Fractional differential equations of a reaction-diffusion SIR model involving the Caputo-fractional time-derivative and a nonlinear diffusion operator

TL;DR

This paper analyzes a fractional reaction-diffusion SIR epidemic model incorporating Caputo fractional derivatives and a nonlinear p-Laplacian operator, focusing on optimal vaccination strategies and numerical solutions.

Contribution

It introduces a novel fractional SIR model with nonlocal derivatives and nonlinear diffusion, establishing existence, uniqueness, and optimal control solutions.

Findings

Existence and uniqueness of nonnegative solutions are proven.

An optimal control framework for vaccination is developed.

Numerical simulations demonstrate the model's effectiveness with different fractional orders.

Abstract

The main aim of this study is to analyze a fractional parabolic SIR epidemic model of a reaction-diffusion, by using the nonlocal Caputo fractional time-fractional derivative and employing the $p$-Laplacian operator. The immunity is imposed through the vaccination program, which is regarded as a control variable. Finding the optimal control pair that reduces the number of sick people, the associated vaccination, and treatment expenses across a constrained time and space is our main study. The existence and uniqueness of the nonnegative solution for the spatiotemporal SIR model are established. It is also demonstrated that an optimal control exists. In addition, we obtain a description of the optimal control in terms of state and adjoint functions. Then, the optimality system is resolved by a discrete iterative scheme that converges after an appropriate test, similar to the forward-backward sweep method. Finally, numerical approximations are given to show the effectiveness of the proposed control program, which provides meaningful results using different values of the fractional order and $p$, respectively the order of the Caputo derivative and the $p$-Laplacian operators.