Fractional differential equations of a reaction-diffusion SIR model involving the Caputo-fractional time-derivative and a nonlinear diffusion operator
Achraf Zinihi, Moulay Rchid Sidi Ammi, Delfim F. M. Torres
TL;DR
The paper addresses spatiotemporal epidemic dynamics by coupling a SIR model with a Caputo fractional time-derivative $^{\mathcal{C}} \mathcal{D}^{\alpha}_{t}$ and a nonlinear diffusion operator $\Delta_p$, incorporating a vaccination control. It develops a rigorous analysis: existence and uniqueness of nonnegative solutions, existence of an optimal vaccination policy, and necessary optimality conditions via a state–adjoint system, solved numerically with a forward-backward sweep. The authors provide a detailed numerical study highlighting how the fractional order $\alpha$ and diffusion exponent $p$ influence disease spread, with and without vaccination, and demonstrate the effectiveness of the vaccination strategy within 80 days. The work offers a versatile framework for memory-effects and nonlinear spatial diffusion in epidemic modeling, with practical implications for spatial vaccination planning and public health strategies.
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.