Stability analysis of a giving up smoking model

  • Authors

    • Zainab Alkhudhari King Abdulaziz University
    • Sarah Al-Sheikh King Abdulaziz University
    • Salma Al-Tuwairqi King Abdulaziz University
    2014-04-26
    https://doi.org/10.14419/ijamr.v3i2.2239

  • Abstract

    In this paper we present and analyze a generalization of the giving up smoking model that was introduced by Sharomi and Gumel [4], in which quitting smoking can be temporary or permanent. In our model, we study a population with peer pressure effect on temporary quitters and we consider also the possibility of temporary quitters becoming permanent quitters and the impact of this transformation on the existence and stability of equilibrium points. Numerical results are given to support the results.

    Keywords: Smoking model, Smoking generation number, Local stability, Global stability, Second additive compound matrix, Liapunov function.

  • References

    1. MM. Bassiony, Smoking in Saudi Arabia, Saudi Med Journal 30 (2009) 876-881.
    2. World Health Organization report on tobacco, 2013. http://www.who.int/tobacco/global_report/2013/en/index.html
    3. C. Castillo-Garsow, G. Jordan-Salivia, and A. Rodriguez Herrera, Mathematical models for the dynamics of tobacco use, recovery, and relapse, Technical Report Series BU-1505-M, Cornell University, Ithaca, NY, USA, 1997.
    4. O. Sharomi, A.B. Gumel, Curtailing smoking dynamics: A mathematical modeling approach, Applied Mathematics and Computation 195 (2008) 475-499.
    5. A. Lahrouz, L. Omari, D. Kiouach, A. Belma‚ti, Deterministic and stochastic stability of a mathematical model of smoking, Statistics and Probability Letters 81 (2011) 1276-1284.
    6. G.Zaman, Qualitative Behavior of Giving Up Smoking Models, Bulletin of the Malaysian Mathematical Sciences Society 34 (2011) 403-415.
    7. G.Zaman, Optimal Campaign in the Smoking Dynamics, Computational and Mathematical Methods in Medicine, 2011(2011) Article ID 163834, 9 pages
    8. V.S.Ert¸rk, G.Zamanb, S.Momanic, A numericñanalytic method for approximating a giving up smoking model containing fractional derivatives, Computers and Mathematics with Applications 64 (2012) 3065-3074
    9. A. Zeb, G. Zaman and S. Momani, Square-root dynamics of a giving up smoking model, Applied Mathematical Modelling 37 (2013) 5326-5334
    10. G.A.K. van Voorn, B.W. Kooi, Smoking epidemic eradication in a eco-epidemiological dynamical model, Ecological Complexity, 14 (2013) 180-189
    11. Z. Alkhudhari, S. Al-Sheikh, S. Al-Tuwairqi, Global Dynamics of a Mathematical Model on Smoking, ISRN Applied Mathematics vol. (2014), Article ID 847075.
    12. P.van den Driessche, J.Watmough, Reproduction numbers and sub-threshhold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences 180 (2002) 29-48.
    13. F.Brauer, P.van den Driessche and J.Wu, Mathematical Epidemiology, Springer-Verlag (2008).
    14. L.Perko, Differential Equations and Dynamic Systems, Springer Verlag(1991).
    15. L.Cai, X.Li, M.Ghosh, B.Guo, Stability analysis of an HIV/AIDS epidemic model with treatment, Journal of Computational and Applied Mathematics 229 (2009) 313-323.
    16. C. C. McCluskey, P.van den Driessche, Global analysis of two tuberculosis models, Journal of Dynamics and Di§erential Equation 16 (2004) 139-166.
    17. J.Arino, C.C. McCluskey, and P.van den Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM, Journal Applied Mathematics 64 (2003) 260-276.
    18. X.Ge, M.Arcak, A su¢ cient condition for additive D-stability and application to reaction-diffusion models, Systems and Control Letters 58 (2009) 736-74.
    19. M.W.Hirsch, S.Smale, and R.L.Devaney, Differential equations, dynamical systems and an introduction to chaos, Elsevier Academic press (1974).
    20. S.Busenberg, P.van den Driessche, Analysis of a disease transmission model in a population with varying size, Journal of Mathematical Biology 28 (1990) 257-270.

  • Downloads

  • How to Cite

    Alkhudhari, Z., Al-Sheikh, S., & Al-Tuwairqi, S. (2014). Stability analysis of a giving up smoking model. International Journal of Applied Mathematical Research, 3(2), 168-177. https://doi.org/10.14419/ijamr.v3i2.2239

    Received date: 2014-03-17

    Accepted date: 2014-04-12

    Published date: 2014-04-26