Chenciner bifurcation, strong resonances and Arnold tongues of a discrete time SIR epidemic model

TL;DR

This paper analyzes a three-dimensional SIR epidemic model, revealing complex bifurcation structures including Chenciner bifurcations, strong resonances, and Arnold tongues, supported by theoretical analysis and numerical simulations.

Contribution

It provides a comprehensive bifurcation analysis of the SIR model using advanced mathematical tools, identifying new bifurcation phenomena and resonance structures.

Findings

Identification of codimension 1 and 2 bifurcations including Chenciner bifurcation.

Existence of Arnold tongues in weak resonances with two periodic orbits.

Numerical verification of bifurcation phenomena using MatcontM.

Abstract

In this paper, we mainly study the dynamic properties of a class of three-dimensional SIR models. Firstly, we use the {\it complete discriminant theory} of polynomials to obtain the parameter conditions for the topological types of each fixed point. Secondly, by employing the center manifold theorem and bifurcation theory, we prove that the system can undergo codimension 1 bifurcations, including transcritical, flip and Neimark-Sacker bifurcations, and codimension 2 bifurcations which contain Chenciner bifurcation, 1:3 and 1:4 strong resonances. Besides, by the theory of normal form, we give theoretically the Arnold tongues in the weak resonances such that the system possesses two periodic orbits on the stable invariant closed curve generated from the Neimark-Sacker bifurcation. Finally, in order to verify the theoretical results, we detect all codimension 1 and 2 bifurcations by using MatcontM and numerically simulate all bifurcation phenomena and the Arnold tongues in the weak resonances.