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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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show all references

References:
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Springer, Berlin, 1967.  Google Scholar

[2]

J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324.  Google Scholar

[3]

Math. Biosci., 246 (2013), 105-112. doi: 10.1016/j.mbs.2013.08.003.  Google Scholar

[4]

Discrete. Cont. Dyn. Sys. B., 19 (2014), 715-733. doi: 10.3934/dcdsb.2014.19.715.  Google Scholar

[5]

Math. Biosci. Eng., 12 (2015), 99-115.  Google Scholar

[6]

P. Am. Math. Soc., 136 (2008), 2793-2802. doi: 10.1090/S0002-9939-08-09341-6.  Google Scholar

[7]

Springer, 1991. doi: 10.1007/BFb0084432.  Google Scholar

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Appl. Math. Lett., 26 (2013), 687-691. doi: 10.1016/j.aml.2013.01.010.  Google Scholar

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Math. Meth. Appl. Sci., 38 (2015), 2703-2718. doi: 10.1002/mma.3252.  Google Scholar

[10]

J. Differ. Equations, 248 (2010), 1-20. doi: 10.1016/j.jde.2009.09.003.  Google Scholar

[11]

J. Math. Anal. Appl., 361 (2010), 38-47. doi: 10.1016/j.jmaa.2009.09.017.  Google Scholar

[12]

Appl. Math. Lett., 24 (2011), 1685-1692. doi: 10.1016/j.aml.2011.04.019.  Google Scholar

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J. Epidemiol. Commun. H., 33 (1979), 299-304. doi: 10.1136/jech.33.4.299.  Google Scholar

[14]

Math. Biosci. Eng., 6 (2009), 603-610. doi: 10.3934/mbe.2009.6.603.  Google Scholar

[15]

Math. Biosci., 218 (2009), 138-141. doi: 10.1016/j.mbs.2009.01.006.  Google Scholar

[16]

Nonlinear Anal. RWA., 14 (2013), 1693-1704. doi: 10.1016/j.nonrwa.2012.11.005.  Google Scholar

[17]

Appl. Math. Lett., 38 (2014), 73-78. doi: 10.1016/j.aml.2014.07.005.  Google Scholar

[18]

J. Appl. Math. Comput., 35 (2011), 161-178. doi: 10.1007/s12190-009-0349-z.  Google Scholar

[19]

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[21]

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[22]

Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[23]

Math. Meth. Appl. Sci., 39 (2016), 1964-1976. doi: 10.1002/mma.3613.  Google Scholar

[24]

Osaka J. Math., 52 (2015), 117-138.  Google Scholar

[25]

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[26]

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[27]

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[28]

Math. Biosci., 208 (2007), 312-324. doi: 10.1016/j.mbs.2006.10.008.  Google Scholar

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