Stability and Convergence Analysis of Smoking Impact in Society with Algorithm Aspects

Main Article Content

Aqeel Ahmad
Maryam Shahid
Muhammad Farman
M.O. Ahmad


In this manuscript, an epidemic model employed the dynamics of drugs usage among adults. Among smokers, often the desire to quit smoking arises. A large number of smokers attempt to quit, but only a few of them are successful. A non-linear mathematical model is employed to study and assess the dynamics of smoking and its impact on public health in a community. We prove the essential properties, bounded, positivity and well-posed, also local and global stability analysis has been made for the epidemic model. The sensitivity analysis of the model is provided by threshold or reproductive number as well as analyzed qualitatively. We develop an unconditionally convergent nonstandard finite difference scheme by applying Mickens approach φ(h) = h + O(h2) instead of h to control the spread of bad impact in society. Finally numerical simulations are also established to investigate the influence of the system parameters on the spread of the smoking impact in society.

Article Details


  1. J. Biazar, Solution of the epidemic model by Adomian decomposition method, Appl. Math. Comput. 173 (2006), 1101-1106.
  2. S. Busenberg and P. Driessche, Analysis of a disease transmission model in a population with varying size, J. Math. Biol. 28 (1990), 65-82.
  3. A.M.A. El-Sayed, S.Z. Rida and A.A.M. Arafa, On the solutions of time-fractional bacterial chemotaxis in a diffusion gradient chamber, Int. J. Nonlinear Sci. 7 (2009), 485-495.
  4. A.A.M. Arafa, S.Z. Rida and M. Khalil, Fractional modeling dynamics of HIV and 4 T-cells during primary infection, Nonlinear Biomed. Phys. 6 (2012), 1-7.
  5. C.M. Kribs-Zaleta, Structured models for heterosexual disease transmission, Math. Biosci. 160 (1999), 83-108.
  6. B. Buonomo and D. Lacitignola, On the dynamics of an SEIR epidemic model with a convex incidence rate, Ricerche Mat. 57 (2008), 261-281.
  7. X. Liu and C. Wang, Bifurcation of a predator-prey model with disease in the prey, Nonlinear Dyn. 62 (2010), 841-850.
  8. F. Haq, K. Shah, G.U Rahman and M. Shahzad. Numerical solution of fractional order smoking model via laplace Adomian decomposition method, Alex. Eng. J. 57 (2018), 1061-1069.
  9. C. Chavez and B. Song; Dynamical models of tuberculosis and their applications; Math. Biosci. Eng. 1 (2004), 361-404.
  10. A. McNeill, M. Raw, J. Whybrow and P. Bailey; National strategy for smoking cessation treatment in England; Addiction 100 (S.2) (2005), 1-11.
  11. R.P. Sargent, R.M. Shepard and S.A. Glantz; Admission for myocardial infarction associated with public smoking bun; Br.Med. J. 1 (2004), 328-977.
  12. Y.M. Terry-McElrath, M.A. Wakefield, S. Emery, H. Saffer, G.M. Szczypka and P. O. Malley P; State antitobacco advertising and smoking outcomes by gender and race/ethnicity; Ethnicity and Health 12 (2007), 339-362.
  13. R. Ullah, M. Khan, G. Zaman, S. Islam, M.A. Khan, S. Jan and T. Gul, Dynamical Featurers of mathemtical model on smoking, J. Appl. Environ. Biol. Sci., 6 (2016), 92-96.
  14. (3th October, 2016).
  15. (17th November, 2016).
  16. 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).
  17. O. Sharomi and A. B. Gumel, Curtailing smoking dynamics: A mathematical modeling approach, Appl. Math. Comput. 195 (2008), 475-499.
  18. G. Zaman, Qualitative behavior of giving up smoking model; Bull. Malaysian Math. Sci. Soc. 2 (2011), 403-415.
  19. S.A. Matintu, Smoking as Epedemic: Modeling and Simulation study, American J. Appl. Math. 5 (2017), 31-38.
  20. A. Ahmad, M. Farman, F. Yasin and M. O. Ahmad, Dynamical transmission and effect of smoking in society, Int. J. Adv. Appl. Sci. 5(2) (2018), 71-75
  21. F. Ashraf, A. Ahmad, M. U. Saleem, M. Farman and M.O. Ahmad, Dynamical behavior of HIV immunology model with non-integer time fractional derivatives, Int. J. Adv. Appl. Sci. 5(3) (2018), 39-45, .
  22. A. Ahmad, M. Farman, M. O Ahmad, N. Raza and Abdullah, Dynamical behavior of SIR epidemic model with non-integer time fractional derivatives: A mathematical analysis, Int. J. Adv. Appl. Sci. 5(1) (2018), 123-129.
  23. J.B. Swartz, Use of a multistage model to predict time trends in smoking induced lung cancer, J. Epidemiol. Commun. Healt. 46 (1992), 11-31.
  24. F. Brauer and C. Castillo-Cha vez, Mathematical Models in Population Biology and Epidemiology, Springer, (2001).
  25. A. Zeb, G. Zaman, V.S. Erturk, B. Alzalg, F. Yousafzai and M. Khan, Approximating a giving up smoking dynamic on adolescent nicotine dependence in fractional order, PLoS ONE, 11 (2016), 10-15.
  26. G. Zaman, Optimal campaign in the smoking dynamics, Comput. Math. Method. Med. 2011 (2011), Article ID 163834.
  27. G. Zaman, Qualitative behavior of giving up smoking models, Bull. Malay. Math. Sci. Soc. 34 (2011), 403-415.
  28. V. Suat Erturk, G. Zamanb and S. Momanic, A numeric analytic method for approximating a giving up smoking model containing fractional derivatives, Comput. Math. Appl. 64 (2012), 3065-3074.
  29. R. E. Mickens, Exact solutions to a finite difference model of a nonlinear reactions advection equation: Implications for numerical analysis, Numer. Methods Partial Differ. Equations, 5 (1989), 313-325.
  30. R. E.Mickens, Applications of Nonstandard finite difference Schemes, World Scientific, Singaporen (2000).
  31. R. Anguelov and J. M.-S. Lubuma, Nonstandard finite difference method by nonlocal approximations, Math. Comput. Simul. 61 (2003), 465-475.
  32. R. E. Mickens, Nonstandard finite difference Models of differential equations, World Scientific, Singapore (1994).
  33. R. Anguelov and J.M.-S. Lubuma, Contributions to the mathematics of the nonstandard nite dierencemethodandapplications, Numer. Methods Partial Differ. Equations, 17 (2001), 518-543.
  34. J.M.-S. Lubuma and K.C. Patidar, Non-standard methods for singularly perturbed problems possessing oscillatory/layer solutions, Appl. Math. Comput. 187(2) (2007), 1147-1160.
  35. L.W. Roeger, Exact difference schemes, in A. B. Gumel Mathematics of Continuous and Discrete Dynamical Systems, Contemp. Math. 618 (2014), 147-161.