Mathematical Analysis of Hepatitis B Virus Dynamics

Abstract

Hepatitis B is a life-threatening liver infection due to the hepatitis B virus. Hepatitis B virus (HBV) infection is one of the predominant public health challenges globally. We develop a deterministic model to understand the underlying dynamics of HBV infection at the population level. The model, which incorporates the vaccination and treatment of individuals, the re-infections of infected classes, is rigorously analyzed to gain insight into its dynamical features. The mathematical analysis reveals that the model exhibits a backward bifurcation due to exogenous re-infection. It is shown that, in the absence of re-infection, the model has a disease-free equilibrium (DFE) which is globally asymptotically stable using Lyapunov function and LaSalle Invariance Principle whenever the associated reproduction threshold is less than unity. Further, the model has a positive unique endemic equilibrium (EEP) which is globally asymptotically stable (GAS) when the associated threshold quantity is greater than one. Next, we incorporate optimal control strategies as vaccination and creating awareness in the model. A system of di erential equations with control variables is considered and Pontryagin's Maximum Principle is applied to characterise the optimal controls. In the optimal control system, the main focus is to minimize the cost of two controls as well as to decrease the disease burden. The numerical simulations indicate that the optimal control strategy is e ective not only to minimize the infection but also the most successful way to control the infection. Furthermore, we have extended the model considering dose-structured vaccination for assessing the impact of vaccines among the population. Here we have analysed the stability of equilibria and threshold analysis for imperfect vaccine impact on population-level. The local sensitivity analysis is done and observed that some parameters play a prominent role to determine the magnitude of the threshold. Latin Hypercube sampling-PRCC analysis illustrates that disease transmission rate, the fraction of the acutely infected individuals who developed the chronic infection, development rate of symptomatic chronic carriers and disease complications are the most in uential parameters in the disease dynamics. Additionally, we have formulated and rigorously analysed a new basic model for HBV in vivo to attain insight into its qualitative aspects. The numerical analyses reveals that the model has a globally-asymptotic stable (GAS) virus-free equilibrium (VFE) and a positive virus persistent equilibrium (VPE) when the basic reproduction number is less than and greater than one, respectively. Finally, the basic HBV model is extended incorporating the e ect of immune systems namely cell-mediated and humoral immune responses. Numerical simulations show that the humoral immune system is more e ective (to control HBV burden in vivo) than the cell-mediated immune system because of the increasing antibody level within the host due to vaccine impact.