Optimal Control of a Diffusive Epidemiological Model Involving the Caputo-Fabrizio Fractional Time-Derivative

TL;DR

The paper addresses optimal vaccination control for a spatial SEIR epidemic model with memory effects captured by the Caputo-Fabrizio fractional time derivative $^{\mathcal{CFC}} \mathcal{D}^{\alpha}_{t}$ and diffusion $-\lambda \Delta$. It establishes well-posedness and existence of an optimal control, derives necessary conditions via an adjoint system, and demonstrates the approach with forward–backward numerical sweeps on a 2D domain over 60 days. The main contributions include rigorous existence/uniqueness and positivity results for the fractional PDE, a variational framework for vaccination optimization, and practical insights showing vaccination’s critical role in memory-affected spread dynamics. This work provides a memory-aware, spatiotemporal optimization framework that can inform public-health strategies under more realistic diffusion and history effects.

Abstract

In this work we study a fractional SEIR biological model of a reaction-diffusion, using the non-singular kernel Caputo-Fabrizio fractional derivative in the Caputo sense and employing the Laplacian operator. In our PDE model, the government seeks immunity through the vaccination program, which is considered a control variable. Our study aims to identify the ideal control pair that reduces the number of infected/infectious people and the associated vaccine and treatment costs over a limited time and space. Moreover, by using the forward-backward algorithm, the approximate results are explained by dynamic graphs to monitor the effectiveness of vaccination.

Figures (8)

• Figure 1: Transmission dynamics in the SEIR model.
• Figure 2: Algorithm organigram for the proposed SEIR epidemic model.
• Figure 3: Numerical approximations without vaccination strategy for $\alpha = 1$.
• Figure 4: Numerical approximations without vaccination strategy for $\alpha = 0.9$.
• Figure 5: Numerical approximations without vaccination strategy for $\alpha = 0.8$.

Theorems & Definitions (18)

• Definition 1: Abdeljawad2017
• Definition 2: Abdeljawad2017
• Lemma 1: Abdeljawad2017
• Lemma 2
• proof
• Corollary 1
• Lemma 3
• proof
• Theorem 1
• proof