On global behavior of a some SIR epidemic model based on the Poincar\'e compactification

TL;DR

This paper introduces a novel approach using Poincaré compactification to analyze the global behavior of SIR epidemic models, providing insights into dynamics near equilibrium and asymptotic behavior.

Contribution

It presents a new method based on Poincaré compactification for studying epidemic models, offering an alternative to Lyapunov functions and enhancing understanding of global dynamics.

Findings

Effective analysis of endemic equilibrium behavior

Refined understanding of dynamics near equilibrium

Asymptotic behavior related to the basic reproduction number

Abstract

It is important to study the global behavior of solutions to systems of ordinary differential equations describing the transmission dynamics of infectious disease. In this paper, we present a different approach from the Lyapunov function used in most of them. This approach is based on the Poincar\'e compactification. We then apply the method to a SIR endemic model as a test case, and discuss its effectiveness and the potential applications of this approach. In addition, we refine the discussion of dynamics near the equilibrium, derive the asymptotic behavior, and mention its relation to the basic reproduction number.