Analysis and Numerical Treatment of Chain Smokers using Advanced Fractional

Qazi Muhammad Farooqa; Abdul  Ghaffar; Fakher  Abbas

doi:10.48165/jmmfc.2024.1201

Authors

Qazi Muhammad Farooqa Department of Mathematics, Ghazi University D G Khan 32200, Pakistan
Abdul Ghaffar Department of Mathematics, Ghazi University D G Khan 32200, Pakistan
Fakher Abbas Department of Mathematics, Ghazi University D G Khan 32200, Pakistan

DOI:

https://doi.org/10.48165/jmmfc.2024.1201

Keywords:

Solution Boundedness, Non-Local Kernel, Mathematical Modeling

Abstract

In this paper, a fractal fractional derivative is used to examine an environmentally friendly approach of understanding the workings of smoking in humans. We proposed a fractional differential equation system to represent a time-fractional order smoking model with illness effects. Studies are conducted using methodologies. Lipschitz circumstances and linear growth are utilized to demonstrate the presence and distinction of the suggested model in relation to the impact of the global offset. It is confirmed that the fractional order model’s solutions are bounded and positive. When conducting the initial and subsequent derivative assessments, the Lyapunov function is employed to verify analysis of global stability. In order to examine the influence of smoking on humans, the fractional operator is studied. To do this, solutions are constructed applying the extended version of the Mittag-Leffler kernel using a two-step Lagrange polynomial method. Numerical simulation is performed to see how the fractional order smoking model behaves. Such research will aid in understanding the behavior of smoking and in developing human defenses.

References

[1] C. S. Chou and A. Friedman,Introduction,in Introduction to Mathematical Biology. Springer Un dergraduate Texts in Mathematics and Technology, Springer, Cham, 2016.

[2] Biazar J (2006). Solution of the epidemic model by Adomian decomposition method. Applied Mathematics and Computation, 173(2): 1101-1106.

[3] Makinde OD (2007). Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy. Applvcied Mathematics and Computation, 184(2): 842-848.

[4] Arafa AAM, Rida SZ, and Khalil M (2012). Fractional modeling dynamics of HIV and CD4+ T-cells during primary infection. Nonlinear Biomedical Physics, 6(1): 1-7.

[5] Erturk VS, Zaman G, and Momani S (2012). A numericanalytic method for approximating a giving up smoking model containing fractional derivatives. Computers and Mathematics with Applications, 64(10): 3065-3074.

[6] Centers for Disease Control and Prevention (CDC), Annual smoking attributable mortality, years of potential life lost, and economic cost-united state 1995-1999, MMWR. Morbidity and Mortal ity Weekly Report, vol. 51, no. 14, pp. 300303, 2002.

[7] A. Jemal, M. M. Center, C. Desantis, and E. M. Ward, Global patterns of cancer incidence and mortality rates and trends, Cancer Epidemiology, Biomarkers & Prevention, vol. 19, pp. 18931907, 2010.

[8] A. Lahrouz, L. Omari, D. Kiouach, A. Belmaati, Deterministic and stochastic stability of a math ematical model of smoking, Statist. Probab. Lett. 81 (8) (2011) 12761284.

[9] World Health Organization: WHO. (2022). Tobacco. www.who. int. https://www.who.int/news room/fact-sheets/detail/tobacco.

[10] H. Singh, D. Baleanu, J. Singh, H. Dutta, Computational study of fractional order smoking model, Chaos Solitons Fractals 142 (2021) 110440.

[11] S. Ucar, E. Ucar, N. Ozdemir, Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative, Chaos Solitons Fractals 118 (2019) 300306

[12] Z.A. Khan, M.U. Rahman, K. Shah, Study of a fractal fractional smoking models with relapse and harmonic mean type incidence rate, J. Function Spaces (2021) 111.

[13] B. Melkamu, B. Mebrate, A fractional model for the dynamics of smoking tobacco using caputo fabrizio derivative, J. Appl. Math. (2022) 110, https://doi.org/10.1155/2022/2009910.

[14] Ahmad, Aqeel, et al. ”Flip bifurcation analysis and mathematical modeling of cholera disease by taking control measures.” Scientific Reports 14.1 (2024): 10927.

[15] Ahmad, Aqeel, et al. ”Stability Analysis of SARSCoV2 with Heart Attack Effected Patients and Bifurcation.” Advanced Biology 8.4 (2024): 2300540.

[16] H. Bulut, H. M. Baskonus, and F. B. M. Belgecam, The analytic solutions of some fractional ordinary differential equation by Sumudu transform method, Abstr. Appl. Anal., vol. 2013, article 203875, 6 pages, 2013.

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

20

[18] A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons and Fractals, vol. 89, pp. 447454, 2016.

[19] L. L. Huang, D. Baleanue, and S. D. Wu Zeng, A new application of the fractional logistic map, Romanian Journal of Physics, vol. 61, pp. 11721179, 2016.

[20] G. Zaman, Qualitative behavior of giving up smoking model, Bulletin of the Malaysian Mathe matical Sciences Society, vol. 34, pp. 403415, 2011.

[21] J. Singh, D. Kumar, M. A. Qurashi, and D. Baleanu, A new fractional model for giving up smok ing dynamics, Advances in Difference Equations, vol. 2017, no. 1, Article ID 88, 2017.

[22] Arafa AAM, Rida SZ, and Khalil M (2012). Fractional modeling dynamics of HIV and CD4+ T-cells during primary infection. Nonlinear Biomedical Physics, 6(1): 1-7.

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

[24] Lubin JH and Caporaso NE (2006). Cigarette smoking and lung cancer: modeling total exposure and intensity. Cancer Epidemiology and Prevention Biomarkers, 15(3): 517-523.

[25] Garsow CC, Salivia GJ, and Herrera AR (2000). Mathematical models for the dynamics of toba coo use, recovery and relapse. Technical Report Series BU-1505-M, Cornell University, UK.

[26] Mickens RE (1989). Exact solutions to a finite-difference model of a nonlinear reaction-advection equation: Implications for numerical analysis. Numerical Methods for Partial Differential Equa tions, 5(4): 313-325.

[27] Sharomi O and Gumel AB (2008). Curtailing smoking dynamics: a mathematical modeling ap proach. Applied Mathematics and Computation, 195(2): 475-499.

[28] Zeb A, Chohan MI, and Zaman G (2012). The homotopy analysis method for approximating of giving up smoking model in fractional order. Applied Mathematics, 3(8): 914-919.

[29] Alkhudhari Z, Al-Sheikh S, and Al-Tuwairqi S (2014). Global dynamics of a mathematical model on smoking. Hindawi Publishing Corporation, ISRN Applied Mathematics, 2014: Article ID 847075, 7 pages. http://doi.org/10.1155/2014/ 847075

[30] C. Castillo-Garsow, G. Jordan-Salivia, A.E. Rodriguez-Herrera, Mathematical models for the dynamics of tobacco use, Recovery and Relapse. (1997). http://ecommons.cornell.edu/bitstream/ 1813/32095/1/BU-1505-M.pdf.

[31] O. Sharomi, A.B. Gumel, Curtailing smoking dynamics: a mathematical modeling approach, Appl. Math Comput. 195 (2) (2008) 475499.

[32] G. Zaman, Qualitative behavior of giving up smoking models, Bulletin of the Malaysian Mathe matical Sciences Soc. Second Series 34 (2) (2011) 403415.

[33] R. Ullah, M. Khan, G. Zaman, S. Islam, M.A. Khan, S. Jan, T. Gul, Dynamical features of a mathematical model on smoking, J. Appl. Environ. Biol. Sci 6 (1) (2016) 9296.

[34] Ahmad, Aqeel, et al. ”Analytical analysis and bifurcation of pine wilt dynamical transmission with host vector and nonlinear incidence using sustainable fractional approach.” Partial Differen tial Equations in Applied Mathematics 11 (2024): 100830.

21

[35] Khurram Faiz, Aqeel Ahmad, Muhammad Suleman Khan, Safdar Abbas, Control of Marburg Virus Disease Spread in Humans under Hypersensitive Response through Fractal-Fractional . 2024. Journal of Mathematical Modeling and Fractional Calculus 1 (1): 88-113

[36] P. Xiao, Z. Zhang, X. Sun, Smoking dynamics with health education effect, AIMS Mathematics 3 (4) (2018) 584599.

[37] B. Fekede, B. Mebrate, Sensitivity and mathematical model analysis on secondhand smoking tobacco, J. Egyptian Math. Soc. 28 (1) (2020) 50.

[38] C. Liu, W. Sun, X. Yi, Optimal control of a fractional smoking system, J. Industrial and Manage. Optimization 19 (4) (2023) 29362954.

[39] P. Veeresha, D.G. Prakasha, H.M. Baskonus, Solving smoking epidemic model of fractional order using a modified homotopy analysis transform method, Mathematical Sciences 13 (2019) 115128.

[40] P. Veeresha, N.S. Malagi, D.G. Prakasha, H.M. Baskonus, An efficient technique to analyze the fractional model of vectorborne diseases, Phys. Scr. 97 (5) (2022) 054004.

[41] P. Veeresha, E. Ilhan, D.G. Prakasha, H.M. Baskonus, W. Gao, Regarding on the fractional math ematical model of Tumour invasion and metastasis, Comput. Model. Eng. Sci. 127 (3) (2021) 10131036.

[42] C. Maji, F. Al Basir, D. Mukherjee, C. Ravichandran, K. Nisar, COVID-19 propagation and the usefulness of awareness-based control measures: a mathematical model with delay, AIMS Math 7 (7) (2022) 1209112105

[43] Ahmad, Aqeel, et al. ”Study on symptomatic and asymptomatic transmissions of COVID-19 including flip bifurcation.” International Journal of Biomathematics (2024): 2450002.

[44] Z. Zhang, W. Zhang, K.S. Nisar, N. Gul, A. Zeb, V. Vijayakumar, Dynamical aspects of a tu berculosis transmission model incorporating vaccination and time delay, Alex. Eng. J. 66 (2023) 287300.

[45] V. Vijayakumar, K.S. Nisar, A. Shukla, B. Hazarika, R. Samidurai, An investigation on the ap proximate controllability of impulsive neutral delay differential inclusions of second order, Math ematical Methods in the Appl. Sciences (2022).

[46] K. Kavitha, V. Vijayakumar, R. Udhayakumar, C. Ravichandran, Results on controllability of Hilfer fractional differential equations with infinite delay via measures of noncompactness, Asian J. Control 24 (3) (2022) 14061415.