Global dynamics of a general class of multi-group epidemic models with latency and relapse

Xiaomei Feng; Zhidong Teng; Fengqin Zhang

doi:10.3934/mbe.2015.12.99

Article Contents

2015, Volume 12, Issue 1: 99-115. Doi: 10.3934/mbe.2015.12.99

This issue Previous Article Delayed population models with Allee effects and exploitation Next Article Construction of Lyapunov functions for some models of infectious diseases in vivo: From simple models to complex models

Global dynamics of a general class of multi-group epidemic models with latency and relapse

1.
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046
2.
Department of Applied Mathematics, Yuncheng University, Yuncheng 044000, Shanxi

Received: March 31, 2014

Revised: October 31, 2014

Published: November 2014

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Abstract

A multi-group model is proposed to describe a general relapse phenomenon of infectious diseases in heterogeneous populations. In each group, the population is divided into susceptible, exposed, infectious, and recovered subclasses. A general nonlinear incidence rate is used in the model. The results show that the global dynamics are completely determined by the basic reproduction number $R_0.$ In particular, a matrix-theoretic method is used to prove the global stability of the disease-free equilibrium when $R_0\leq1,$ while a new combinatorial identity (Theorem 3.3 in Shuai and van den Driessche [29]) in graph theory is applied to prove the global stability of the endemic equilibrium when $R_0>1.$ We would like to mention that by applying the new combinatorial identity, a graph of 3n (or 2n+m) vertices can be converted into a graph of n vertices in order to deal with the global stability of the endemic equilibrium in this paper.

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Mathematics Subject Classification: Primary: 34D23; Secondary: 92B05.

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