Optimal Vaccination for Contagious Diseases with Seasonal Transmission

We study optimal periodic time-dependent vaccination strategies for seasonal epidemics with waning immunity, aiming to minimize the basic reproduction number under a fixed annual vaccine supply. Our analysis, which does not pre-assume the structure of the vaccination profile, reveals that the optimal time-periodic vaccination profile typically alternates between intervals of active, time-varying vaccination and intervals with no vaccination. Within the active periods, we derive an explicit expression for the optimal vaccination rate. For a wide class of continuous seasonal transmission rate functions, we show that the optimal solution does not exhibit pulse components. In contrast, when the transmission rate exhibits upward jump discontinuities, such as those arising from school-term forcing, the optimal strategy includes pulse components. These theoretical findings are supported by numerical computations of optimal vaccination profiles across a range of scenarios.