Spatiotemporal SEIQR Epidemic Modeling with Optimal Control for Vaccination, Treatment, and Social Measures.

This paper introduces a spatiotemporal SEIQR epidemic model governed by a system of reaction-diffusion partial differential equations that incorporates optimal control strategies. The model captures the transmission dynamics of an infectious disease across space and time. The model incorporates the quarantined compartment Q as part of the epidemic structure. It also includes three time-dependent control variables representing vaccination, treatment of quarantined individuals, and non-pharmaceutical interventions that reduce transmission. The study has four main objectives: (i) to prove the existence, uniqueness, and positivity of global strong solutions using analytic semigroup theory, (ii) to demonstrate the existence of optimal control strategies through functional analysis techniques, (iii) to derive first-order necessary optimality conditions via convex perturbation methods and adjoint equations, and (iv) to perform numerical simulations to assess the effectiveness of different combinations of control interventions. The simulation results emphasize the advantages of combining pharmaceutical and non-pharmaceutical interventions to minimize disease prevalence and control-related expenses.