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October  2016, 21(8): 2615-2630. doi: 10.3934/dcdsb.2016064

## Threshold dynamics of a delayed multi-group heroin epidemic model in heterogeneous populations

 1 Key Laboratory of Eco-environments in Three Gorges Reservoir Region, (Ministry of Education), School of Mathematics and Statistics, Southwest University, Chongqing 400715, China, China 2 School of Mathematical Science, Heilongjiang University, Harbin 150080, China

Received  April 2015 Revised  June 2016 Published  September 2016

The aim of this paper is to investigate the threshold dynamics of a heroin epidemic in heterogeneous populations. The model is described by a delayed multi-group model, which allows us to model interactions both within-group and inter-group separately. Here we are able to prove the existence of heroin-spread equilibrium and the uniform persistence of the model. The proofs of main results come from suitable applications of graph-theoretic approach to the method of Lyapunov functionals and Krichhoff's matrix tree theorem. Numerical simulations are performed to support the results of the model for the case where $n=2$.
Citation: Lili Liu, Xianning Liu, Jinliang Wang. Threshold dynamics of a delayed multi-group heroin epidemic model in heterogeneous populations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2615-2630. doi: 10.3934/dcdsb.2016064
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##### References:
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