Bifurcation analysis of the Microscopic Markov Chain Approach to contact-based epidemic spreading in networks

TL;DR

This paper analyzes the bifurcation behavior of the Microscopic Markov Chain Approach in modeling contact-based epidemic spreading, revealing a transcritical bifurcation that explains the transition to endemic states.

Contribution

It provides a dynamical systems perspective on the MMCA model, demonstrating the existence of a stable endemic state through bifurcation analysis.

Findings

Existence of a stable endemic state as a global attractor

The endemic state arises from a transcritical bifurcation

Mathematical grounding for practical epidemic modeling

Abstract

The dynamics of many epidemic compartmental models for infectious diseases that spread in a single host population present a second-order phase transition. This transition occurs as a function of the infectivity parameter, from the absence of infected individuals to an endemic state. Here, we study this transition, from the perspective of dynamical systems, for a discrete-time compartmental epidemic model known as Microscopic Markov Chain Approach, whose applicability for forecasting future scenarios of epidemic spreading has been proved very useful during the COVID-19 pandemic. We show that there is an endemic state which is stable and a global attractor and that its existence is a consequence of a transcritical bifurcation. This mathematical analysis grounds the results of the model in practical applications.