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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

Lili Liu 1, , Xianning Liu 1, and  Jinliang Wang 2,

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 $n=2$.
Keywords: delay, graph-theoretic approach, global stability., Multi-group heroin epidemic model, heterogeneous populations.
Mathematics Subject Classification: Primary: 34D23; Secondary: 92B3.
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, Berlin, 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), 365-382. 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), 105-112. 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), 715-733. 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), 99-115.  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), 2793-2802. 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), 687-691. 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), 2703-2718. 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), 1-20. 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), 38-47. 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), 1685-1692. 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), 299-304. 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), 603-610. doi: 10.3934/mbe.2009.6.603.  Google Scholar

[15]

G. Mulone and B. Straughan, A note on heroin epidemics, Math. Biosci., 218 (2009), 138-141. 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), 1693-1704. 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), 73-78. 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), 161-178. 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), 1581-1592. 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), 1513-1532. 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), 407-435. 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), 29-48. 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), 1964-1976. 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), 117-138.  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), 99-116. 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), 209-225. 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), 1454-1472. doi: 10.1137/050622948.  Google Scholar

[28]

E. White and C. Comiskey, Heroin epidemics, treatment and ODE modelling, Math. Biosci., 208 (2007), 312-324. 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, Berlin, 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), 365-382. 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), 105-112. 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), 715-733. 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), 99-115.  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), 2793-2802. 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), 687-691. 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), 2703-2718. 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), 1-20. 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), 38-47. 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), 1685-1692. 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), 299-304. 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), 603-610. doi: 10.3934/mbe.2009.6.603.  Google Scholar

[15]

G. Mulone and B. Straughan, A note on heroin epidemics, Math. Biosci., 218 (2009), 138-141. 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), 1693-1704. 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), 73-78. 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), 161-178. 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), 1581-1592. 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), 1513-1532. 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), 407-435. 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), 29-48. 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), 1964-1976. 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), 117-138.  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), 99-116. 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), 209-225. 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), 1454-1472. doi: 10.1137/050622948.  Google Scholar

[28]

E. White and C. Comiskey, Heroin epidemics, treatment and ODE modelling, Math. Biosci., 208 (2007), 312-324. doi: 10.1016/j.mbs.2006.10.008.  Google Scholar

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