A dynamically consistent computational method to solve numerically a mathematical model of polio propagation with spatial diffusion

Abstract

Background and objective: In this work, a mathematical model based on differential equations is proposed to describe the propagation of polio in a human population. The motivating system is a compartmental nonlinear model which is based on the use of ordinary differential equations and four compartments, namely, susceptible, exposed, infected and vaccinated individuals.

Methods: In this manuscript, the mathematical model is extended in order to account for spatial diffusion in one dimension. Nonnegative initial conditions are used, and we impose homogeneous Neumann conditions at the boundary. We determine analytically the disease-free and the endemic equilibria of the system along with the basic reproductive number.

Results: We establish thoroughly the nonnegativity and the boundedness of the solutions of this problem, and the stability analysis of the equilibrium solutions is carried out rigorously. In order to confirm the validity of these results, we propose an implicit and linear finite-difference method to approximate the solutions of the continuous model.

Conclusions: The numerical model is stable in the sense of von Neumann, it yields consistent approximations to the exact solutions of the differential problem, and that it is capable of preserving unconditionally the positivity of the approximations. For illustration purposes, we provide some computer simulations that confirm some theoretical results derived in the present manuscript.

Keywords: Basic reproductive number; Diffusive mathematical model; Global stability analysis; Neumann stability analysis; Positivity-preserving scheme.