Mathematical Analysis for an Age-Structured Heroin Epidemic Model

  • Lili Liu
  • Xianning LiuEmail author


In this paper, an age-structured heroin epidemic model, where the susceptibility of individuals and the relapse of heroin users in treatment are described by two age-dependent variables, is formulated and analyzed. The basic reproduction ratio of the model is derived and proved to be a threshold condition, which completely determines the global behaviors of the model. The asymptotic smoothness of the semiflow generated by the family of solutions, uniform persistence and existence of an interior global attractor have been presented for establishing and defining a Lyapunov functional on this attractor. Some control strategies of heroin and two special cases of the model formulation are addressed.


Lyapunov functional Global stability Age-structure Heroin model 

Mathematics Subject Classification

34D23 34K20 92D30 



L. Liu is supported by the National Natural Science Foundation of China (11601239). X. Liu is supported by the National Natural Science Foundation of China (11671327). We are grateful to the editors and the anonymous referees for their careful reading and helpful comments which led to great improvement of our manuscript.


  1. 1.
    Adams, R.A., Fournier, J.J.: Sobolev Spaces, vol. 140. Academic Press, San Diego (2003) zbMATHGoogle Scholar
  2. 2.
    Brauer, F., Shuai, Z., van den Driessche, P.: Dynamics of an age-of-infection cholera model. Math. Biosci. Eng. 10, 1335–1349 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Burattini, M., Massad, E., Coutinho, F., Azevedo-Neto, R., Menezes, R., Lopes, L.: A mathematical model of the impact of crack-cocaine use on the prevalence of HIV/AIDS among drug users. Math. Comput. Model. 28(3), 21–29 (1998) CrossRefzbMATHGoogle Scholar
  4. 4.
    Comiskey, C.: National prevalence of problematic opiate use in Ireland. Tech. rep., EMCDDA Tech. Report (1999) Google Scholar
  5. 5.
    Hale, J.K.: Functional Differential Equations. Springer, Berlin (1971) CrossRefzbMATHGoogle Scholar
  6. 6.
    Hale, J.K.: Asymptotic Behavior of Dissipative Systems. Am. Math. Soc., Providence (1988) zbMATHGoogle Scholar
  7. 7.
    Hao, W., Su, Z., Xiao, S., Fan, C., Chen, H., Liu, T., Young, D.: Longitudinal surveys of prevalence rates and use patterns of illicit drugs at selected high-prevalence areas in China from 1993 to 2000. Addiction 99(9), 1176–1180 (2004) CrossRefGoogle Scholar
  8. 8.
    Huang, G., Liu, A.: A note on global stability for a heroin epidemic model with distributed delay. Appl. Math. Lett. 26, 687–691 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Huang, G., Liu, X., Takeuchi, Y.: Lyapunov functions and global stability for age-structured HIV infection model. SIAM J. Appl. Math. 72(1), 25–38 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kelly, A.W., Carvalho, M., Teljeur, C.: Prevalence of Opiate Use in Ireland 2000–2001: A 3-Source Capture Recapture Study: A Report to the National Advisory Committee on Drugs Sub-Committee on Prevalence. Stationery Office, London (2003) Google Scholar
  11. 11.
    Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics (Part I). Proc. R. Soc. A 115, 700–721 (1927) CrossRefzbMATHGoogle Scholar
  12. 12.
    Li, X., Zhou, Y., Stanton, B.: Illicit drug initiation among institutionalized drug users in China. Addiction 97(5), 575–582 (2002) CrossRefGoogle Scholar
  13. 13.
    Liu, X., Wang, J.: Epidemic dynamics on a delayed multi-group heroin epidemic model with nonlinear incidence rate. J. Nonlinear Sci. Appl. 9(5), 2149–2160 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Liu, J., Zhang, T.: Global behaviour of a heroin epidemic model with distributed delays. Appl. Math. Lett. 24, 1685–1692 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lyapunov, A.M.: The general problem of the stability of motion. Int. J. Control 55, 531–534 (1992) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Magal, P.: Compact attractors for time periodic age-structured population models. Electron. J. Differ. Equ. 2001, 1–35 (2001) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Magal, P., Zhao, X.Q.: Global attractors and steady states for uniformly persistent dynamical systems. SIAM J. Math. Anal. 37, 251–275 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Melnik, A.V., Korobeinikov, A.: Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility. Math. Biosci. Eng. 10, 369–378 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mulone, G., Straughan, B.: A note on heroin epidemics. Math. Biosci. 218, 138–141 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nyabadza, F., Hove-Musekwa, S.D.: From heroin epidemics to methamphetamine epidemics: modelling substance abuse in a South African province. Math. Biosci. 225(2), 132–140 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rossi, C.: Operational models for epidemics of problematic drug use: the Mover–Stayer approach to heterogeneity. Socio-Econ. Plan. Sci. 38(1), 73–90 (2004) CrossRefGoogle Scholar
  22. 22.
    Samanta, G.P.: Dynamic behaviour for a nonautonomous heroin epidemic model with time delay. J. Appl. Math. Comput. 35, 161–178 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Smith, H.L., Thieme, H.R.: Dynamical Systems and Population Persistence. Am. Math. Soc., Providence (2011) zbMATHGoogle Scholar
  24. 24.
    Webb, G.F.: Theory of Nonlinear Age-Dependent Population Dynamics. Dekker, New York (1985) zbMATHGoogle Scholar
  25. 25.
    White, E., Comiskey, C.: Heroin epidemics, treatment and ODE modelling. Math. Biosci. 208, 312–324 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yang, J., Li, X., Zhang, F.: Global dynamics of a heroin epidemic model with age structure and nonlinear incidence. Int. J. Biomath. 9(03), 1650033 (2016) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Key Laboratory of Eco-environments in Three Gorges Reservoir Region (Ministry of Education), School of Mathematics and StatisticsSouthwest UniversityChongqingChina
  2. 2.Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis, Complex Systems Research CenterShanxi UniversityTaiyuanChina

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