Mathematical Biosciences and Engineering, 2015, 12(1): 99-115. doi: 10.3934/mbe.2015.12.99

Primary: 34D23; Secondary: 92B05.

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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

A multi-group model is proposed to describe a general relapse phenomenon of infectious diseasesin heterogeneous populations.In each group, the population is divided intosusceptible, exposed, infectious, and recovered subclasses. A generalnonlinear incidence rate is used in the model. The results show that the global dynamics are completelydetermined by the basic reproduction number $R_0.$ In particular, a matrix-theoretic method is used to provethe global stability of the disease-free equilibrium when $R_0\leq1,$while a new combinatorial identity (Theorem 3.3 in Shuai and vanden Driessche [29]) in graph theory is applied to provethe 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 intoa graph of n vertices in order to deal with the global stability of the endemic equilibrium in this paper.
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