COVID-19 spatiotemporal SIR model: Regional optimal control approach

Abstract

In the present work, we consider a spatio-temporal model to describe the evolution of covid19 in an area Ω (Ω can be a city, a country,..). Taking into account the financial means of the considered country, we suppose that the number of available vaccines is destined to a region ω₁⊆Ω (ω₁ can be an industrial city, a university city...) and we suppose that the available treatments are dedicated to a region ω₂⊆Ω (ω₂ can be a military city,..), it is not excluded that ω₁=ω₂. To minimize the number of infection with minimal cost, we apply an optimal regional control strategy to stop the death of infected individuals in the considered area. Much of this work has been devoted to mathematical study, where the existence of the optimal controls and the solutions of the state system are proven, an optimal control characterization in terms of state and adjoint functions are provided, and the optimality system is solved numerically using a forward-backward sweep method. Our numerical results suggest that when vaccination and treatment procedures are used together, the control approach becomes more effective in protecting a specific region from epidemic transmission from neighboring regions.