American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density
  • DCDS-B Home
  • This Issue
  • Next Article
    Phase transition of oscillators and travelling waves in a class of relaxation systems
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 and 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.

[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.

[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.

[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.

[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.

[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.

[7]

Y. Hino, T. Naito and S. Murakami, Functional Differential Equations with Infinite Delay, Springer, 1991. doi: 10.1007/BFb0084432.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications, Springer, Berlin, 1967.

[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.

[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.

[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.

[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.

[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.

[7]

Y. Hino, T. Naito and S. Murakami, Functional Differential Equations with Infinite Delay, Springer, 1991. doi: 10.1007/BFb0084432.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[1]

Chunmei Zhang, Wenxue Li, Ke Wang. Graph-theoretic approach to stability of multi-group models with dispersal. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 259-280. doi: 10.3934/dcdsb.2015.20.259
[2]

Toshikazu Kuniya, Yoshiaki Muroya. Global stability of a multi-group SIS epidemic model for population migration. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1105-1118. doi: 10.3934/dcdsb.2014.19.1105
[3]

Huan Su, Pengfei Wang, Xiaohua Ding. Stability analysis for discrete-time coupled systems with multi-diffusion by graph-theoretic approach and its application. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 253-269. doi: 10.3934/dcdsb.2016.21.253
[4]

Yoshiaki Muroya, Toshikazu Kuniya, Yoichi Enatsu. Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3057-3091. doi: 10.3934/dcdsb.2015.20.3057
[5]

Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with vaccination age, distributed delay and random perturbation. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1083-1106. doi: 10.3934/mbe.2015.12.1083
[6]

Jinliang Wang, Xianning Liu, Toshikazu Kuniya, Jingmei Pang. Global stability for multi-group SIR and SEIR epidemic models with age-dependent susceptibility. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2795-2812. doi: 10.3934/dcdsb.2017151
[7]

Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3515-3550. doi: 10.3934/dcdsb.2016109
[8]

Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with generalized nonlinear incidence and vaccination age. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 977-996. doi: 10.3934/dcdsb.2016.21.977
[9]

Jinliang Wang, Hongying Shu. Global analysis on a class of multi-group SEIR model with latency and relapse. Mathematical Biosciences & Engineering, 2016, 13 (1) : 209-225. doi: 10.3934/mbe.2016.13.209
[10]

Xiaomei Feng, Zhidong Teng, Fengqin Zhang. Global dynamics of a general class of multi-group epidemic models with latency and relapse. Mathematical Biosciences & Engineering, 2015, 12 (1) : 99-115. doi: 10.3934/mbe.2015.12.99
[11]

Gunduz Caginalp, Mark DeSantis. Multi-group asset flow equations and stability. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 109-150. doi: 10.3934/dcdsb.2011.16.109
[12]

Yoshiaki Muroya. A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model). Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 999-1008. doi: 10.3934/dcdss.2015.8.999
[13]

Bin Fang, Xue-Zhi Li, Maia Martcheva, Li-Ming Cai. Global stability for a heroin model with two distributed delays. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 715-733. doi: 10.3934/dcdsb.2014.19.715
[14]

C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837
[15]

Maya Mincheva, Gheorghe Craciun. Graph-theoretic conditions for zero-eigenvalue Turing instability in general chemical reaction networks. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1207-1226. doi: 10.3934/mbe.2013.10.1207
[16]

Miljana Jovanović, Vuk Vujović. Stability of stochastic heroin model with two distributed delays. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2407-2432. doi: 10.3934/dcdsb.2020016
[17]

Lin Zhao, Zhi-Cheng Wang, Liang Zhang. Threshold dynamics of a time periodic and two–group epidemic model with distributed delay. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1535-1563. doi: 10.3934/mbe.2017080
[18]

Abdennasser Chekroun, Mohammed Nor Frioui, Toshikazu Kuniya, Tarik Mohammed Touaoula. Mathematical analysis of an age structured heroin-cocaine epidemic model. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4449-4477. doi: 10.3934/dcdsb.2020107
[19]

Rui Wang, Xiaoyue Li, Denis S. Mukama. On stochastic multi-group Lotka-Volterra ecosystems with regime switching. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3499-3528. doi: 10.3934/dcdsb.2017177
[20]

Yukihiko Nakata, Yoichi Enatsu, Yoshiaki Muroya. On the global stability of an SIRS epidemic model with distributed delays. Conference Publications, 2011, 2011 (Special) : 1119-1128. doi: 10.3934/proc.2011.2011.1119

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (196)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy