Global stability of multi-group epidemic models with distributed delays

Abstract

We investigate a class of multi-group epidemic models with distributed delays. We establish that the global dynamics are completely determined by the basic reproduction number R 0 . More specifically, we prove that, if R 0 1, then the disease-free equilibrium is globally asymptotically stable; if R 0 > 1, then there exists a unique endemic equilibrium and it is globally asymptotically stable. Our proof of global stability of the endemic equilibrium utilizes a graph-theoretical approach to the method of Lyapunov functionals.

Epidemic dynamics on a delayed multi-group heroin epidemic model with nonlinear incidence rate

“…It should be pointed here that for a class of multi-group SEIR models described by ordinary differential equations, a graph-theoretic approach to the method of global Lyapunov functions was proposed and used to obtain the global stability of a unique endemic equilibrium in [6]. During the past decades, the multi-group epidemic models have been extensively studied by many authors (see, for example, [5,6,9,12,13,21,24,25,27] and the references therein). These studies have enriched our knowledge of epidemic models with heterogeneity.…”

Global analysis of a multi-group SIR epidemic model with nonlinear incidence rates and distributed moving delays between patches

“…On the other hand, Guo et al [8] have first succeeded in the proof of global stability for a multi-group SIR epidemic model by making use of the theory of non-negative matrices, Lyapunov functions and a subtle grouping technique in estimating the derivatives of Lyapunov functions guided by graph theory. To analyze the global stability of various multi-group epidemic models, many authors literature on multi-group models follow to use this graph theoretic approach (see for example, [4,10,15,16,29,30,36,37]). …”

A graph-theoretic approach to exponential stability of stochastic BAM neural networks with time-varying delays

“…Thus a Lyapunov function for neural network can be built by a linear combination of Lyapunov functions of vertex systems according to a weighted digraph. Graph-theoretic approach is widely applied to infectious disease models [24][25][26], chemical model [27] and biology models [28][29][30][31][32].…”