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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
##### References:
 [1] N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications,, Springer, (1967). Google Scholar [2] O. Diekmann, J. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365. doi: 10.1007/BF00178324. Google Scholar [3] M. C. Eisenberg, Z. Shuai, J. H. Tien and P. van den Driessche, A cholera model in a patchy environment with water and human movement,, Math. Biosci., 246 (2013), 105. doi: 10.1016/j.mbs.2013.08.003. Google Scholar [4] B. Fang, X. Li, M. Martcheva and L. Cai, Global stability for a heroin model with two distributed delays,, Discrete. Cont. Dyn. Sys. B., 19 (2014), 715. doi: 10.3934/dcdsb.2014.19.715. Google Scholar [5] X. Feng, Z. Teng and F. Zhang, Global dynamics of a general class of multi-group epidemic models with latency and relapse,, Math. Biosci. Eng., 12 (2015), 99. Google Scholar [6] H. Guo, M. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions,, P. Am. Math. Soc., 136 (2008), 2793. doi: 10.1090/S0002-9939-08-09341-6. Google Scholar [7] Y. Hino, T. Naito and S. Murakami, Functional Differential Equations with Infinite Delay,, Springer, (1991). doi: 10.1007/BFb0084432. Google Scholar [8] G. Huang and A. Liu, A note on global stability for a heroin epidemic model with distributed delay,, Appl. Math. Lett., 26 (2013), 687. doi: 10.1016/j.aml.2013.01.010. Google Scholar [9] G. Huang, J. Wang and J. Zu, Global dynamics of multi-group dengue disease model with latency distributions,, Math. Meth. Appl. Sci., 38 (2015), 2703. doi: 10.1002/mma.3252. Google Scholar [10] M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Differ. Equations, 248 (2010), 1. doi: 10.1016/j.jde.2009.09.003. Google Scholar [11] M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays,, J. Math. Anal. Appl., 361 (2010), 38. doi: 10.1016/j.jmaa.2009.09.017. Google Scholar [12] J. Liu and T. Zhang, Global behaviour of a heroin epidemic model with distributed delays,, Appl. Math. Lett., 24 (2011), 1685. doi: 10.1016/j.aml.2011.04.019. Google Scholar [13] D. R. MacKintosh and G. T. Stewart, A mathematical model of a heroin epidemic: Implications for control policies,, J. Epidemiol. Commun. H., 33 (1979), 299. doi: 10.1136/jech.33.4.299. Google Scholar [14] C. C. McCluskey, Global stability for an {SEIR} epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 6 (2009), 603. doi: 10.3934/mbe.2009.6.603. Google Scholar [15] G. Mulone and B. Straughan, A note on heroin epidemics,, Math. Biosci., 218 (2009), 138. doi: 10.1016/j.mbs.2009.01.006. Google Scholar [16] Y. Muroya, Y. Enatsu and T. Kuniya, Global stability for a multi-group SIRS epidemic model with varying population sizes,, Nonlinear Anal. RWA., 14 (2013), 1693. doi: 10.1016/j.nonrwa.2012.11.005. Google Scholar [17] Y. Muroya and T. Kuniya, Further stability analysis for a multi-group {SIRS} epidemic model with varying total population size,, Appl. Math. Lett., 38 (2014), 73. doi: 10.1016/j.aml.2014.07.005. Google Scholar [18] G. Samanta, Dynamic behaviour for a nonautonomous heroin epidemic model with time delay,, J. Appl. Math. Comput., 35 (2011), 161. doi: 10.1007/s12190-009-0349-z. Google Scholar [19] H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission,, Nonlinear Anal. RWA., 13 (2012), 1581. doi: 10.1016/j.nonrwa.2011.11.016. Google Scholar [20] Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions,, SIAM J. Appl. Math., 73 (2013), 1513. doi: 10.1137/120876642. Google Scholar [21] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model),, SIAM J. Math. Anal., 24 (1993), 407. doi: 10.1137/0524026. Google Scholar [22] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [23] J. Wang and X. Liu, Modeling diseases with latency and nonlinear incidence rates: Global dynamics of a multi-group model,, Math. Meth. Appl. Sci., 39 (2016), 1964. doi: 10.1002/mma.3613. Google Scholar [24] J. Wang, X. Liu, J. Pang and D. Hou, Global dynamics of a multi-group epidemic model with general exposed distribution and relapse,, Osaka J. Math., 52 (2015), 117. Google Scholar [25] J. Wang, J. Pang and X. Liu, Modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model,, J. Biol. Dynam., 8 (2014), 99. doi: 10.1080/17513758.2014.912682. Google Scholar [26] J. Wang and H. Shu, Global dynamics of a multi-group epidemic model with latency, relapse and nonlinear incidence rate,, Math. Biosci. Eng., 13 (2016), 209. doi: 10.3934/mbe.2016.13.209. Google Scholar [27] W. Wang and X.-Q. Zhao, An epidemic model with population dispersal and infection period,, SIAM J. Appl. Math., 66 (2006), 1454. doi: 10.1137/050622948. Google Scholar [28] E. White and C. Comiskey, Heroin epidemics, treatment and ODE modelling,, Math. Biosci., 208 (2007), 312. doi: 10.1016/j.mbs.2006.10.008. Google Scholar

show all references

##### References:
 [1] N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications,, Springer, (1967). Google Scholar [2] O. Diekmann, J. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365. doi: 10.1007/BF00178324. Google Scholar [3] M. C. Eisenberg, Z. Shuai, J. H. Tien and P. van den Driessche, A cholera model in a patchy environment with water and human movement,, Math. Biosci., 246 (2013), 105. doi: 10.1016/j.mbs.2013.08.003. Google Scholar [4] B. Fang, X. Li, M. Martcheva and L. Cai, Global stability for a heroin model with two distributed delays,, Discrete. Cont. Dyn. Sys. B., 19 (2014), 715. doi: 10.3934/dcdsb.2014.19.715. Google Scholar [5] X. Feng, Z. Teng and F. Zhang, Global dynamics of a general class of multi-group epidemic models with latency and relapse,, Math. Biosci. Eng., 12 (2015), 99. Google Scholar [6] H. Guo, M. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions,, P. Am. Math. Soc., 136 (2008), 2793. doi: 10.1090/S0002-9939-08-09341-6. Google Scholar [7] Y. Hino, T. Naito and S. Murakami, Functional Differential Equations with Infinite Delay,, Springer, (1991). doi: 10.1007/BFb0084432. Google Scholar [8] G. Huang and A. Liu, A note on global stability for a heroin epidemic model with distributed delay,, Appl. Math. Lett., 26 (2013), 687. doi: 10.1016/j.aml.2013.01.010. Google Scholar [9] G. Huang, J. Wang and J. Zu, Global dynamics of multi-group dengue disease model with latency distributions,, Math. Meth. Appl. Sci., 38 (2015), 2703. doi: 10.1002/mma.3252. Google Scholar [10] M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Differ. Equations, 248 (2010), 1. doi: 10.1016/j.jde.2009.09.003. Google Scholar [11] M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays,, J. Math. Anal. Appl., 361 (2010), 38. doi: 10.1016/j.jmaa.2009.09.017. Google Scholar [12] J. Liu and T. Zhang, Global behaviour of a heroin epidemic model with distributed delays,, Appl. Math. Lett., 24 (2011), 1685. doi: 10.1016/j.aml.2011.04.019. Google Scholar [13] D. R. MacKintosh and G. T. Stewart, A mathematical model of a heroin epidemic: Implications for control policies,, J. Epidemiol. Commun. H., 33 (1979), 299. doi: 10.1136/jech.33.4.299. Google Scholar [14] C. C. McCluskey, Global stability for an {SEIR} epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 6 (2009), 603. doi: 10.3934/mbe.2009.6.603. Google Scholar [15] G. Mulone and B. Straughan, A note on heroin epidemics,, Math. Biosci., 218 (2009), 138. doi: 10.1016/j.mbs.2009.01.006. Google Scholar [16] Y. Muroya, Y. Enatsu and T. Kuniya, Global stability for a multi-group SIRS epidemic model with varying population sizes,, Nonlinear Anal. RWA., 14 (2013), 1693. doi: 10.1016/j.nonrwa.2012.11.005. Google Scholar [17] Y. Muroya and T. Kuniya, Further stability analysis for a multi-group {SIRS} epidemic model with varying total population size,, Appl. Math. Lett., 38 (2014), 73. doi: 10.1016/j.aml.2014.07.005. Google Scholar [18] G. Samanta, Dynamic behaviour for a nonautonomous heroin epidemic model with time delay,, J. Appl. Math. Comput., 35 (2011), 161. doi: 10.1007/s12190-009-0349-z. Google Scholar [19] H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission,, Nonlinear Anal. RWA., 13 (2012), 1581. doi: 10.1016/j.nonrwa.2011.11.016. Google Scholar [20] Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions,, SIAM J. Appl. Math., 73 (2013), 1513. doi: 10.1137/120876642. Google Scholar [21] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model),, SIAM J. Math. Anal., 24 (1993), 407. doi: 10.1137/0524026. Google Scholar [22] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [23] J. Wang and X. Liu, Modeling diseases with latency and nonlinear incidence rates: Global dynamics of a multi-group model,, Math. Meth. Appl. Sci., 39 (2016), 1964. doi: 10.1002/mma.3613. Google Scholar [24] J. Wang, X. Liu, J. Pang and D. Hou, Global dynamics of a multi-group epidemic model with general exposed distribution and relapse,, Osaka J. Math., 52 (2015), 117. Google Scholar [25] J. Wang, J. Pang and X. Liu, Modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model,, J. Biol. Dynam., 8 (2014), 99. doi: 10.1080/17513758.2014.912682. Google Scholar [26] J. Wang and H. Shu, Global dynamics of a multi-group epidemic model with latency, relapse and nonlinear incidence rate,, Math. Biosci. Eng., 13 (2016), 209. doi: 10.3934/mbe.2016.13.209. Google Scholar [27] W. Wang and X.-Q. Zhao, An epidemic model with population dispersal and infection period,, SIAM J. Appl. Math., 66 (2006), 1454. doi: 10.1137/050622948. Google Scholar [28] E. White and C. Comiskey, Heroin epidemics, treatment and ODE modelling,, Math. Biosci., 208 (2007), 312. doi: 10.1016/j.mbs.2006.10.008. Google Scholar
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