Analysis of a New Delayed HBV Model with Exposed State and Immune Response to Infected Cells and Viruses

We propose a comprehensive delayed HBV model, which not only considers the immune response to both infected cells and viruses and a time delay for the immune system to clear viruses but also incorporates an exposed state and the proliferation of hepatocytes. We prove the positivity and boundedness of solutions and analyze the global stability of two boundary equilibria and then study the local asymptotic stability and Hopf bifurcation of the positive (infection) equilibrium and also the stability of the bifurcating periodic solutions. Moreover, we illustrate how the factors such as the time delay, the immune response to infected cells and viruses, and the proliferation of hepatocytes affect the dynamics of the model by numerical simulation.


Introduction
Hepatitis B virus (HBV) has become one of the serious infectious diseases threatening global human health, which can cause chronic liver infection and further result in liver inflammation, fibrosis, cirrhosis, or even cancer [1]. Each year more than 1 million people die of end-stage liver diseases like cancer due to the HBV infection [2].
Mathematical modeling and analysis of the dynamics of such infectious viruses as HBV play important roles in understanding the factors that govern the infectious disease progression and offering insights into developing treatment strategies and guiding antiviral drug therapies [3]. So far, there have been plenty of mathematical models proposed to describe and analyze virus infection, immune responses, and antiretroviral treatment [4][5][6][7][8][9][10].
Among these works, the development of virus models with immune responses is gaining much attention [3,11,12]. The immune system is essential in controlling the level of virus reproduction in terms of the strength of the Cytotoxic T Lymphocyte (CTL) response. A small change of the CTL response may have a large effect on virus production and infected cells load. As to this aspect, typical work can be summarized as follows. Chen et al. [12] indicated that the immunity system can not only clear free viruses but also kill infected cells. Elaiw and AlShamrani [3] proposed a fourdimensional model with humoral immunity response and general function and analyzed the global asymptotic stability of all equilibria based on the general function. However, both models in [3,12] do not consider time delay. In order to characterize the time of a body's immune response after the virus infection of target cells, time delay has been taken into account [13][14][15][16]. For example, Zhu et al. [16] proposed an HIV infection model with CTL response delay and analyzed the effect of time delay on the stability of equilibria. Besides, a latent period would be necessary to be incorporated into a virus model because when viruses infect a healthy organ like liver, it will not be pathogenetic at once, as it takes about six weeks to six months from the infection to the incidence [17][18][19][20]. For example, Medley et al. [17] proposed an HBV model with an exposed state, namely, infected but not yet infectious. Moreover, it was modeled in [21,22] that the liver can regenerate cells and compensate the lost infected hepatocytes by the proliferation of hepatocytes.
In this paper, we will propose a more comprehensive model than those existing ones, which not only considers the immune response to both infected cells and viruses and a time delay for the immune system to clear viruses but also incorporates an exposed state and the proliferation of hepatocytes. We first discuss the existence of two boundary equilibria and one positive (infection) equilibrium. We then analyze the global stability of the two boundary equilibria, the local asymptotic stability and Hopf bifurcation of the positive equilibrium and also the stability of the bifurcating periodic solutions. Moreover, we perform numerical simulations to illustrate some of the theoretical results we obtain and also illustrate how the factors such as the immune response to infected cells and viruses and the proliferation of hepatocytes affect the dynamics of the model under time delay. The paper is structured as follows. In Section 2, a delayed mathematical model is proposed, and the positivity and boundedness of solutions, existence of two boundary equilibria, and one positive equilibrium are discussed, followed by the global stability analysis of these two boundary equilibria and the local asymptotic stability and Hopf bifurcation of the positive equilibrium in Section 3. The stability of the bifurcating periodic solutions is studied in Section 4. In Section 5, some numerical simulations and discussions are given. Finally, a conclusion is given in Section 6.

Mathematical Model
Wang et al. [23] proposed a virus infection model of four-dimensional equations with delayed humoral immune response, which, however, does not involve an exposed state and consider the proliferation of hepatocytes. Although it considers the immune response to viruses, it does not involve the immune response to infected cells.
Based on this model, we propose a new and comprehensive HBV model, which not only considers the immune response to both infected cells and viruses and a time delay for the immune system to clear viruses but also incorporates an exposed state and the proliferation of hepatocytes. To better understand our model, we illustrate its mechanism in Figure 1.
The model is then given as follows: where , , , V, , and denote the number of uninfected cells, exposed cells, infected cells, free viruses, CTLs, and alanine aminotransferases (ALT), respectively. The parameter represents the natural production rate of uninfected cells. is a new term which is introduced to represent the proliferation of hepatocytes, where is the proliferation rate. Parameters , (and the following) 1 , , 4 , and 7 represent the natural death rate of uninfected cells, exposed cells, infected cells, free viruses, CTLs, and ALT, respectively. represents the infection rate from uninfected cells to exposed cells and 2 the transfer rate from exposed cells to infected cells. The production rate of free viruses from infected cells is denoted by , and the production rate of CTLs by 3 . 5 represents the production rate of ALT from the extrahepatic tissue and 1 6 the production rate of ALT when the infected hepatocytes are killed by CTL. The immunityinduced clearance for infected cells is modeled by a term 1 , where 1 represents the clearance rate of infected cells. Similarly, the immunity-induced clearance for free viruses is modeled by 2 V , where 2 represents the clearance rate of free viruses. is time delay. All the parameters in this paper are positive and > . For convenience, we define new parameter = − .
Based on integral operator spectral radius, the basic reproductive number is 0 = ( −1 ), where Hence, we have the basic reproductive number being 0 = 2 0 / 1 ( 1 + 2 ). The equilibrium without immune response is 11 = ( 1 , Similarly, we have the basic reproductive number is

Global Stability Analysis of the Two Boundary Equilibria.
In this section, we will employ the direct Lyapunov method and LaSalle's invariance principle to establish the global asymptotic stability of the two boundary equilibria. In this section, we will discuss the local asymptotic stability and Hopf bifurcation of the positive equilibrium 22 . The characteristic equation of system (1) at 22 is as follows: Define BioMed Research International 5 10 = 12 + 8 9 , Then the characteristic equation ( ; ) above becomes When = 0, (9) further becomes Using the Routh-Hurwitz criterion [23], we obtain the following lemma. Proof. By the Routh-Hurwitz criterion, if the four conditions are satisfied, then all roots of (10) have negative real parts.
Therefore, the positive equilibrium 22 is locally asymptotically stable when = 0. For more details, we refer the readers to [25,26]. From Lemma 3, we know that all roots of ( ; ) lie to the left of the imaginary axis when = 0. However, with increasing from zero, some of its roots may cross the imaginary axis to the right. In this case, there are some roots having positive real parts, and therefore the equilibrium 22 becomes unstable. Next, we will discuss the stability of system (1) at 22 when > 0.

Theorem 6.
If 0 is a simple root of ( ) = 0, then there is a Hopf bifurcation for the system as increases past (0) 0 . See Appendix C for proof.

Stability of the Bifurcating Periodic Solutions
In this section, we will continue to derive the explicit formulas for determining the stability, direction, and other properties of the Hopf bifurcation at a critical value (0) 0 by means of the normal form and the center manifold theory [27].
First, we make the following hypotheses.
(2) The remaining roots of (22) have strictly negative real parts.
In the following, we will compute the coefficients, 20 , 11 , 02 , and 21 , using the method given in [27]. The detailed computation of 20 , 11 , 02 , and 21 is presented in Appendix E.

Simulation and Discussions
In this section, we will numerically illustrate the theoretical results obtained above and also discuss how the factors such as the immune response to infected cells and viruses and the proliferation of hepatocytes affect the dynamics of the model under time delay.
For the following simulations, we choose the parameter values for system (1) as follows:  Figure 2 indicates that a stable limit cycle is obtained as expected and Figure 3 indicates the state dynamics of uninfected cells, exposed cells, infected cells, free viruses, CTLs, and ALT, which are periodically oscillating.
When < 0.041, the bifurcating periodic solutions are unstable. For example, when = 0.03, the simulation result is shown in Figures 4 and 5. From Figures 4 and 5, we know that the positive equilibrium 22 is asymptotically stable and the system will converge to 22 .
With the increasing of time delay ( ), the radius of limit cycle will increase. The simulation result is shown Figure 6.

The Immune Response to Infected Cells.
Here we will investigate the effect of the immune response to infected cells  Figure 7 when 1 = 0.01, we can see that, with the decrease of 1 , the stable periodic solution becomes unstable, that is, asymptotically stable.

The Immune Response to Viruses.
We continue to investigate the effect of the immune response to viruses on the model dynamics under time delay. When we change 2 = 0.1963 to 2 = 0.001 by fixing other values given in (31), we obtain a simulation result at = 0.05, illustrated in Figure 8. Similarly, comparing Figures 2 and 8, we can see that, with the decrease of 2 , the stable periodic solution also becomes unstable, that is, asymptotically stable. We further can see that the effect of the immune response to infected cells on the model dynamics is similar to that of the immune response to viruses.

Proliferation of Hepatocytes.
Then we investigate the effect of proliferation of hepatocytes on the model dynamics under time delay. For this, we still keep = 0.05 and change the value of parameter . When we change = 0.6933 to = 0.001 by fixing other values given in (31), we obtain a simulation result at = 0.05, illustrated in Figure 9. Figure 9 shows that when = 0.001, the bifurcating periodic solution is stable, compared with Figure 2 when = 0.6933. We also try other values of parameter rand obtaining similar results. Thus, we can see parameter has a small effect on the model dynamics, which reflect in periodicity and the positive equilibrium 22 .

Conclusions
In this paper, we consider a comprehensive delayed HBV model. Different from other existing models, our model not only considers the immune response to both infected cells and viruses and a time delay for the immune system to clear viruses but also incorporates an exposed state and the proliferation of hepatocytes. We then prove the positivity and boundedness of solutions and analyze the global stability of two boundary equilibria and investigate the local asymptotic stability and Hopf bifurcation of the positive (infection) equilibrium and also the stability of the bifurcating periodic solutions. We also numerically illustrate the Hopf bifurcation and the stability of the bifurcating periodic solutions.
Moreover, we numerically illustrate how the factors such as the time delay, the immune response to infected cells and viruses, and the proliferation of hepatocytes affect the dynamics of the model, which shows that the former two factors have a big effect on the model dynamics, while the latter one does not have a big effect.
Calculating the derivative of | 0 ( )| 2 to , we have

(C.6)
Calculating the derivative of both sides of (C.1) to , we havė Then we have (C.8) ] . As 0 is a simple root of ( ) = 0, we know( 0 ) ̸ = 0. From (C.11), we further know we obtain that the roots of (9) have positive real part when ∈ [0, (0) 0 ), which contrasted with Theorem 5. Hence, we can see that When = (0) 0 , except for the pair of purely imaginary roots, the remaining roots of ( ; ) have strictly negative real parts, so the system has Hopf bifurcation.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.