Journal of Applied Mathematics and Computing

, Volume 35, Issue 1–2, pp 161–178 | Cite as

Dynamic behaviour for a nonautonomous heroin epidemic model with time delay

Article
First Online:
Received:

Abstract

In this paper, we have modified the White and Comiskey heroin epidemic model (White and Comiskey in Math. Biosci. 208:312–324, 2007) into a nonautonomous heroin epidemic model with distributed time delay. We have introduced some new threshold values R * and R * and further obtained that the heroin-using career will be permanent when R *>1 and the heroin-using career will be going to extinct when R *<1. Using the method of Lyapunov functional, some sufficient conditions are derived for the global asymptotic stability of the system. The aim of this modification is to identify the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.

Keywords

Heroin epidemic Time delay Permanence Lyapunov functional Global stability 

Mathematics Subject Classification (2000)

93C15 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anderson, R.M., May, R.M.: Population biology of infectious diseases. Part I. Nature 280, 361–367 (1979) CrossRefGoogle Scholar
  2. 2.
    Capasso, V.: Mathematical Structures of Epidemic Systems. Lectures Notes in Biomathematics, vol. 97. Springer, Berlin (1993) MATHCrossRefGoogle Scholar
  3. 3.
    Comiskey, C., Cox, G.: Research outcome study in Ireland (ROSIE): evaluating drug treatment effectiveness, baseline findings, March 2005. www.nuim.ie/ROSIE/ResearchHistory.shtml
  4. 4.
    Diekmann, O., Heesterbeek, J.A.P.: Mathematical Epidemiology of Infectious Diseases: Model Building Analysis, and Interpretation. Wiley, New York (2000) Google Scholar
  5. 5.
    Fairchild, C.K.: Modeling trends in emergency room drug abuse episodes. In: Proceedings of the Survey Research Method Section, pp. 305–309. American Statistical Association (1994) Google Scholar
  6. 6.
    Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993) MATHGoogle Scholar
  7. 7.
    Hay, G., Gannon, M., MacDougall, J., Millar, T., Eastwood, C., McKeganey, N.: Local and national estimates of the prevalence of opiate use and/or crack cocaine use. In: R. Murray, L. Tinsley (eds.) Measuring Different Aspects of Problem in Drug Use: Methodological Developments, N. Singleton, Online Report OLR 16/06. London, Home Office (2006) Google Scholar
  8. 8.
    Kermark, W.O., Mckendrick, A.G.: Contributions to the mathematical theory of epidemics. Part I. Proc. R. Soc. A 115(5), 700–721 (1927) CrossRefGoogle Scholar
  9. 9.
    Ma, W., Song, M., Takeuchi, Y.: Global stability of an SIR epidemic model with time delay. Appl. Math. Lett. 17, 1141–1145 (2004) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ma, Z., Zhou, Y., Wang, W., Jin, Z.: Mathematical Modelling and Research of Epidemic Dynamical Systems. Science Press, Beijing (2004) Google Scholar
  11. 11.
    Meng, X., Chen, L., Cheng, H.: Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination. Appl. Math. Comput. 186, 516–529 (2007) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Mulone, G., Straughan, B.: A note on heroin epidemics. Math. Biosci. 218, 138–141 (2009) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Sanchez, F., Wang, X., Castillo-Cahvez, C., Gorman, D.M., Gruenwald, P.J.: Drinking as an epidemic: a simple mathematical model with recovery and relapse. In: Witkiewitz, K., Marlett, G.A. (eds.) Therapist’s Guide to Evidence-Based Relapse Prevention, p. 353. Academic Press, New York (2007) CrossRefGoogle Scholar
  14. 14.
    Sporer, K.A.: Acute heroin overdose. Ann. Intern. Med. 130, 584–590 (1999) Google Scholar
  15. 15.
    Teng, Z., Chen, L.: The positive periodic solutions of periodic Kolmogorov type systems with delays. Acta Math. Appl. Sin. 22, 446–456 (1999) MATHMathSciNetGoogle Scholar
  16. 16.
    White, E., Comiskey, C.: Heroin epidemics, treatment and ODE modelling. Math. Biosci. 208, 312–324 (2007) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Zhang, T., Teng, Z.: On a nonautonomous SEIRS model in epidemiology. Bull. Math. Biol. 69, 2537–2559 (2007) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Zhang, T., Teng, Z.: Permanence and extinction for a nonautonomous SIRS epidemic model with time delay. Appl. Math. Model. 33, 1058–1071 (2009) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2009

Authors and Affiliations

  1. 1.Mathematical InstituteSlovak Academy of SciencesBratislavaSlovak Republic
  2. 2.Department of MathematicsBengal Engineering and Science UniversityShibpurIndia

Personalised recommendations