This research describes a compartmental epidemic model of Hepatitis B virus (HBV) transmission that includes vertical transmission and spontaneous recovery in acute patients. The model incorporates a saturated treatment response for persistently infected populations and a vaccination mechanism for susceptible populations. The basic reproduction number, R0, is calculated using the next-generation matrix approach, which provides important information on disease dynamics. To find out the most influential parameter of the model dynamics, a sensitivity analysis is carried out with the help of Latin Hypercube Sampling (LHS) along with the Partial Rank Correlation Coefficient (PRCC). The qualitative behavior of the model is investigated using stability analysis of disease-free and endemic equilibria. It is established that the disease-free equilibrium is globally asymptotically stable when R0 < 1, but the endemic equilibrium achieves global stability when R0 > 1. Pontryagin’s Maximum Principle is used to optimize public health initiatives, resulting in three optimal control techniques that attempt to reduce the combined cost of treatment and immunization. The results provide a rigorous theoretical foundation for designing cost-effective interventions against HBV transmission. This work contributes a more biologically accurate and analytically rich model to the literature, offering new insights into the strategic control of HBV infection, particularly in regions where vaccination and treatment resources are limited.