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 τ 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.


Introduction
Recently, a hepatitis B virus (HBV) model with time delay that was proposed and investigated in the literature [1][2][3][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 [1].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. > 0 is an incidence rate coefficient describing the infection process.The initial conditions for the system (1) are where  0 ,  0 , and  0 are nonnegative functions.Based on some observations of virus particles , the system (1) is simplified in [1] 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   = (0, 0, 0)  and the infected equilibrium  * = ( * ,  * ,  * )  , where The following results, Theorems 1 and 2, come from [1].
The initial conditions for the system (5) are where It is straightforward to show the following.Lemma 3. The solution of (5) with an initial condition (6) is nonnegative for all  ≥ 0.
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][6][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. [8].To verify the obtained results, some numerical simulations are included in Section 4. The paper ends with a brief discussion.

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   (0, 0, 0) for all nonnegative parameters.The characteristic equation of ( 5) at   is Hence,   is a saddle with dim   (  ) = 1, dim   (  ) = 2 for  0 < 1;   (  ) and   (  ) are the local unstable and stable manifolds of   , respectively.  is locally asymptotically stable for  0 < 1.In fact,   is globally asymptotically stable for  0 < 1, see [1,4].Now, we will investigate the stability of the chronic infected equilibrium  * .Linearizing the system (5) whose characteristic equation reads as For  = 0, 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  = 0. We now give a definition, which can be found in [2,9].
From Lemmas 5 and 6, we can have the following lemma.

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  > 0.
(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  = 0 such that the characteristic equation (10) has a pair of purely imaginary roots  0 .
Then, one turns to an investigation of local stability of the chronic infected equilibrium  * in the case of  0 < 1. Theorem 8.For  0 < 1 holds, there exists a sequence of values for : such that (10) has a pair of purely imaginary roots  0 when  =   ,  = 0, 1, 2, . ... 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  = 0. Next, we will show that there is a unique pair of purely imaginary roots  0 ( 0 > 0) for characteristic equation (10).
Assume that for some  > 0,  ( > 0) is a root of ( 10), which implies that Note that  > 0,  > 0 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  0 satisfying (26).From (26), we get the corresponding    > 0 such that (26) has a pair of purely imaginary roots Therefore, by using Rouché's theorem [3], there is a unique positive   =    + (1/) satisfying (26), that is, the characteristic equation ( 10) has a pair of purely imaginary roots of the form ± 0 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  1 =  + 2 + ,  2 = (2 +  + ) + , and  3 = ( + ), and (28) can be written as follows: Therefore, 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 [3].Therefore, the transversality condition holds, and the conditions for Hopf bifurcation [11] are then satisfied at  =   .In conclusion, we have the following stability and bifurcation results to (5).
Theorem 10.The properties of the Hopf bifurcation are determined by the values in (75).
In Figure 4, we try to reflect the changes of stability of chronic infected equilibrium  * as  increases from 0 to 9. In Figure 4, each vertical blue strip corresponds to component of (), (), and () at each , respectively.From Figure 3, we see that if  ∈ (0, 8.1786 4293), approximately, the vertical amplitudes of (), (), and () are as small as a point, suggesting that  * is asymptotically stable; if  increases, the vertical amplitudes of (), (), and () will become larger and larger, showing that  * becomes more and more unstable.In particular, if  = 8.1787, points are well distributed around the positive equilibrium, and their amplitudes are all equivalent.This shows that periodic solution near positive equilibrium may occur.

Conclusions
In this paper, a delay differential equation of hepatitis B virus infection is formulated.We analyzed the stability of the equilibria; a sufficient condition was given to guarantee the global stability of the origin.Local stability of the chronic infected equilibrium was considered.By choosing time delay  as a bifurcated parameter, a sufficient condition has been presented for checking the existence of the Hopf bifurcation.The explicit formulae determining the direction, stability, and other properties of bifurcating periodic solutions were given  by using the normal form theory and the center manifold theorem.Some numerical simulations were performed to support the analytical results found.Although bifurcations in a population dynamics with delay has been investigated by many researchers.However, to the best of our knowledge, there are few papers on the bifurcation of delay differential equation of hepatitis B virus infection dynamics with delay.