Constrained Optimal Control Problem Applied to Vaccination for COVID-19 Epidemic

In this study, we propose a dynamic mathematical model framework governed by a system of differential equations that integrates COVID-19 outbreaks. We extend the standard SEAIR model to incorporate the vaccination component. We analyze the existence and uniqueness of the solution, compute the basic reproduction number R₀, and study the equilibrium state’s local stability. We formulate an optimal control problem to minimize the number of infected individuals while considering intervention costs. Our optimal control problem integrates two realistic constraints: total vaccine administration and maximum daily vaccine administration. We use a penalty method to handle these constraints to convert this problem to a more familiar form. We approximate the obtained constrained optimization problem and derive an optimality system that characterizes the optimal control. Finally, we perform numerical simulations using the reported data on COVID-19 infections and vaccination in France to compare the optimal intervention strategies under different settings.