  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.

Physics and Society;Statistical Mechanics;Social and Information Networks;Dynamical Systems