Optimal vaccination and treatment policies for regional approximate controllability of the time-fractional reaction-diffusion SIR epidemic systems

Abstract

This paper addresses the optimal vaccination and treatment control problems for regional approximate controllability of a new spatiotemporal epidemic model that is developed by afterwards adding at the basic susceptible-infected-recovered (SIR) epidemic system with the Caputo time-fractional derivative of order α∈(0,1] and the diffusion term in each compartment. The obtained results can be used by policy-makers in any nation to optimally plan the limited vaccination and treatment resources ahead of ongoing outbreaks. Toward this aim the Sakawa-type controller is introduced, which is a finite-dimensional controller that prevents and controls the spreading of infection. Using the semigroup theory, we provide a framework to analyze the sufficient conditions under which the considered time-fractional reaction-diffusion SIR epidemic system is regionally approximately controllable with the proposed control actuation architecture. An approach on finding the optimal solution that minimizes the cost of corresponding vaccination and treatment control programs for the studied regional controllability problem is then presented. Finally, we finish with numerical results to illustrate our theoretical results.

Keywords: Optimal control; Regional approximate controllability; Spatiotemporal SIR epidemic model; Time-fractional reaction–diffusion systems.