Optimization Theory: Concepts, Methods, and Applications

This study develops a compartmental model to characterize the transmission dynamics of the Hepatitis B virus (HBV) with two distinct viral strains. The basic reproduction number is derived via the next-generation matrix method, and three biologically relevant equilibria are identified and analyzed. To evaluate intervention policies, an optimal control problem is formulated with two controls—vaccination and treatment—aimed at minimizing both infection prevalence and implementation costs. Within a deterministic framework, Pontryagin’s Maximum Principle is employed to establish the necessary conditions for optimality. Numerical simulations conducted in MATLAB corroborate the analytical results and quantify the impact of the proposed interventions under baseline parameter values. A cost-effectiveness assessment indicates that, relative to vaccination alone, treatment yields the most economically efficient reduction in disease burden. The modeling framework offers practical guidance for HBV-endemic settings, particularly in regions facing constrained healthcare resources.