Abstract
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.
1. 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 [4–10].
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 four-dimensional 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 [13–16]. 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 [17–20]. 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.
2. 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 , , , , , 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) , , , and 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 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 . represents the production rate of ALT from the extrahepatic tissue and the production rate of ALT when the infected hepatocytes are killed by CTL. The immunity-induced clearance for infected cells is modeled by a term , where represents the clearance rate of infected cells. Similarly, the immunity-induced clearance for free viruses is modeled by , where 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 .
2.1. Positivity and Boundedness of Solutions
In this subsection, we prove the positivity and the boundedness of solutions of system (1).
We denote , , , , , , . From the first equation of system (1), we have ; therefore, for if . Next, we consider the second, third, and fourth equation in system (1) as a nonautonomous system for :
Based on Theorem in [24], we have , , if , , .
, we have , if , .
, because , so we have , if .
Hence, the nonnegative is proved. In what follows, we will study the boundedness of solutions. We define as a linear combination of , , , , :
Therefore, we obtain , namely, . So we have Because the boundedness of , .
The boundedness is proved.
2.2. Equilibrium
In this subsection, we study the equilibria of system (1). The method to obtain equilibria is setting and computes the following:
The system (1) has two boundary equilibria (an infection-free equilibrium in which , , and an equilibrium without immune response in which , , , , , ) and a positive (infection) equilibrium in which , , , , , .
The infection-free equilibrium is , where , and , and the basic reproductive number is obtained by the following method.
Based on integral operator spectral radius, the basic reproductive number is , whereHence, we have the basic reproductive number being .
The equilibrium without immune response is , where , , , , and . Similarly, we have the basic reproductive number is at .
The infected positive equilibrium is , where
3. Analysis
3.1. 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.
Theorem 1. The infection-free equilibrium is globally asymptotically stable if and only if .
See Appendix A for proof.
Theorem 2. The equilibrium without immune response is globally asymptotically stable if and only if .
See Appendix B for proof.
3.2. Local Asymptotic Stability and Hopf Bifurcation of the Positive Equilibrium
In this section, we will discuss the local asymptotic stability and Hopf bifurcation of the positive equilibrium .
The characteristic equation of system (1) at is as follows:
Define
Then the characteristic equation above becomesWhen , (9) further becomeswhereUsing the Routh-Hurwitz criterion [23], we obtain the following lemma.
Lemma 3. If (10) satisfies , , and , then the positive equilibrium is locally asymptotically stable when .
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 is locally asymptotically stable when . 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 . 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 becomes unstable. Next, we will discuss the stability of system (1) at when .
We first divide (9) into two parts and obtainSuppose (9) has a purely imaginary root . Substituting into (12) yieldsBy separating the real part and imaginary part, the following real part is obtained:whereLetTherefore, if (9) has a purely imaginary root , it is equivalent to the fact that has a positive real root .
Theorem 4. If has no positive real roots, then the positive equilibrium is locally asymptotically stable for any .
Proof. If has no positive real roots, then obviously (9) has no positive real roots. Therefore, the positive equilibrium is locally asymptotically stable for any .
Substituting into (22), we obtain the real part,and imaginary part,Assuming that has positive real roots, denoted by . As , we then haveLetwhere and .
Therefore, the characteristic equation has a pair of purely imaginary roots . For every integer and , define as the root of (9) near , satisfying and . Then the following theorem is obtained.
Theorem 5. If has some positive real roots, then is locally asymptotically stable for , where
Proof. For , has no positive real roots when , which means that all the roots of (9) have strictly negative real parts when . Therefore, is locally asymptotically stable for .
Theorem 6. If is a simple root of , then there is a Hopf bifurcation for the system as increases past .
See Appendix C for proof.
4. 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 by means of the normal form and the center manifold theory [27].
First, we make the following hypotheses.
(1) Equation (9) has a pair of purely imaginary roots at , where .
(2) The remaining roots of (22) have strictly negative real parts.
(3) is a simple root of .
We use to represent a new bifurcation parameter. Let , and , where . Therefore, system (1) can be written as the following functional differential equation:where
By the Riesz representation theorem [28], there exists a matrix-valued function such thatwhere .
For , we further defineThen system (22) can be written asFor , defineand an inner product of where and . Then and are adjoint operators.
Let and be the eigenvectors of and corresponding to the eigenvalues and , respectively. We choose and as so that is satisfied. We give the detailed computation of (29) in Appendix D.
In the following, we will compute the coefficients, , , , and , using the method given in [27]. The detailed computation of , , , and is presented in Appendix E.
Then the following values can be computed:The signs of , determine the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions, respectively [27]. From (C.11) of Appendix C, we obtain .
Let . We obtain the following theorem.
Theorem 7. Assume the hypotheses (1), (2), and (3) at the beginning of Section 4 hold.
(1) If , then the bifurcating periodic solutions exist for in a -neighborhood.
(2) If , the bifurcating periodic solutions are orbitally asymptotically stable as .
Proof. (1) If , where , and the hypotheses (1), (2), and (3) hold, then, from Theorem 6, we draw the conclusion that the existence and stability of the bifurcating periodic solutions are only determined by .
(2) If , namely, , then there exist stable periodic solutions for in a -neighborhood. So the bifurcating periodic solutions are orbitally asymptotically stable as .
5. 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:
We set the initial values to , , , , , and , where .
5.1. Hopf Bifurcation and the Stability of Periodic Solutions
With the parameter values given in (31), we have the positive equilibrium and the critical time value .
When , we obtain stable bifurcating periodic solutions. For example, when , the simulation result is shown Figures 2 and 3. 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.
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When , the bifurcating periodic solutions are unstable. For example, when , the simulation result is shown in Figures 4 and 5. From Figures 4 and 5, we know that the positive equilibrium is asymptotically stable and the system will converge to .
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With the increasing of time delay , the radius of limit cycle will increase. The simulation result is shown Figure 6.
5.2. The Immune Response to Infected Cells
Here we will investigate the effect of the immune response to infected cells on the model dynamics under time delay. When we change to by fixing other values given in (31), we obtain a simulation result at , illustrated in Figure 7. Comparing Figure 2 when and Figure 7 when , we can see that, with the decrease of , the stable periodic solution becomes unstable, that is, asymptotically stable.
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5.3. 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 to by fixing other values given in (31), we obtain a simulation result at , illustrated in Figure 8. Similarly, comparing Figures 2 and 8, we can see that, with the decrease of , 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.
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5.4. Proliferation of Hepatocytes
Then we investigate the effect of proliferation of hepatocytes on the model dynamics under time delay. For this, we still keep and change the value of parameter . When we change to by fixing other values given in (31), we obtain a simulation result at , illustrated in Figure 9. Figure 9 shows that when , the bifurcating periodic solution is stable, compared with Figure 2 when . We also try other values of parameter and obtaining similar results. Thus, we can see parameter has a small effect on the model dynamics, which reflect in periodicity and the positive equilibrium .
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6. 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.
Appendix
A. The Proof of Theorem 1
We construct a Lyapunov functional as follows:Calculating the derivative of along with the trajectories of system (1), we obtainTherefore, when , holds true. Furthermore, if and only if , and , the Lyapunov functional satisfies . According to LaSalle’s invariance principle [29], the infection-free equilibrium is globally asymptotically stable when . Apparently, when , is unstable.
B. The Proof of Theorem 2
Similarly, we define the following Lyapunov functional:The derivative of along with the trajectories of system (1) can be calculated asTherefore, ensures that holds true. Furthermore, if and only if , , , , , and , the Lyapunov functional satisfies . Using the LaSalle’s invariance principle, we can see that the equilibrium without immune response is globally asymptotically stable.
C. The Proof of Theorem 6
The characteristic equation (9) can be written into the following form:whereand and are continuously differentiable to .
Suppose that one of the roots of (C.1) is , satisfying and for a positive real number .
Let Calculating the derivative of to , we have Then we haveBecause , we haveCalculating the derivative of both sides of (C.1) to , we haveThen we haveSince , we obtainThus, we haveWhen , obviously, we haveAs is a simple root of , we know . From (C.11), we further know . If ; we obtain that the roots of (9) have positive real part when , which contrasted with Theorem 5. Hence, we can see that . When , except for the pair of purely imaginary roots, the remaining roots of have strictly negative real parts, so the system has Hopf bifurcation.
D. The Detailed Computation of (29)
In what follows, we will compute (29).
Substituting into (26), we haveTherefore, is the characteristic equation of .
As , we have whereSo we have
The eigenvectors of areSimilarly,The eigenvectors of areSince we have that
E. The Detailed Computation of , , , and
In the following, we will compute coefficients, , , , and , using the method given in [27].
LetwhereThen we haveWe further obtainUsing the method [27], we obtain the following coefficients:wherewhere
So far, we have obtained , , , and .
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported by National Natural Science Foundation of China (61273226).