Necessary optimality conditions of a reaction-diffusion SIR model with ABC fractional derivatives

TL;DR

This paper develops and analyzes a reaction-diffusion SIR epidemic model using ABC fractional derivatives, establishing existence, uniqueness, and optimal control conditions, and demonstrating the effectiveness of fractional control strategies through numerical simulations.

Contribution

It introduces a novel fractional SIR model with ABC derivatives, proving solution existence, deriving optimality conditions, and comparing fractional and classical models numerically.

Findings

Fractional model solutions exist and are unique.

Optimal control strategies are effectively characterized.

Numerical results highlight the importance of fractional order selection.

Abstract

The main aim of the present work is to study and analyze a reaction-diffusion fractional version of the SIR epidemic mathematical model by means of the non-local and non-singular ABC fractional derivative operator with complete memory effects. Existence and uniqueness of solution for the proposed fractional model is proved. Existence of an optimal control is also established. Then, necessary optimality conditions are derived. As a consequence, a characterization of the optimal control is given. Lastly, numerical results are given with the aim to show the effectiveness of the proposed control strategy, which provides significant results using the AB fractional derivative operator in the Caputo sense, comparing it with the classical integer one. The results show the importance of choosing very well the fractional characterization of the order of the operators.