Optimal control of a diffusive epidemiological model involving the Caputo–Fabrizio fractional time-derivative

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.