Spatiotemporal optimal control of vaccination in a generalized reaction-diffusion model for infectious disease dynamics

In this article, we formulate a generalized reaction-diffusion model for infectious disease dynamics that accounts for multiple disease progression stages and incorporates the spatial movement of populations. Vaccination, which varies both temporally and spatially, is introduced as a control variable aimed at conferring immunity to susceptible individuals. The primary objective is to characterize an optimal vaccination strategy that minimizes the combined costs of infection spread and vaccine deployment. By leveraging the state, sensitivity, and adjoint systems, we derive a comprehensive framework for the optimal control problem. This work provides a rigorous mathematical foundation for spatiotemporal optimization of vaccination strategies in reaction-diffusion models, with potential applications to a wide range of infectious disease scenarios.