Mathematical Modeling and Analysis for the Dynamics of a Two Stage Hepatitis B Virus with Vaccination and Vertical Transmission

Abstract

In this thesis, we proposed a deterministic mathematical model for acute and chronic hepatitis B virus with vaccination and vertical transmission. We examined the well-posedness of the model by proved its positivity, uniqueness, and boundedness of the solutions to the system equation. We computed both the disease-free (DFE) and endemic equilibrium (EEP) points of the model and analyzed their local and global stability by computing its basic reproduction number. We have shown that when the reproduction number R0 < 1, the illness will die out or go extinct, as the disease-free (DFE) equilibrium point is stable for R0 < 1. The global stability of the disease-free equilibrium point is also proved by applying the castillo-Chavez theorem. The local stability of the endemic equilibrium point was determined using center manifold theory and its bifurcation analy sis, and it was found that the model exhibits a forward bifurcation at R0 = 1. The global stability of the endemic equilibrium point is also determined by using the Lyapounov function. We have also performed a sensitivity analysis of the model with respect to the model parameters. We observed that, the recruitment rate, and the transmission rate of hepatitis B virus have positive correlation with R0, while the vaccination rate of newborns(pediatric vaccination) and vaccination rate have negative correlation with R0. Numerical simulation of the model is also performed using some assumed and literaturized values of the parameters