On qualitative analysis of a discrete time SIR epidemical model

TL;DR

This paper analyzes the local dynamics and bifurcations of a discrete-time SIR epidemic model, revealing complex behaviors including stability conditions, bifurcations, and the existence of cycles, supported by analytical and numerical methods.

Contribution

It provides a comprehensive classification of bifurcations and stability conditions in a discrete SIR model, including the existence of cycles of arbitrary length.

Findings

Existence and stability conditions for disease-free and endemic states

Identification of bifurcations and their codimensions

Presence of cycles of arbitrary length via Sharkovskii's theorem

Abstract

The main purpose of this paper is to study the local dynamics and bifurcations of a discrete-time SIR epidemiological model. The existence and stability of disease-free and endemic fixed points are investigated along with a fairly complete classification of the systems bifurcations, in particular, a complete analysis on local stability and codimension 1 bifurcations in the parameter space. Sufficient conditions for positive trajectories are given. The existence of a 3-cycle is shown, which implies the existence of cycles of arbitrary length by the celebrated Sharkovskii's theorem. Generacity of some bifurcations is examined both analytically and through numerical computations. Bifurcation diagrams along with numerical simulations are presented. The system turns out to have both rich and interesting dynamics.