A survey on Lyapunov functions for epidemic compartmental models

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

Boundedness and global exponential stability for delayed differential equations with applications

The boundedness of solutions for a class of n-dimensional differential equations with distributed delays is established by assuming the existence of instantaneous negative feedbacks which dominate the delay effect. As an important by-product, some criteria for global exponential stability of equilibria are obtained. The results are illustrated with applications to delayed neural networks and population dynamics models.POCI 2010CMATFundação para a Ciência e a Tecnologia (FCT) - SFRH/BD/29563/2006CMAFFEDE

Persistence and Global Dynamics of an Extended Rosenzweig-MacArthur Model

This paper investigates persistence and global dynamics of a tritrophic food chain model consisting of prey, predator, and super-predator. We establish dissipativeness, ultimate boundedness of an invariant region in the state space of this model via the notion of omega-limit sets, absorbing region and global attractor. We explore Freedman-Waltman theorem, and Bendixson-Dulac theorem to guarantee persistence conditions of the model. Lyapunov’s functionals and Lyapunov-LaSalle invariance principle ensure the existence of global asymptotic stability of the system. Numerical responses, phase-portrait and phase-flows were used to illustrate propositions and lemmas. Key words: Global asymptotic stability; Lyapunov functional; Persistence