On the computational simulation of a temporally nonlocal and nonlinear diffusive epidemic model of disease transmission

This paper reports on a nonlocal mathematical system to model the spreading of polio among a community of human individuals. Our model consists of four compartments: susceptible, exposed, infected and vaccinated individuals. The mathematical model considers spatial diffusion, and temporal fractional derivatives of the Caputo type. We use nonnegative initial conditions and homogeneous data of the Neumann type on the boundary. We determine analytically the disease-free and the endemic equilibria along with the basic reproductive number. We establish thoroughly the nonnegativity of the solutions, and the stability of the equilibria. Computationally, we propose an implicit finite-difference method to approximate the solutions of the model. The numerical model is stable in the sense of von Neumann, it yields consistent approximations to the exact solutions of the differential problem, and it is capable of preserving unconditionally the positivity of the approximations. For the sake of illustration, we provide some computer simulations that confirm some of the theoretical results derived in the present paper.