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

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.

Automated summary

The dynamics of epidemic compartmental models for infectious diseases show a second-order phase transition as a function of the infectivity parameter, transitioning from the absence of infected individuals to an endemic state. We study this transition using a discrete-time compartmental epidemic model called Microscopic Markov Chain Approach, which has been proven to be useful for forecasting epidemic spreading scenarios. Our analysis reveals the existence of a stable and globally attractive endemic state, which is a consequence of transcritical bifurcation. This mathematical analysis validates the practical applications of the model.