A survey on Lyapunov functions for epidemic compartmental models

Abstract

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.

Positivity and boundedness preserving nonstandard finite difference schemes for solving Volterra’s population growth model

“…(1) NSFD methods preserving general Lyapunov functions: One of the limitations of the proposed NSFD methods is that they only preserve general quadratic Lyapunov functions, whereas many well-known Lyapunov functions have been proposed (see, for example, [5,13,25,26,27,28,40,43,44,45]). For example, general Voltera-type Lyapunov functions of the form [43] V V L (y) := (2) High-order NSFD methods preserving Lyapunov functions: The convergence of order 1 of the proposed NSFD methods can be considered as another limitation of them.…”

“…It is well-known that the global asymptotic stability (GAS) analysis of equilibrium points is a very important problem with various useful applications in fields of theory and practice, for example in mathematical physics, economics, control theory, biology, ecology and so on (see, for instance, [24,29,30,32,42]). The Lyapunov stability theory with the help of suitable Lyapunov functions has been recognized as one of the most successful approaches to this problem (see, for example, [5,13,25,26,27,28,40,43,44,45]). For the sake of convenience, below we recall the Lyapunov stability theorem, which is also known as Barbashin-Krasovskii theorem, for the GAS of continuous-time dynamical systems [24,Theorem 4.2].…”