Abstract
The qualitative study of a smoking model with parametric conditions for diseases controlling under the influence of smoking is investigated through rigorous mathematical study. The mathematical modeling of epidemiological smoking model having six compartments is traced out. Mathematical expressions for smoke-free and smoke-present equilibrium points have been developed. The strength of Lyapunov functional theory has been exploited to show that smoke-free equilibrium point is globally asymptotically stable whenever basic reproduction number \(R_\circ <1\). The competency of graphical and theoretic process is utilized to observe the global behavior of unique smoke equilibrium point. The sensitivity analysis of the model is performed through the basic reproduction number and diseased classes effectively to design reliable, robust and stable control strategies.
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R. Ahmed, R. Rashid, P.W. McDonald, S.W. Ahmed, Prevalence of cigarette smoking among young adults in Pakistan. J. Pak. Med. Assoc. 58, 597–601 (2008)
World Health Organization. Tobacco. https://www.who.int/news-room/fact-sheets/detail/tobacco (2019)
C.C. Garsow, G.J. Salivia, A.R. Herrera, Mathematical models for dynamics of tobacco use, Recovery and Relapse, Technical Report Series BU-1505-M, January (1997)
Z. Alkhudhari, S. Sheikh, S. Al-Tuwairqi, Global dynamics of a mathematical model on smoking. Int. Sch. Res. Not. Appl. Math. 2014, Article ID: 847075, 2014, 7 (2014)
F.J. Chaloupka, Curbing the epidemic: governments and the economics of tobacco control. Tobacco Control 8, 196–201 (1999)
C.C. Garsow, G.J. Salivia, A.R. Herrera, Mathematical Models for the Dynamics of Tobacco Use, Recovery, and Relapse, Technical Report Series BU- 1505-M (Cornell University, Ithaca, NY, 2000)
A.T. Merchant, S.P. Luby, G. Parveen, Smoking among males in a low socioeconomic area of Karachi. J. Pak. Med. Assoc. 48, 62–63 (1998)
A.T. Merchant, S.P. Luby, G. Parveen, Smoking in Pakistan: more than cancer and heart disease. J. Pak. Med. Assoc. 48, 77–79 (1998)
M.A. Shaikh, A. Kamal, Prevalence and pattern of smoking in university students—perspective from Islamabad. J. Coll. Phys. Surg. Pak. 14, 194–194 (2004)
K. Nasir, N. Rehan, Epidemiology of cigarette smoking in Pakistan. Addiction 96, 1847–1854 (2001)
M. Zaman, U. Irfan, Mukhtiar, E. Irshad, Prevalence of cigarette smoking among Peshawar University students. Pak. J. Chest Med. 8, 9–18 (2002)
O. Sharomi, A.B. Gumel, Curtailing smoking dynamics, a mathematical modeling approach. Appl. Math. Comput. 195, 475–499 (2008)
A. Lahrouz, L. Omari, D. Kiouach, A. Belmaati, Deterministic and stochastic stability of a mathematical model of smoking. Stat. Probab. Lett. 81, 1276–1284 (2011)
G. Zaman, Qualitative behavior of giving up smoking models. Bull. Mal. Sci. Soc. 34, 403–415 (2011)
G. Zaman, Optimal campaign in the smoking dynamics. Comput. Math. Methods Med. Vol. 2011, Article ID: 163834, 2011, 9 (2011)
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, 92–96 (2016)
A.U. Awan, A. Sharif, T. Hussain, M. Ozair, Smoking model with cravings to smoke. Adv. Stud. Biol. 9, 31–41 (2017)
A.L. Mojeeb, I.K. Adu, Modelling the dynamics of smoking epidemic. J. Adv. Math. Comput. Sci. 1–19 (2017)
N.H. Shah, F.A. Thakkar, B.M. Yeolekar, Stability analysis of tuberculosis due to smoking. Int. J. Innov. Sci. Res. Technol. 3(1) (2018)
A.M. Pulecio-Montoya, L.E. López-Montenegro, L.M. Benavides, Analysis of a mathematical model of smoking. Contemp. Eng. Sci. 12, 117–129 (2019)
Z. Zhang, R. Wei, W. Xia, Dynamical analysis of a giving up smoking model with time delay. Adv. Differ. Equ. 2019(1), 505 (2019)
S.A. Khan, K. Shah, G. Zaman, F. Jarad, Existence theory and numerical solutions to smoking model under Caputo–Fabrizio fractional derivative. Chaos Interdiscip. J. Nonlinear Sci. 29(1), 013128 (2019)
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, 300–306 (2019)
G. ur Rahman, R.P. Agarwal, Q. Din, Mathematical analysis of giving up smoking model via harmonic mean type incidence rate. Appl. Math. Comput. 354, 128–148 (2019)
C. Sun, J. Jia, Optimal control of a delayed smoking model with immigration. J. Biol. Dyn. 13(1), 447–460 (2019)
P. Veeresha, D.G. Prakasha, H.M. Baskonus, Solving smoking epidemic model of fractional order using a modified homotopy analysis transform method. Math. Sci. 13(2), 115–128 (2019)
A.M.S. Mahdy, N.H. Sweilam, M. Higazy, Approximate solution for solving nonlinear fractional order smoking model. Alex. Eng. J. 59(2), 739–752 (2020)
A.A. Alshareef, H.A. Batarfi, Stability analysis of chain, mild and passive smoking model. Am. J. Comput. Math. 10(01), 31 (2020)
Z. Zhang, J. Zou, R.K. Upadhyay, A. Pratap, Stability and Hopf bifurcation analysis of a delayed tobacco smoking model containing snuffing class. Adv. Differ. Equ. 2020(1), 1–19 (2020)
A.A. Kashif, I.Q. Memon, A. Siyal, Thermal transmittance and thermo-magnetization of unsteady free convection viscous fluid through non-singular differentiations. Physica Scripta (2020). https://doi.org/10.1088/1402-4896/abc981
A.A. Kashif, Abdon Atangana, Dual fractional modeling of rate type fluid through non-local differentiation. Numer. Methods Partial Differ. Equ. 1–16 (2020). https://doi.org/10.1002/num.22633
J.F. Gómez-Aguilar, A. Atangana, V.F. Morales-Delgado, Electrical circuits RC, LC, and RL described by Atangana Baleanu fractional derivatives. Int. J. Circuit Theory Appl. 45(11), 1514–33 (2017)
K.A. Abro, A. Atangana, Numerical and mathematical analysis of induction motor by means of AB-fractal-fractional differentiation actuated by drilling system. Numer. Methods Partial Differ. Equ. (2020). https://doi.org/10.1002/num.22618
K.A. Abro, A. Siyal, B. Souayeh, A. Atangana, Application of statistical method on thermal resistance and conductance during magnetization of fractionalized free convection flow. Int. Commun. Heat Mass Transfer 119, 104971 (2020). https://doi.org/10.1016/j.icheatmasstransfer.2020.104971
K.M. Owolabi, A. Atangana, Robustness of fractional difference schemes via the Caputo subdiffusion–reaction equations. Chaos Solitons Fractals 111, 119–127 (2018)
A.A. Kashif, M. Soomro, A. Atangana, J.F.G. Aguilar, Thermophysical properties of Maxwell Nanoluids via fractional derivatives with regular kernel. J. Therm. Anal. Calorim. (2020). https://doi.org/10.1007/s10973-020-10287-9
K.A. Abro, B. Das, A scientific report of non-singular techniques on microring resonators: an application to optical technology. Optik Int. J. Light Electron Opt. 224, 165696 (2020). https://doi.org/10.1016/j.ijleo.2020.165696
A. Khan, J.F. Gómez-Aguilar, T.S. Khan, H. Khan, Stability analysis and numerical solutions of fractional order HIV/AIDS model. Chaos Solitons Fractals 122, 119–128 (2019)
Q. Ali, S. Riaz, A.U. Awan, K.A. Abro, Thermal investigation for electrified convection flow of Newtonian fluid subjected to damped thermal flux on a permeable medium. Physica Scripta (2020). https://doi.org/10.1088/1402-4896/abbc2e
A.A. Kashif, Role of fractal-fractional derivative on ferromagnetic fluid via fractal Laplace transform: a first problem via fractal-fractional differential operator. Eur. J. Mech. B Fluids 85, 76–81 (2021). https://doi.org/10.1016/j.euromechflu.2020.09.002
A. Coronel-Escamilla, F. Torres, J.F. Gómez-Aguilar, R.F. Escobar-Jiménez, G.V. Guerrero-Ramírez, On the trajectory tracking control for an SCARA robot manipulator in a fractional model driven by induction motors with PSO tuning. Multibody Syst. Dyn. 43(3), 257–277 (2018)
A.K. Ali, M.H. Laghari, J.F. Gomez-Aguilar, Application of Atangana-Baleanu fractional derivative to carbon nanotubes based non-Newtonian nanofluid: applications in nanotechnology. J. Appl. Comput. Mech. 6(SI), 1260–1269 (2020). https://doi.org/10.22055/JACM.2020.33461.2229
A. Yoku, H. Durur, A.K. Ali, D. Kaya, Role of Gilson-Pickering equation for the different types of soliton solutions: a nonlinear analysis. Eur. Phys. J. Plus 135, 657 (2020). https://doi.org/10.1140/epjp/s13360-020-00646-8
A. Khan, H. Khan, J.F. Gómez-Aguilar, T. Abdeljawad, Existence and Hyers–Ulam stability for a nonlinear singular fractional differential equations with Mittag–Leffler kernel. Chaos Solitons Fractals 127, 422–427 (2019)
A.U. Awan, M. Tahir, A.A. Kashif, Multiple soliton solutions with chiral nonlinear Schrödinger’s equation in (2+1)-dimensions. Eur. J. Mech. B Fluids (2020). https://doi.org/10.1016/j.euromechflu.2020.07.014
A.K. Ali, A. Atangana, Porous effects on the fractional modeling of magnetohydrodynamic pulsatile flow: an analytic study via strong kernels. J. Therm. Anal. Calorim. (2020). https://doi.org/10.1007/s10973-020-10027-z
V.F. Morales-Delgado, J.F. Gómez-Aguilar, K.M. Saad, M. AltafKhan, P. Agarwal, Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: a fractional calculus approach. Physica A Stat. Mech. Appl. 523, 45–65 (2019)
A.A. Kashif, A. Atangana, Numerical study and chaotic analysis of meminductor and memcapacitor through fractal-fractional differential operator. Arab. J. Sci. Eng. (2020). https://doi.org/10.1007/s13369-020-04780-4
A.A. Kashif, A. Abdon, A comparative analysis of electromechanical model of piezoelectric actuator through Caputo–Fabrizio and Atangana–Baleanu fractional derivatives. Math. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6638
K.A. Abro, J.F. Gomez-Aguilar, Role of Fourier sine transform on the dynamical model of tensioned carbon nanotubes with fractional operator. Math. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6655
J.E. Solís-Pérez, J.F. Gómez-Aguilar, A. Atangana, Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws. Chaos Solitons Fractals 114, 175–185 (2018)
A.K. Ali, A. Atangana, Mathematical analysis of memristor through fractal-fractional differential operators: a numerical study. Math. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6378
B. Lohana, A.A. Kashif, A.W. Shaikh, Thermodynamical analysis of heat transfer of gravity-driven fluid flow via fractional treatment: an analytical study. J. Therm. Anal. Calorim. (2020). https://doi.org/10.1007/s10973-020-09429-w
K.A. Abro, A. Atangana, A comparative study of convective fluid motion in rotating cavity via Atangana–Baleanu and Caputo–Fabrizio fractal-fractional differentiations. Eur. Phys. J. Plus 135, 226–242 (2020). https://doi.org/10.1140/epjp/s13360-020-00136-x
A.A. Kashif, A. Siyal, A. Atangana, Thermal stratification of rotational second-grade fluid through fractional differential operators. J. Therm. Anal. Calorim. (2020). https://doi.org/10.1007/s10973-020-09312-8
A.K. Ali, A. Atangana, Role of non-integer and integer order differentiations on the relaxation phenomena of viscoelastic fluid. Physica Scripta 95, 035228 (2020). https://doi.org/10.1088/1402-4896/ab560c
K.A. Abro, A fractional and analytic investigation of thermo-diffusion process on free convection flow: an application to surface modification technology. Eur. Phys. J. Plus 135(1), 31–45 (2019). https://doi.org/10.1140/epjp/s13360-019-00046-7
Q. Din, M. Ozair, T. Hussain, U. Saeed, Qualitative behavior of a smoking model. Adv. Differ. Equ. 2016(1), 1–12 (2016)
P.V.D. Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)
D.J. Struik (ed.), A Source Book in Mathematics 1200–1800 (Princeton University Press, Princeton, 1986), pp. 89–93
Z. Shuai, P.V.D. Driessche, Global stability of infectious diseases models using lyapunov functions. SIAM J. Appl. Math. 73, 1513–1532 (2013)
J.P. LaSalle, The stability of dynamical systems, regional conference series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA (1976)
F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969)
D.B. West, Introduction to Graph Theory (Prentice-Hall, Upper Saddle River, NJ, 1996)
M.Y. Li, Z. Shuai, Global-stability problems for coupled systems of differential equations on networks. J. Differ. Equ. 248, 1–20 (2010)
N. Chitnis, J.M. Hyman, J.M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull. Math. Biol. 70(5), 1272 (2008)
L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko, The Mathematical Theory of Optimal Processes, vol. 4 (Gordon and Breach Science Publishers, New York, NY, 1986)
W.H. Fleming, R.W. Rishel, Deterministic and Stochastic Optimal Control (Springer, New York, 1975)
D.L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering (Academic Press, New York, 1982)
S. Lenhart, J.T. Workman, Optimal Control Applied to Biological Models, Mathematical and Computational Biology Series (Chapman and Hall/CRC Press, London/Boca Raton, 2007)
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Hussain, T., Awan, A.U., Abro, K.A. et al. A mathematical and parametric study of epidemiological smoking model: a deterministic stability and optimality for solutions. Eur. Phys. J. Plus 136, 11 (2021). https://doi.org/10.1140/epjp/s13360-020-00979-4
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DOI: https://doi.org/10.1140/epjp/s13360-020-00979-4