[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Discontinuity, Nonlinearity and Complexity
ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Dynamical Behavior and Mathematical Analysis of Fractional Order Smoking Model

Discontinuity, Nonlinearity, and Complexity 12(1) (2023) 57--74 | DOI:10.5890/DNC.2023.03.005

Aqeel Ahmad$^1$, Muhammad Farman$^2$, Ali Akgul$^3$, Adnan Khan$^4$

Download Full Text PDF

 

Abstract

In this paper the fractional order smoking model is represent with Caputo and Caputo Fabrizio fractional derivative operator of order $\phi \in (0, 1]$ for dynamical transmission of smoking. Human beings face dangerous diseases caused by smoking, including arms, lungs, stomach, cervix, breast, pancreatic cancer and many others. Stability and qualitative analysis of model is studied to show the dynamical behaviour of the model in feasible region. It's important to note that a more powerful approach for computing convergent solutions is applied for mathematical models based on a fractional order differential equation structure. Study of the convergence is often provided to demonstrate the process's effectiveness. It shows the stability, uniqueness and applicability of the model for the control of smoking in the society. Numerical simulation are established to show the actual behavior of the smoking spread.

References

  1. [1]  Biazar, J. (2006), Solution of the epidemic model by Adomian decomposition method, Applied Mathematics and Computation, 173, 1101-1106.
  2. [2]  El-Sayed, A.M.A., Rida, S.Z., and Arafa, A.A.M. (2009), On the solutions of time-fractional bacterial chemotaxis in a diffusion gradient chamber, International Journal of Nonlinear Science, 7, 485-495.
  3. [3]  Makinde, O.D. (2007), Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy, Applied Mathematics And Computation, 184, 842-848.
  4. [4]  Arafa, A.A.M., Rida, S.Z. and Khalil, M. (2012), Fractional modeling dynamics of HIV and 4 T-cells during primary infection, Nonlinear Biomedical Physics, 6, 1-7.
  5. [5]  Buonomo, B. and Lacitignola, D. (2008), On the dynamics of an SEIR epidemic model with a convex incidence rate, Ricerche di Matematica, 57, 261-281.
  6. [6]  Liu, X. and Wang, C. (2010), Bifurcation of a predator-prey model with disease in the prey, Nonlinear Dynamics, 62, 841-850.
  7. [7]  Erturk, V.S., Zaman, G., and Momani, S. (2012), A numeric analytic method for approximating a giving up smoking model containing fractional derivatives, Computers \& Mathematics with Applications, 64, 3068-3074.
  8. [8]  Zaman, G. (2011), Optimal campaign in the smoking dynamics, Computational and Mathematical Methods in Medicine, 2011, 1-9, Article ID 163834.
  9. [9]  Zaman, G. (2011), Qualitative behavior of giving up smoking models, Bulletin of the Malaysian Mathematical Sciences Society, 34, 403-415.
  10. [10]  Ahmad, A., Farman, M., Yasin, F., and Ahmad, M.O.(2018), Dynamical transmission and effect of smoking in society, International Journal of Advanced and Applied Sciences, 5(2), 71-75.
  11. [11]  Haq, F., Shah, K., Khan, A., Shahzad, M., and Rahman, G.U. (2017), Numerical solution of fractional order epidemic model of a vector born disease by laplace adomian decomposition method, Punjab University Journal of Mathematics, 49(2), 13-22.
  12. [12]  He, J.H. (2015), A new fractional derivative and its application to explanation of polar bear hairs, Journal of King Saud University Science, 28(2), 190-192.
  13. [13]  Saleem, M.U., Farman, M., Ahmad, A., Naeem, M., and Ahmad, M.O. (2019), Stability analysis and control of fractional order diabetes mellitus model for artificial pancreas, Punjab University Journal of Mathematics, 51(4), 85-101.
  14. [14]  Ahmad, A., Farman, M., Ahmad, M.O., Raza, N., and Abdullah, M. (2018), Dynamical behavior of SIR epidemic model with non-integer time fractional derivatives, A mathematical analysis, International Journal of Advanced and Applied Sciences, 5(1), 123-129.
  15. [15]  Carvalho, A. and Pinto, C.M.A. (2016), A delay fractional order model for the co-infection of malaria and HIV/AIDS, International Journal of Dynamics and Control, 5(1), 168-186.
  16. [16]  Liu, X. and Wang, C. (2010), Sentry Health Monitors, http://www.lifeclinic.com/focus/diabetes/finger.asp (2016). Accessed 5 Apr 2016., Nonlinear Dynamics, 62, 841-850.
  17. [17]  He, J.H. (2014), A tutorial review on fractal spacetime and fractional calculus, International Journal of Theoretical Physics, 53, 3698-3718.
  18. [18]  Wang, K.L. and Liu, S.Y. (2016), He's fractional derivative for non-linear fractional heat transfer equation, Thermal Science, 20, 793-796.
  19. [19]  Zafar, A.A., Riaz, M.B., and Imran, M.A. (2019), Unsteady rotational flow of fractional maxwell fluid in a cylinder subject to shear stress on the boundary, Punjab University Journal of Mathematics, 50(2), 21-32.
  20. [20]  He, J.H. (2011), A new fractal derivation, Thermal Science, 15, S145-S147.
  21. [21]  Kumar, D., Singh, J., and Baleanu, D. (2016), Numerical computation of a fractional model of differential-difference equation, Journal of Computational and Nonlinear Dynamics, 11(6), DOI:10.1115/1.4033899.
  22. [22]  Kumar, D., Singh, J., and Baleanu, D. (2018), A new fractional model for convective straight fins with temperature dependent thermal conductivity, Thermal Science, 22(6), 2791-2802.
  23. [23]  He, J.H. (2014), A tutorial review on fractal spacetime and fractional calculus, International Journal of Theoretical Physics, 53, 3698-3718.
  24. [24]  He, J.H. (2006), Homotopy perturbation method for solving boundary value problems, Physics letters A, 350, 87-88.
  25. [25]  He, J.H., Elagan, S.K., and Li, Z.B. (2012), Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Physics letters A, 376(4), 257-259.
  26. [26]  Srivastava, H.M., Kumar, D., and Singh, J. (2017), An efficient analytical technique for fractional model of vibration equation, Applied Mathematical Modelling, 45, 192-204.
  27. [27]  Ahmad, A., Farman, M., Ahmad, M.O., Raza, N., and Abdullah, M. (2018), Dynamical behavior of SIR epidemic model with non-integer time fractional derivatives: A mathematical analysis, International Journal of Advanced and Applied Sciences, 5(1), 123-129.
  28. [28]  Ashraf, F., Ahmad, A., Saleem, M.U., Farman, M., and Ahmad, M.O. (2018), Dynamical behavior of HIV immunology model with non-integer time fractional derivatives, International Journal of Advanced and Applied Sciences, 5(3), 39-45.
  29. [29]  Ahmad, H., Aly, R., Seadawy, Khan, T.A., and Thounthong, P. (2020), Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations, Journal of Taibah University for Science, 14(1), 346-358.
  30. [30]  Akgul, A. and Ahmad, H. (2020), Reproducing kernel method for Fangzhu's oscillator forwater collection from air, Mathematical Methods in the Applied Sciences, 2020, 1-10.
  31. [31]  Ahmad, H., Khan, T.A., Ahmad, I., Stanimirovic, P.S., and Chu, Y.M. (2020), A new analyzing technique for nonlinear time fractional Cauchy reaction-diffusion model equations, Results in Physics, 19, 1-8.
  32. [32]  Ghanbari, B., Kumar, S., and Kumar, R. (2020), A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative, Chaos, Solitons and Fractals, 133, 109619.
  33. [33]  Emile, F., Goufoa, D., Kumar, S., and Mugisha, S.B. (2020), Similarities in a fifth-order evolution equation with and with no singular kernel, Chaos, Solitons and Fractals, 130, 109467.
  34. [34]  Kumar, S., Kumar, R., Osman, M.S., and Samet, B. (2021), A wavelet based numerical scheme for fractional order SEIR epidemic of measles by using Genocchi polynomials, Numerical Methods Partial Differential Equations, 37, 1250-1268.
  35. [35]  Ali, K.K., Salam, M.A.A.E., Mohamed, E.M.H., Samet, B., Kumar, S., and Osman, M.S. (2020), Numerical solution for generalized nonlinear fractional integro-differential equations with linear functional arguments using Chebyshev series, Advances in Difference Equations, 494(1), 1-23.
  36. [36]  Losada, J. and Nieto, J.J. (2015), Properties of a new fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1, 8792.
  37. [37]  Bushnaq, S., Khan, S.A., Shah, K., and Zaman, G. (2018), Mathematical analysis of HIV/AIDS infection model with Caputo-Fabrizio fractional derivative, Cogent Mathematics $\&$ Statistics, 5, 1-12.
  38. [38]  Caputo, M. and Fabrizio, M. (2015), A new definition offractional derivative with out singular kernel, Progress in Fractional Differentiation \& Applications, 1, 73-85.
  39. [39]  Ullah, R., Khan, M., and Zaman, G. (2010), Dynamical features of a mathematical model on smoking, J. Appl. Environmental \& Biological Sciences, 6, 92-96.
  40. [40]  Ahmad, H., Khan, T.A., Stanimirović, P.S., Chu, Y.M., and Ahmad, I. (2020), Modified variational iteration algorithm-II: convergence and applications to diffusion models, Complexity, 2020, 1-14.

Copyright © 2011-2024 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates