Abstract
This article aims to study a hepatitis B virus (HBV) infection model incorporating two nonlinear incidences and spatial diffusion in capsids, virus, and cytotoxic T lymphocyte (CTL) immune response. Three equilibria which are infection-free, immune-free, and infection with CTL immunity are calculated under rational assumptions. Furthermore, two reproduction numbers are verified to assert the global stability of the HBV model. In the end, the theoretical results on HBV dynamics are further illustrated by performing numerical simulations.
1. Introduction
Hepatitis B is a disease induced by HBV attacking the hepatocytes [1–3], which has attracted worldwide attention. Mathematical models play an important role in practical applications, such as the Kopel model [4], Hindmarsh–Rose model [5], and Lotka–Volterra model [6], especially in viral infection mechanism, trends, and control strategy of infectious diseases. Recently, Manna and Chakrabarty [7] considered capsids into the HBV infection model and discussed the global properties. It was supposed in above models that the cells and virus are fully mixed in space. Besides, the effect of spatial heterogeneity is neglected. Therefore, the movement of cells and virus is essential. Recently, mathematical models with reaction diffusion have been designed to study its impact of the mobility of cells and viruses [8–10].
Compared with virus-to-cell infection, an available way of virus transmission is cell-to-cell transmission, which is mentioned in [11, 12]. Virus models considering two infection modes have been studied [13, 14]. Meanwhile, the time delay cannot be ignored in numerous biological phenomena. On the basis of the model in [15], Manna et al. [16, 17] introduced HBV models with capsids in which the cell-to-cell transmission has not been included. As discussed in [18], a delayed HBV model presented by Guo et al. [19] has neglected CTL immune response and cell-to-cell transmission. Since then, many complicated dynamical behaviors about delayed HBV infection models are revealed in [20–28]. Thus, it is necessary to introduce a diffused HBV model with two viral infection modes.
Motivated by Shu et al. [14], Manna et al. [8, 15], Connell and Yang [29], Yang and Xu [30], we establish the following diffused HBV model:with initial conditionsand homogeneous Neumann boundary conditionswhere is a bounded domain in with smooth boundary and denotes the outward normal derivative on . is the Laplacian operator where . , , , , and denote the densities of the uninfected hepatocytes, infected hepatocytes, intracellular HBV DNA-containing capsids, virus, and CTL cells at position and at time , respectively, and other parameters are described in Table 1.
We assume that the incidences and satisfy the following conditions:
is continuously differentiable; , ; or .
and , .
Specifically, the main contributions of this work are as follows. Firstly, the novelty of this model is that it includes two viral transmission modes, two types of delays, and spatial diffusion. Meanwhile, the global stability of feasible equilibrium basis of is investigated. Secondly, to understand the viral pathogenesis and disease diffusion better, the spatial effects and Fickian diffusion for capsids, virus, and CTL cells are introduced. Compared with existing works [31], it is more general to consider spatial diffusion in this paper. Thirdly, the cell-to-cell transmission in the HBV model helps to increase . So, the effect of cell-to-cell transmission is assumed as a key factor.
This paper is organized as follows. In Section 2, we study the existence of feasible equilibria which depend on two reproduction numbers. In Section 3, the global stabilities of three equilibria are established. In Section 4, numerical simulations are presented to validate the theoretical results. In Section 5, a summary is given.
2. Positivity, Boundedness, and Equilibrium
Let be the Banach space with the supremum norm. For , define , which is a Banach space of continuous functions from into with the norm . If and , then is defined by .
Theorem 1. For any given initial condition satisfying (2), there exists a unique nonnegative solution of models (1)–(3) defined on and this solution remains bounded for all .
Proof. For any and , we define byAfter that, we rewrite models (1)–(3) as follows:where , and . Obviously, is locally Lipschitz in . From [32–34], we deduce that model (4) has a unique local solution on , where is the maximal existence time for solution of model (4).
It is obvious that a lower solution of models (1)–(3) is . So, we have , and .
Letand then we can obtainwhere . Therefore,and for , , and are bounded.
Using the boundedness of and models (1)–(3), we obtainIf be a solution to the following equation:Then, we have By the comparison principle [35], . Hence,Similarly, we have
Summarizing the inference above and applying [36], we have shown that , and are bounded on . Therefore, by the standard theory for semilinear parabolic systems [37], we have .
Clearly, model (1) always has an infection-free equilibrium , where . Denotewhich is the basic reproductive number of model (1).
Any equilibrium of model (1) satisfies the following equations:If , from model (6), we getDefineThen, it follows from that and . This, together with the expression of in (12), yieldsThen, if , which implies that there exists such that . Hence, model (1) has a unique immune-free equilibrium , whereIf , a short calculation shows thatDefine yields . Thus, we haveSo, there exists a unique that satisfies .
Denotewhich is the CTL immunity reproduction number. Further, from model (6), we can obtainThus, if , model (1) has a unique infection equilibrium with CTL immunity , where
3. Stability Analysis
For convenience, for any solution of model (1), we let
Theorem 2. If , then the infection-free equilibrium is globally asymptotically stable.
Proof. Define a Lyapunov functionalCalculating the time derivative of along the solution, we obtainCondition and the expression of given in (12) imply thatUsing the divergence theorem, we getThus, we haveTherefore, . . From LaSalle’s invariance principle [38], is globally asymptotically stable when .
Assume that and satisfy
Theorem 3. If , , and – hold, then the immune-free equilibrium is globally asymptotically stable.
Proof. Let , and we have , . Define a Lyapunov functionalCalculating the time derivative of along the solution, we obtainBy divergence theorem, we getThus, we haveTherefore, using condition , we have . . From LaSalle’s invariance principle [38], is globally asymptotically stable when and .
Theorem 4. If and hold, then the infection equilibrium with CTL immunity is globally asymptotically stable.
Proof. Define a Lyapunov functionalSimilarly, we haveTherefore, . . From LaSalle’s invariance principle [38], is globally asymptotically stable when .
4. Numerical Simulations
In this section, we numerically validate the results obtained in Section 3. Let and . One can easily verify that and satisfy . Following [7], we select , and . Moreover, , and are chosen as free parameters. In Figures 1–3, part labels (a), (b), (c), (d), and (e) denote time-series figures of , , , , and .
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and also have two impacts on the dynamical behavior of the HBV model. For one thing, we can observe from Figure 4 that and become large enough when and approach 0. For another, by the expressions of and , as we can see in Figure 5, and become smaller as and increase. Therefore, the strategy of controlling HBV should refer to drugs that can lengthen the two delays.
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5. Discussion
This paper investigated a diffused HBV model with two time delays and CTL immune response, in which two infection modes and are considered. Under on two general incidence functions, the global dynamics of model (1) are established by basic reproduction numbers and . If , is globally asymptotically stable; if and , is globally asymptotically stable; if , is globally asymptotically stable. and are irrelevant to , , and , Meanwhile, two delays and have no influence on the global stability of , , and . Hence, the results obtained extend the work in [7].
Numerical simulations graphically show the stability analysis for model (1). It is worth pointing out that and are decreasing functions on death rates and , delaying and . Therefore, for virus clearance, and exert a prominent role. This has brought some effects to explore new drugs to stop the virus infection or inhibit virus production. Since and lead to the virus infection, ignoring one of them would contribute to an underestimated basic reproduction number. Based on this, the strategy of prevention and treatment cannot eliminate virus infection. As discussed in [28, 39, 40], applying these methods to fractional-order or age-structured HBV models is our future work.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
HM proved the stability of equilibria. MJ was responsible for validation.
Acknowledgments
This work was supported by the NSFC (no. 11901363), Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (no. 2021L279), and Youth Research Fund for Shanxi Basic Research Project (nos. 2015021025 and 202103021224291).