay reads as the following: where and represent the number of uninfected cells and infected cells, respectively. represents the number of exposed cells, that is, the cells that have acquired the virus but are not yet producing new virions. denotes the number of free virions. is the time delay for virion production. Here, the positive constant is the rate at which new uninfected live cells are generated. The positive constant is the per capita death rate of uninfected live cells. Infected live cells are killed by immune cells at rate and produce free virions at rate , where is what so-called “burst” constant. Free virions are cleared by lymphatic and other mechanisms at rate , where is a constant. is an incidence rate coefficient describing the infection process. The initial conditions for the system (1) are where , , and are nonnegative functions. Based on some observations of virus particles , the system (1) is simplified in  as the following:
A direct computation shows that the basic infection reproduction number for the system (2) is
For the sake of simplicity, let , and the system (3) is equivalent to the following system: which has two equilibria: the infection-free equilibrium and the infected equilibrium , where
Theorem 1. If , the infection-free equilibrium of the system (5) is globally asymptotically stable.
Theorem 2. If , the chronic infected equilibrium of the system (5) is locally asymptotically stable.
The initial conditions for the system (5) are where .
It is straightforward to show the following.
It is well known that the studies on the dynamical systems not only include the discussion of stabilities, attractivity, and persistence, but also include many dynamical behaviors such as periodic solutions, bifurcations, and chaos. Particularly, the properties of periodic solutions appearing through the Hopf bifurcation in delayed systems are of great interest [5–7]. In the present paper, our main objective is to investigate the bifurcation phenomena of the modified hepatitis B virus (HBV) model with time delay .
This paper is organized as follows. In Section 2, by analyzing the characteristic equation of the linearized system of the system (5) at the equilibria, we discuss the stability of the origin and the positive equilibrium and the existence of the Hopf bifurcations occurring at the chronic infected equilibrium. In Section 3, the formulae determining the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions on the center manifold are obtained by using the normal form theory and the center manifold theorem due to Hassard et al. . To verify the obtained results, some numerical simulations are included in Section 4. The paper ends with a brief discussion.
2. Stability of Equilibria and Existence of the Hopf Bifurcation
In this section, we will investigate the stability of the equilibria and the existence of the Hopf bifurcations occurring at the chronic infected equilibrium. Then, it is easy to check that the system (5) has an equilibrium for all nonnegative parameters. The characteristic equation of (5) at is
Now, we will investigate the stability of the chronic infected equilibrium . Linearizing the system (5) at yields the following linear system: whose characteristic equation reads as with
For , characteristic equation (10) reduces to
By the Routh-Hurwitz criterion, we know that all the roots of (12) have negative real parts, that is, the chronic infected equilibrium is locally asymptotically stable for . We now give a definition, which can be found in [2, 9].
Definition 4. The equilibrium is called asymptotically stable if there is an such that
where is the solution of the system (5) with an initial condition (6).
Song et al.  investigated the distribution of roots of the following equation: where and . When , (15) reduces to
Obviously, is a root of (16) if and only if satisfies
Separating the real and the imaginary parts, we have
Squaring the two equations and adding them give where , , and . Song et al.  obtained the following results on the distribution of roots of (15) and (19).
Lemma 5. For the polynomial equation (19),(i)if , then (19) has at least one positive root,(ii)if and , then (19) has no positive roots,(iii)if and , then (19) has positive roots if and only if and , where
Lemma 6. For the transcendental equation (19),(i)if and , then all roots with positive real parts of (19) have the same sum as those of the polynomial equation (16), for all ,(ii)if or , and , then all roots with positive real parts of (19) have the same sum as those of the polynomial equation (16), for .
Lemma 7. (i) The chronic infected equilibrium of the system (5) is absolutely stable if and only if the equilibrium of the corresponding ordinary differential equation (ODE) system is asymptotically stable, and the characteristic equation (10) has no purely imaginary roots for any .
(ii) The chronic infected equilibrium of the system (5) is conditionally stable if and only if all roots of the characteristic equation (10) have negative real parts at such that the characteristic equation (10) has a pair of purely imaginary roots .
Then, one turns to an investigation of local stability of the chronic infected equilibrium in the case of .
Theorem 8. For holds, there exists a sequence of values for : such that (10) has a pair of purely imaginary roots when , . That is, the chronic infected equilibrium of the system (5) is conditionally stable.
Proof. From the above arguments, we know that all roots of characteristic equation (10) have negative real parts at . Next, we will show that there is a unique pair of purely imaginary roots for characteristic equation (10).
Assume that for some , is a root of (10), which implies that
Note that , because of the positivities of the parameters , , , , and the properties , .
Separating the real and imaginary parts and using Euler’s formula give which is equivalent to where
In order to solve (24), we first consider the following: where
By Lemma 5, there is a unique positive satisfying (26). From (26), we get the corresponding such that (26) has a pair of purely imaginary roots
Therefore, by using Rouché’s theorem , there is a unique positive satisfying (26), that is, the characteristic equation (10) has a pair of purely imaginary roots of the form as . By Lemma 7, we complete the proof of Theorem 8.
Next, we turn to show that
This will signify that there exists at least one eigenvalue with positive real part . We first consider the following:
Differentiating (30) with respect to , we have
For the sake of simplicity, let , , and , and (28) can be written as follows:
This root of characteristic equation (9) crosses the imaginary axis from the left to the right as continuously varies from a number less than to one greater than again by Rouché’s theorem . Therefore, the transversality condition holds, and the conditions for Hopf bifurcation  are then satisfied at . In conclusion, we have the following stability and bifurcation results to (5).
Theorem 9. Suppose that holds. Then, for each fixed , there exists a sequence of values for : such that the positive equilibrium is asymptotically stable when , and unstable when . Furthermore, (5) undergoes a Hopf bifurcation at when , where is defined by (28).
3. Properties of the Hopf Bifurcations
In this section, we will study the properties of the Hopf bifurcations by using the normal theory and the center manifold theorem due to Hassard et al. . Let , , , , , and . Then, is the Hopf bifurcation value of system. We drop the tildes for simplification of notations, then the system (5) becomes an functional differential equation in as where is the Banach space of continuous functions mapping the interval into , , for and , are read, respectively, as Obviously, is a continuous linear function mapping into , by the Riesz representation theorem, there exists a matrix function of bounded variation for , such that
In fact, we can choose where denotes the Dirac delta function.
If is any given function in and is the unique solution of the linearized equation of (35) with initial function at zero, then the solution operator is defined by , for all . It is obvious that , , is a strongly continuous semigroup of linear transformation on and the infinitesimal generator of , is For , the space of functions mapping the interval into which have a continuous first derivative also defines
Then, system (35) is equivalent to
For , the space of functions mapping interval into the three-dimensional row vectors which have continuous first derivative defines and a bilinear inner product where . Then, and are adjoint operators. By the discussion in Section 2, we know that are eigenvalues of . Hence, they are also eigenvalues of . We first need to compute the eigenvectors of and corresponding to and , respectively. Suppose that is the eigenvectors of corresponding to , then . Then, from the definition of and (36), (38), and (39), we have
For , then we obtain
Similarly, we can obtain the eigenvector of corresponding to , where
We need to determine the value of to ensure that . By (44), we have
Therefore, we can choose as
Next, we will compute the coordinate to describe the center manifold at . Let be the solution of (42) when . Define
On the center manifold , we have where and are local coordinates for the center manifold in the direction of and . Note that is real if is real. We only consider real solutions. For solution of (42), since , we have
For convenience of calculation, let
Taking (37) into account, we can obtain that
Straightforward calculation leads to
Comparing the coefficients with (54), we have