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July  2021, 14(7): 2571-2589. doi: 10.3934/dcdss.2020178

Existence and uniqueness results for a smoking model with determination and education in the frame of non-singular derivatives

Department of Mathematics, Balıkesir University, Balıkesir 10145, Turkey

* Corresponding author: Sümeyra Uçar

Received  April 2019 Revised  January 2021 Published  July 2021 Early access  May 2021

These days, it is widely known that smoking causes numerous diseases, as well as resulting in many avoidable losses of life globally, and therefore encumbers the society with enormous unnecessary burdens. The aim of this study is to examine in-depth a smoking model that is mainly influenced by determination and educational actions via CF and AB derivatives. For both fractional order models, the fixed point method is used, which allows us to follow the proof of existence and the results of uniqueness. The effective properties of the above-mentioned fractional models are theoretically exhibited, their results are confirmed by numerical graphs by various fractional orders.

Citation: Sümeyra Uçar. Existence and uniqueness results for a smoking model with determination and education in the frame of non-singular derivatives. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2571-2589. doi: 10.3934/dcdss.2020178
References:
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A. Atangana, On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation, Appl. Math. Comput., 273 (2016), 948-956.  doi: 10.1016/j.amc.2015.10.021.  Google Scholar

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A. Atangana and B. S. T. Alkahtani, Analysis of the Keller-Segel model with a fractional derivative without singular kernel, Entropy, 17 (2015), 4439-4453.  doi: 10.3390/e17064439.  Google Scholar

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A. Atangana and B. S. T. Alkahtani, New model of groundwater flowing within a confine aquifer: application of Caputo-Fabrizio derivative, Arabian Journal of Geosciences, 9 (2016), 6pp. Google Scholar

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A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.  Google Scholar

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A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.  Google Scholar

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A. Atangana and I. Koca, On the new fractional derivative and application to nonlinear Baggs and Freedman model, J. Nonlinear Sci. Appl., 9 (2016), 2467-2480.  doi: 10.22436/jnsa.009.05.46.  Google Scholar

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A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (2018), Paper No. 3, 21 pp. doi: 10.1051/mmnp/2018010.  Google Scholar

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D. Baleanu and A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Commun. Nonlinear Sci. Numer. Simul., 59 (2018), 444-462.  doi: 10.1016/j.cnsns.2017.12.003.  Google Scholar

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D. Baleanu, Z. B. Guvenc and J. A. T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, Dordrecht, 2010. Google Scholar

[12]

H. M. BaskonusT. MekkaouiZ. Hammouch and H. Bulut, Active control of a chaotic fractional order economic system, Entropy, 17 (2015), 5771-5783.  doi: 10.3390/e17085771.  Google Scholar

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T. J. BrinkerS. S. BalderjahnW. Seeger and D. A. Groneberg, Education Against Tobacco (EAT): A quasi-experimental prospective evaluation of a programme for preventing smoking in secondary schools delivered by medical students: A study protocol, BMJ Open, 4 (2014), 1-7.  doi: 10.1136/bmjopen-2014-004909.  Google Scholar

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C. Bullen, Impact of tobacco smoking and smoking cessation on cardiovascular risk and disease, Expert Review of Cardiovascular Therapy, 6 (2008), 883-895.  doi: 10.1586/14779072.6.6.883.  Google Scholar

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M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2015), 73-85.   Google Scholar

[16]

F. Evirgen, Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM, Int. J. Optim. Control. Theor. Appl. IJOCTA, 6 (2016), 75-83.  doi: 10.11121/ijocta.01.2016.00317.  Google Scholar

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F. Evirgen and N. Özdemir, A fractional order dynamical trajectory approach for optimization problem with HPM, Fractional Dynamics and Control, Springer, New York, (2012), 145–155. doi: 10.1007/978-1-4614-0457-6_12.  Google Scholar

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A. FernandezD. Baleanu and H. M. Srivastava, Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 517-527.  doi: 10.1016/j.cnsns.2018.07.035.  Google Scholar

[19]

J. F. Gómez-Aguilar, Analytical and numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations, Phys. A, 494 (2018), 52-75.  doi: 10.1016/j.physa.2017.12.007.  Google Scholar

[20]

O. K. Ham, Stages and processes of smoking cessation among adolescents, Western Journal of Nursing Research, 29 (2007), 301-315.  doi: 10.1177/0193945906295528.  Google Scholar

[21]

F. HaqK. ShahG. ur Rahman and M. Shahzad, Numerical solution of fractional order smoking model via laplace Adomian decomposition method, Alexandria Engineering Journal, 57 (2018), 1061-1069.  doi: 10.1016/j.aej.2017.02.015.  Google Scholar

[22]

K. O. Haustein and D. Groneberg, Tobacco or Health?: Physiological and Social Damages Caused by Tobacco Smoking, Springer-Verlag Berlin, 2010. Google Scholar

[23]

M. KhalidF. S. Khan and A. Iqbal, Perturbation-iteration algorithm to solve fractional giving up smoking mathematical model, International Journal of Computer Applications, 142 (2016), 1-6.  doi: 10.5120/ijca2016909891.  Google Scholar

[24]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.  Google Scholar

[25]

İ. Koca, Analysis of rubella disease model with non-local and non-singular fractional derivatives, Int. J. Optim. Control. Theor. Appl. IJOCTA, 8 (2018), 17-25.  doi: 10.11121/ijocta.01.2018.00532.  Google Scholar

[26]

S. Kumar, A. Ahmadian, R. Kumar, D. Kumar, J. Singh, D. Baleanu and M. Salimi, An efficient numerical method for fractional SIR epidemic model of infectious disease by using Bernstein wavelets, Mathematics, 8 (2020), 558. doi: 10.3390/math8040558.  Google Scholar

[27]

J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2015), 87-92. Google Scholar

[28]

J. T. Machado and A. M. Lopes, Artistic painting: A fractional calculus perspective, Applied Mathematical Modelling, 65 (2019), 614-626.   Google Scholar

[29]

N. ÖzdemirO. P. AgrawalB. B. İskender and D. Karadeniz, Fractional optimal control of a 2-dimensional distributed system using eigenfunctions, Nonlinear Dynamics, 55 (2009), 251-260.  doi: 10.1007/s11071-008-9360-4.  Google Scholar

[30]

N. Özdemir, D. Karadeniz and B. B. İskender, Fractional optimal control problem of a distributed system in cylindrical coordinates, Phys. Lett. A, 373 (2009), 221-226. doi: 10.1016/j.physleta.2008.11.019.  Google Scholar

[31]

N. Özdemir and M. Yavuz, Numerical solution of fractional Black-Scholes equation by using the multivariate Padé approximation, Acta Physica Polonica A, 132 (2017), 1050-1053.   Google Scholar

[32]

J. Singh, Analysis of fractional blood alcohol model with composite fractional derivative, Chaos Solitons Fractals, 140 (2020), 110127, 6 pp. doi: 10.1016/j.chaos.2020.110127.  Google Scholar

[33]

J. Singh, D. Kumar and D. Baleanu, A new analysis of fractional fish farm model associated with Mittag-Leffler-type kernel, Int. J. Biomath., 13 (2020), 2050010, 17 pp. doi: 10.1142/S1793524520500102.  Google Scholar

[34]

J. SinghD. KumarZ. Hammouch and A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316 (2018), 504-515.  doi: 10.1016/j.amc.2017.08.048.  Google Scholar

[35]

J. Singh, D. Kumar, M. A. Qurashi and D. Baleanu, A new fractional model for giving up smoking dynamics, Adv. Difference Equ., (2017), Paper No. 88, 16 pp. doi: 10.1186/s13662-017-1139-9.  Google Scholar

[36]

N. H. SweilamA. M. Nagy and A. A. El-Sayed, Second kind shifted Chebyshev polynomials for solving space fractional order diffusion equation, Chaos Solitons Fractals, 73 (2015), 141-147.  doi: 10.1016/j.chaos.2015.01.010.  Google Scholar

[37]

M. Toufik and A. Atangana, New numerical approaximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, Eur. Phys. J. Plus, 132 (2017), Article number: 444. doi: 10.1140/epjp/i2017-11717-0.  Google Scholar

[38]

E. Uçar, N. Özdemir and E. Altun, Fractional order model of immune cells influenced by cancer cells, Math. Model. Nat. Phenom., 14 (2019), Paper No. 308, 12 pp. doi: 10.1051/mmnp/2019002.  Google Scholar

[39]

S. Ucar, N. Ozdemir, I. Koca and E. Altun, Novel analysis of the fractional glucose-insulin regulatory system with non-singular kernel derivative, European Physical Journal Plus, 135 (2020), 414. Google Scholar

[40]

S. UçarE. UçarN. Özdemir and Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative, Chaos Solitons Fractals, 118 (2019), 300-306.  doi: 10.1016/j.chaos.2018.12.003.  Google Scholar

[41]

S. UllahM. A. KhanM. FarooqZ. Hammouch and D. Baleanu, A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 975-993.  doi: 10.3934/dcdss.2020057.  Google Scholar

[42]

P. Veeresha, D. G. Prakasha, J. Singh, I. Khan and D. Kumar, Analytical approach for fractional extended Fisher-Kolmogorov equation with Mittag-Leffler kernel, Adv. Difference Equ., (2020), Paper No. 174, 17 pp. doi: 10.1186/s13662-020-02617-w.  Google Scholar

[43]

A. Yadav, P. K. Srivastava and A. Kumar, Mathematical model for smoking: Effect of determination and education, Int. J. Biomath., 8 (2015), 1550001, 14 pp. doi: 10.1142/S1793524515500011.  Google Scholar

[44]

M. Yavuz and E. Bonyah, New approaches to the fractional dynamics of schistosomiasis disease model, Phys. A, 525 (2019), 373-393.  doi: 10.1016/j.physa.2019.03.069.  Google Scholar

show all references

References:
[1]

B. S. T. AlkahtaniA. Atangana and I. Koca, Huge analysis of Hepatitis C model within the scope of fractional calculus, J. Nonlinear Sci. Appl., 9 (2016), 6195-6203.  doi: 10.22436/jnsa.009.12.24.  Google Scholar

[2] A. Atangana, Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology, Academic Press, London, 2018.   Google Scholar
[3]

A. Atangana, On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation, Appl. Math. Comput., 273 (2016), 948-956.  doi: 10.1016/j.amc.2015.10.021.  Google Scholar

[4]

A. Atangana and B. S. T. Alkahtani, Analysis of the Keller-Segel model with a fractional derivative without singular kernel, Entropy, 17 (2015), 4439-4453.  doi: 10.3390/e17064439.  Google Scholar

[5]

A. Atangana and B. S. T. Alkahtani, New model of groundwater flowing within a confine aquifer: application of Caputo-Fabrizio derivative, Arabian Journal of Geosciences, 9 (2016), 6pp. Google Scholar

[6]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.  Google Scholar

[7]

A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.  Google Scholar

[8]

A. Atangana and I. Koca, On the new fractional derivative and application to nonlinear Baggs and Freedman model, J. Nonlinear Sci. Appl., 9 (2016), 2467-2480.  doi: 10.22436/jnsa.009.05.46.  Google Scholar

[9]

A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (2018), Paper No. 3, 21 pp. doi: 10.1051/mmnp/2018010.  Google Scholar

[10]

D. Baleanu and A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Commun. Nonlinear Sci. Numer. Simul., 59 (2018), 444-462.  doi: 10.1016/j.cnsns.2017.12.003.  Google Scholar

[11]

D. Baleanu, Z. B. Guvenc and J. A. T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, Dordrecht, 2010. Google Scholar

[12]

H. M. BaskonusT. MekkaouiZ. Hammouch and H. Bulut, Active control of a chaotic fractional order economic system, Entropy, 17 (2015), 5771-5783.  doi: 10.3390/e17085771.  Google Scholar

[13]

T. J. BrinkerS. S. BalderjahnW. Seeger and D. A. Groneberg, Education Against Tobacco (EAT): A quasi-experimental prospective evaluation of a programme for preventing smoking in secondary schools delivered by medical students: A study protocol, BMJ Open, 4 (2014), 1-7.  doi: 10.1136/bmjopen-2014-004909.  Google Scholar

[14]

C. Bullen, Impact of tobacco smoking and smoking cessation on cardiovascular risk and disease, Expert Review of Cardiovascular Therapy, 6 (2008), 883-895.  doi: 10.1586/14779072.6.6.883.  Google Scholar

[15]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2015), 73-85.   Google Scholar

[16]

F. Evirgen, Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM, Int. J. Optim. Control. Theor. Appl. IJOCTA, 6 (2016), 75-83.  doi: 10.11121/ijocta.01.2016.00317.  Google Scholar

[17]

F. Evirgen and N. Özdemir, A fractional order dynamical trajectory approach for optimization problem with HPM, Fractional Dynamics and Control, Springer, New York, (2012), 145–155. doi: 10.1007/978-1-4614-0457-6_12.  Google Scholar

[18]

A. FernandezD. Baleanu and H. M. Srivastava, Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 517-527.  doi: 10.1016/j.cnsns.2018.07.035.  Google Scholar

[19]

J. F. Gómez-Aguilar, Analytical and numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations, Phys. A, 494 (2018), 52-75.  doi: 10.1016/j.physa.2017.12.007.  Google Scholar

[20]

O. K. Ham, Stages and processes of smoking cessation among adolescents, Western Journal of Nursing Research, 29 (2007), 301-315.  doi: 10.1177/0193945906295528.  Google Scholar

[21]

F. HaqK. ShahG. ur Rahman and M. Shahzad, Numerical solution of fractional order smoking model via laplace Adomian decomposition method, Alexandria Engineering Journal, 57 (2018), 1061-1069.  doi: 10.1016/j.aej.2017.02.015.  Google Scholar

[22]

K. O. Haustein and D. Groneberg, Tobacco or Health?: Physiological and Social Damages Caused by Tobacco Smoking, Springer-Verlag Berlin, 2010. Google Scholar

[23]

M. KhalidF. S. Khan and A. Iqbal, Perturbation-iteration algorithm to solve fractional giving up smoking mathematical model, International Journal of Computer Applications, 142 (2016), 1-6.  doi: 10.5120/ijca2016909891.  Google Scholar

[24]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.  Google Scholar

[25]

İ. Koca, Analysis of rubella disease model with non-local and non-singular fractional derivatives, Int. J. Optim. Control. Theor. Appl. IJOCTA, 8 (2018), 17-25.  doi: 10.11121/ijocta.01.2018.00532.  Google Scholar

[26]

S. Kumar, A. Ahmadian, R. Kumar, D. Kumar, J. Singh, D. Baleanu and M. Salimi, An efficient numerical method for fractional SIR epidemic model of infectious disease by using Bernstein wavelets, Mathematics, 8 (2020), 558. doi: 10.3390/math8040558.  Google Scholar

[27]

J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2015), 87-92. Google Scholar

[28]

J. T. Machado and A. M. Lopes, Artistic painting: A fractional calculus perspective, Applied Mathematical Modelling, 65 (2019), 614-626.   Google Scholar

[29]

N. ÖzdemirO. P. AgrawalB. B. İskender and D. Karadeniz, Fractional optimal control of a 2-dimensional distributed system using eigenfunctions, Nonlinear Dynamics, 55 (2009), 251-260.  doi: 10.1007/s11071-008-9360-4.  Google Scholar

[30]

N. Özdemir, D. Karadeniz and B. B. İskender, Fractional optimal control problem of a distributed system in cylindrical coordinates, Phys. Lett. A, 373 (2009), 221-226. doi: 10.1016/j.physleta.2008.11.019.  Google Scholar

[31]

N. Özdemir and M. Yavuz, Numerical solution of fractional Black-Scholes equation by using the multivariate Padé approximation, Acta Physica Polonica A, 132 (2017), 1050-1053.   Google Scholar

[32]

J. Singh, Analysis of fractional blood alcohol model with composite fractional derivative, Chaos Solitons Fractals, 140 (2020), 110127, 6 pp. doi: 10.1016/j.chaos.2020.110127.  Google Scholar

[33]

J. Singh, D. Kumar and D. Baleanu, A new analysis of fractional fish farm model associated with Mittag-Leffler-type kernel, Int. J. Biomath., 13 (2020), 2050010, 17 pp. doi: 10.1142/S1793524520500102.  Google Scholar

[34]

J. SinghD. KumarZ. Hammouch and A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316 (2018), 504-515.  doi: 10.1016/j.amc.2017.08.048.  Google Scholar

[35]

J. Singh, D. Kumar, M. A. Qurashi and D. Baleanu, A new fractional model for giving up smoking dynamics, Adv. Difference Equ., (2017), Paper No. 88, 16 pp. doi: 10.1186/s13662-017-1139-9.  Google Scholar

[36]

N. H. SweilamA. M. Nagy and A. A. El-Sayed, Second kind shifted Chebyshev polynomials for solving space fractional order diffusion equation, Chaos Solitons Fractals, 73 (2015), 141-147.  doi: 10.1016/j.chaos.2015.01.010.  Google Scholar

[37]

M. Toufik and A. Atangana, New numerical approaximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, Eur. Phys. J. Plus, 132 (2017), Article number: 444. doi: 10.1140/epjp/i2017-11717-0.  Google Scholar

[38]

E. Uçar, N. Özdemir and E. Altun, Fractional order model of immune cells influenced by cancer cells, Math. Model. Nat. Phenom., 14 (2019), Paper No. 308, 12 pp. doi: 10.1051/mmnp/2019002.  Google Scholar

[39]

S. Ucar, N. Ozdemir, I. Koca and E. Altun, Novel analysis of the fractional glucose-insulin regulatory system with non-singular kernel derivative, European Physical Journal Plus, 135 (2020), 414. Google Scholar

[40]

S. UçarE. UçarN. Özdemir and Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative, Chaos Solitons Fractals, 118 (2019), 300-306.  doi: 10.1016/j.chaos.2018.12.003.  Google Scholar

[41]

S. UllahM. A. KhanM. FarooqZ. Hammouch and D. Baleanu, A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 975-993.  doi: 10.3934/dcdss.2020057.  Google Scholar

[42]

P. Veeresha, D. G. Prakasha, J. Singh, I. Khan and D. Kumar, Analytical approach for fractional extended Fisher-Kolmogorov equation with Mittag-Leffler kernel, Adv. Difference Equ., (2020), Paper No. 174, 17 pp. doi: 10.1186/s13662-020-02617-w.  Google Scholar

[43]

A. Yadav, P. K. Srivastava and A. Kumar, Mathematical model for smoking: Effect of determination and education, Int. J. Biomath., 8 (2015), 1550001, 14 pp. doi: 10.1142/S1793524515500011.  Google Scholar

[44]

M. Yavuz and E. Bonyah, New approaches to the fractional dynamics of schistosomiasis disease model, Phys. A, 525 (2019), 373-393.  doi: 10.1016/j.physa.2019.03.069.  Google Scholar

Figure 1.  Numerical simulations for the model (7) at $ \sigma = 0.93 $, $ \sigma = 0.75 $ and $ \sigma = 0.6 $, respectively
Figure 2.  Numerical simulations for the model (8) at $ \sigma = 0.93 $, $ \sigma = 0.75 $ and $ \sigma = 0.6 $ respectively
Figure 3.  The effect of the parameters $ a_{4} $ on the smokers population $ s $ of the model (7) for the fractional order $ \sigma = 0.95 $ and $ \sigma = 0.75 $, respectively
Figure 4.  The effect of the parameters $ a_{4} $ on the smokers population $ s $ of the model (8) for the fractional order $ \sigma = 0.95 $ and $ \sigma = 0.75 $, respectively
Figure 5.  The effect of the parameters $ a_{5} $ on the smokers population $ s $ of the model (7) for the fractional order $ \sigma = 0.95 $ and $ \sigma = 0.75 $, respectively
Figure 6.  The effect of the parameters $ a_{5} $ on the smokers population $ s $ of the model (8) for the fractional order $ \sigma = 0.95 $ and $ \sigma = 0.75 $, respectively
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