  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.
