A consistent discrete version of a non-autonomous SIRVS model

TL;DR

This paper develops a discrete non-autonomous SIRVS epidemiological model from a continuous one using Mickens' non-standard discretization, analyzing disease dynamics and ensuring consistency between models.

Contribution

It introduces a novel discrete SIRVS model with general incidence derived via Mickens' method, establishing conditions for disease persistence and extinction, and comparing discretizations with real data.

Findings

Discrete models align with continuous models when time steps are small.

Conditions for disease extinction and permanence are established.

Numerical simulations validate the discretization approach.

Abstract

A family of discrete non-autonomous SIRVS models with general incidence is obtained from a continuous family of models by applying Mickens non-standard discretization method. Conditions for the permanence and extinction of the disease and the stability of disease-free solutions are determined. The consistency of those discrete models with the corresponding continuous model is discussed: if the time step is sufficiently small, when we have extinction (respectively permanence) for the continuous model we also have extinction (respectively permanence) for the corresponding discrete model. Some numerical simulations are carried out to compare the different possible discretizations of our continuous model using real data.