A Novel Hepatitis B Epidemic Model with Vertical Transmission, Spontaneous Recovery and Optimal Control Analysis

This study develops a compartmental epidemic model for Hepatitis B virus (HBV) transmission incorporating vertical transmission, spontaneous recovery in acutely infected individuals, saturated treatment response for chronically infected individuals, and vaccination of susceptible individuals. The basic reproduction number, $\mathcal{R}_0$, is derived using the next-generation matrix method, providing a threshold quantity for understanding disease dynamics. To identify the most influential parameters, sensitivity analysis is performed using Latin Hypercube Sampling (LHS) together with Partial Rank Correlation Coefficients (PRCCs). The qualitative behavior of the model is investigated through stability analysis of the disease-free and endemic equilibria. It is shown that the disease-free equilibrium is globally asymptotically stable when $\mathcal{R}_0 < 1$, while the endemic equilibrium is globally stable when $\mathcal{R}_0 > 1$. Pontryagin's Maximum Principle is applied to optimize public health interventions, leading to three optimal control strategies aimed at reducing the combined cost of treatment and vaccination. The results provide a rigorous theoretical framework for designing cost-effective interventions against HBV transmission.