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In this survey, we propose an overview on Lyapunov functions for a variety of compartmental models in epidemiology. We exhibit the most widely employed functions, and provide a commentary on their use. Our aim is to provide a comprehensive starting point to readers who are attempting to prove global stability of systems of ODEs. The focus is on mathematical epidemiology, however some of the functions and strategies presented in this paper can be adapted to a wider variety of models, such as prey–predator or rumor spreading.

We propose a novel slow–fast SIRS compartmental model with demography, by coupling a slow disease spreading model and a fast information and misinformation spreading model. Beside the classes of susceptible, infected and recovered individuals of a common SIRS model, here we define three new classes related to the information spreading model, e.g. unaware individuals, misinformed individuals and individuals who are skeptical to disease-related misinformation. Under our assumptions, the system evolves on two time scales. We completely characterize its asymptotic behavior with techniques of Geometric Singular Perturbation Theory (GSPT). We exploit the time scale separation to analyze two lower dimensional subsystem separately. First, we focus on the analysis of the fast dynamics and we find three equilibrium point which are feasible and stable under specific conditions. We perform a theoretical bifurcation analysis of the fast system to understand the relations between these three equilibria when varying specific parameters of the fast system. Secondly, we focus on the evolution of the slow variables and we identify three branches of the critical manifold, which are described by the three equilibria of the fast system. We fully characterize the slow dynamics on each branch. Moreover, we show how the inclusion of (mis)information spread may negatively or positively affect the evolution of the epidemic, depending on whether the slow dynamics evolves on the second branch of the critical manifold, related to the skeptical-free equilibrium or on the third one, related to misinformed-free equilibrium, respectively. We conclude with numerical simulations which showcase our analytical results. • We propose a SIRS model paired to an information spreading model. • The system has two time-scales: fast for information spread and slow for disease spread. • Geometrical Singular Perturbation Theory (GSPT) is used to study the model. • We completely characterize the asymptotic behavior of the system in each time scale. • We include theoretical bifurcation analysis and extensive numerical simulations. [ABSTRACT FROM AUTHOR]