Stability Analysis of Mathematical Models of Hepatitis B Virus Dynamics with Vaccination in Plateau State

This study formulates and performs stability analysis on a mathematical model of hepatitis B virus transmission dynamics incorporating vaccination and chronic carrier compartments, with epidemiological parameters calibrated to seroprevalence and immunisation data from Plateau State, North Central Nigeria. Hepatitis B infection remains a major public health burden in Nigeria, with national seroprevalence estimates suggesting that approximately 8 to 12 percent of the general adult population carries the hepatitis B surface antigen, and Plateau State has been identified as a high-endemicity state in recent seroprevalence surveys. A six-compartment ordinary differential equation model partitioning the population into susceptible, vaccinated, acutely infected, chronically infected, recovered, and immune classes is formulated, incorporating waning vaccine immunity and vertical transmission pathways. The basic reproduction number is derived using the next-generation matrix approach and expressed as a function of vaccination coverage, vaccine efficacy, and natural immunity rates. Local and global stability of the disease-free equilibrium are analysed using linearisation and a Lyapunov function constructed from a combination of linear and nonlinear terms, respectively. The persistence of endemicity is established when the basic reproduction number exceeds unity through uniform persistence theory. Numerical simulations illustrate the impact of vaccination coverage on transmission dynamics and estimate the vaccination threshold required for hepatitis B elimination in Plateau State. Results indicate that a vaccination coverage of 85 percent with a 90 percent efficacious vaccine would reduce the basic reproduction number below unity. Keywords: hepatitis B, stability analysis, vaccination model, Lyapunov function, Plateau State