Backward Bifurcation and Optimal Control in an Age-Structured SVI Epidemic Model

Abstract

This study investigates the dynamics and control of an infectious disease using an age-structured Susceptible-Vaccinated-Infected (SVI) model, incorporating imperfect vaccination and therapeutic treatment. We identify a backward bifurcation phenomenon, driven by treatment rates, which allows disease persistence even when the basic reproduction number R0 < 1, complicating eradication e orts. To address this, we formulate an optimal control problem with time-dependent vaccination and treatment rates to minimize infected individuals and control costs. Using bifurcation theory and integrated semigroup methods, we establish the model's well-posedness, equilibria stability, and bistability conditions. First-order optimality conditions are derived to characterize optimal controls. Numerical simulations,solved via the Forward-Backward Sweep method, demonstrate that combined vaccination and treatment signi cantly reduces disease prevalence, with early intervention being critical. These  ndings underscore the importance of strategic resourceallocation in managing complex epidemic dynamics.