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Discontinuity, Nonlinearity, and Complexity 15(2) (2026) 241--251 | DOI:10.5890/DNC.2026.06.008

V. Kavitha$^{1}$, R. Sowmiya$^{1}$, R. Deepa$^{2}$, D. Baleanu$^{3}$, M. Mallika Arjunan$^{4}$

$^{1}$ Division of Mathematics and Robotics Engineering, School of Sciences, Arts & Media, Karunya Institute of Technology and Sciences,Karunya Nagar, Coimbatore-641114, Tamil Nadu, India

$^{2}$ Department of Mathematics, Panimalar Engineering College, Chennai–600123, Tamil Nadu, India

$^{3}$ Department of Computer Science and Mathematics, Labanese American University, Beirut, Lebanon

$^{4}$ Department of Mathematics, School of Arts, Sciences and Humanities, SASTRA Deemed to be University, Thanjavur-613401, Tamil Nadu, India

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Abstract

This study examines the dynamics of a smoking model using the Caputo ($\c$) fractional derivative, which effectively captures memory effects inherent in complex systems. We conduct a mathematical analysis of the fractional model, ensuring the positivity of solutions, invariant region and demonstrating the existence and uniqueness of solutions through fixed-point theory. For numerical simulations, we employ a generalized predictor-corrector method tailored for the $\c$ derivative. The model is computationally solved, and results are graphically illustrated across various fractional-order values.

References

1. [1]	Castillo-Garsow, C., Jordan-Salivia, G., and Herrera, A.R. (1997), Mathematical models for the dynamics of tobacco use, recovery, and relapse, Technical Report Series BU-1505-M, Cornell University, Ithaca, NY.

2. [2]	Sharomi, O. and Gumel, A.B. (2008), Curtailing smoking dynamics: a mathematical modeling approach, Applied Mathematics and Computation, 195(2), 475-499.

3. [3]	Zaman, G. (2011), Qualitative behavior of giving up smoking models, Bulletin of the Malaysian Mathematical Sciences Society, 34(2), 403-415.

4. [4]	Zaman, G. (2011), Optimal campaign in the smoking dynamics, Computational and Mathematical Methods in Medicine, 163834.

5. [5]	Lahrouz, A., Omari, L., Kiouach, D., and Belmaati, A. (2011), Deterministic and stochastic stability of a mathematical model of smoking, Statistics $\&$ Probability Letters, 81(8), 1276-1284.

6. [6]	Roy, P.K., Mondal, J., Bhattacharyya, R., Bhattacharya, S., and Szabados, T. (2013), Extinction of disease pathogenesis in infected population and its subsequent recovery: a stochastic approach, Journal of Applied Mathematics, 381286.

7. [7]	Sofia, I.R. and Mini Ghosh (2023), Mathematical modeling of smoking habits in the society, Stochastic Analysis and Applications, 41(5), 918-937.

8. [8]	Van Voorn, G.A.K. and Kooi, B.W. (2013), Smoking epidemic eradication in an eco-epidemiological dynamical model, Ecological Complexity, 14, 180-189.

9. [9]	Alkhudhari, Z., Al-sheikh, S., and Al-Tuwairqi, S. (2014), Global dynamics of a mathematical model on smoking, ISRN Mathematical Analysis, 847075.

10. [10]	Yadav, A., Srivastava, P.K., and Kumar, A. (2015), Mathematical model for smoking: effect of determination and education, International Journal of Biomathematics, 8(1), 1550001.

11. [11]	Sharma, A. and Misra, A.K. (2015), Backward bifurcation in a smoking cessation model with media campaigns, Applied Mathematical Modelling, 39, 1087-1098.

12. [12]	Din, Q., Ozair, M., Hussain, T., and Saeed, U. (2016), Qualitative behavior of a smoking model, Advances in Difference Equations, 96.

13. [13]	Singh, J., Kumar, D., Qurashi, M.A., and Baleanu, D. (2017), A new fractional model for giving up smoking dynamics, Advances in Difference Equations, 88.

14. [14]	Jung, J.H., Park, A., and Jung, I.H. (2018), Qualitative and sensitivity analysis of the effect of electronic cigarettes on smoking cessation, Computational and Mathematical Methods in Medicine, 3738584.

15. [15]	Zhang, Z., Wei, R., and Xia, W. (2019), Dynamical analysis of a giving up smoking model with time delay, Advances in Difference Equations, 505.

16. [16]	Odibat, Z. and Baleanu, D. (2023), New solutions of the fractional differential equations with modified Mittag-Leffler kernel, Journal of Computational and Nonlinear Dynamics, 18(9), 091007.

17. [17]	Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.

18. [18]	Xuan, L., Ahmad, S., Ullah, A., Saifullah, S., Akgul, A., and Qu, H. (2022), Bifurcations, stability analysis and complex dynamics of Caputo fractal-fractional cancer model, Chaos, Solitons $\&$ Fractals, 159, 112113.

19. [19]	Iskakova, K., Alam, M.M., Ahmad, S., Saifullah, S., Akgul, A., and Yilmaz, G. (2023), Dynamical study of a novel 4D hyperchaotic system: an integer and fractional order analysis, Mathematics and Computers in Simulation, 208, 219-245.

20. [20]	Alqahtani, R.T., Ahmad, S., and Akgul, A. (2021), Dynamical analysis of bio-ethanol production model under generalized nonlocal operator in Caputo sense, Mathematics, 9(19), 2370.

21. [21]	Kreyszig, E. (1978), Introductory Functional Analysis with Applications, Wiley, New York.