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July  2021, 14(7): 2199-2212. doi: 10.3934/dcdss.2020145

System response of an alcoholism model under the effect of immigration via non-singular kernel derivative

Fırat Evirgen 1,, , Sümeyra Uçar 1, , Necati Özdemir 1, and  Zakia Hammouch 2,

1. 

Department of Mathematics, Balıkesir University, Balıkesir 10145, Turkey
2. 

Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, 75000, Vietnam
3. 

Department of Medical Research, China Medical University Hospital, Taichung 40402, Taiwan
4. 

Department of Sciences, École normale supérieure, Moulay Ismail University of Meknes, Meknes 50000, Morocco

* Corresponding author: Fırat Evirgen.

Received  April 2019 Revised  November 2020 Published  May 2021

In this study, we aim to comprehensively investigate a drinking model connected to immigration in terms of Atangana-Baleanu derivative in Caputo type. To do this, we firstly extend the model describing drinking model by changing the derivative with time fractional derivative having Mittag-Leffler kernel. The existence and uniqueness of the drinking model solutions together with the stability analysis is shown by the help of Banach fixed point theorem. The special solution of the model is investigated using the Sumudu transformation and then, we present some numerical simulations for the different fractional orders to emphasize the effectiveness of the used derivative.

Keywords: Fractional derivatives and integrals, Atangana-Baleanu derivative, existence and uniqueness theory, fixed point theory, mathematical model.
Mathematics Subject Classification: Primary: 26A33, 34A12, 00A71; Secondary: 65R10.
Citation: Fırat Evirgen, Sümeyra Uçar, Necati Özdemir, Zakia Hammouch. System response of an alcoholism model under the effect of immigration via non-singular kernel derivative. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2199-2212. doi: 10.3934/dcdss.2020145
References:
[1]

B. S. T. Alkahtani, I. Koca and A. Atangana, Analysis of a new model of H1N1 spread: Model obtained via Mittag-Leffler function, Advances in Mechanical Engineering, 9 (2017). doi: 10.1177/1687814017705566.  Google Scholar

[2]

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

[3]

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

[4]

A. Atangana and I. Koca, New direction in fractional differentiation, Mathematics in Natural Science, 1 (2017), 18-25.  doi: 10.22436/mns.01.01.02.  Google Scholar

[5]

D. Baleanu and A. Mendes Lopes, Handbook of Fractional Calculus with Applications, Volume 7-8, De Gruyter, 2019. doi: 10.3389/978-2-88945-958-2.  Google Scholar

[6]

F. B. M. Belgacem, R. Silambarasan, H. Zakia and T. Mekkaoui, New and extended applications of the natural and sumudu transforms: Fractional diffusion and stokes fluid flow realms, in Advances in Real and Complex Analysis with Applications (eds. M. Ruzhansky, Y. Cho, P. Agarwal and I. Area), Birkhäuser, (2017), 107–120. doi: 10.1007/978-981-10-4337-6_6.  Google Scholar

[7]

H. Bulut, D. Kumar, J. Singh, R. Swroop and H. M. Baskonus, Analytic study for a fractional model of HIV infection of CD4+T lymphocyte cells, Mathematics in Natural Science, 2 (2018), 33-43.  doi: 10.22436/mns.02.01.04.  Google Scholar

[8]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404.  doi: 10.3934/mbe.2004.1.361.  Google Scholar

[9]

M. A. Dokuyucu and E. Celik, Nonlinear diffusion for chemotaxis and birth-death process for Keller-Segel model, New Trends Math. Sci., 4 (2016), 204-211. doi: 10.20852/ntmsci.2016318931.  Google Scholar

[10]

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

[11]

F. Evirgen and N. Özdemir, Multistage adomian decomposition method for solving NLP problems over a nonlinear fractional dynamical system, J. Comput. Nonlinear Dynam., 6 (2011), 021003. doi: 10.1115/1.4002393.  Google Scholar

[12]

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

[13]

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

[14]

J. F. Gómez-Aguilar and A. Atangana, Fractional derivatives with the power-law and the Mittag-Leffler kernel applied to the nonlinear Baggs-Freedman model, Fractal and Fractional, 2 (2018), 10. doi: 10.3390/fractalfract2010010.  Google Scholar

[15]

J. F. Gómez-Aguilar, A. Atangana and V. F. Morales-Delgado, Electrical circuits RC, LC, and RL described by Atangana-Baleanu fractional derivatives, International Journal of Circuit Theory and Applications, 45 (2017), 1514-1533.   Google Scholar

[16]

H.-F. Huo and L.-X. Feng, Global stability for an HIV/AIDS epidemic model with different latent stages and treatment, Appl. Math. Model., 37 (2013), 1480-1489.  doi: 10.1016/j.apm.2012.04.013.  Google Scholar

[17]

F. Jarad, T. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos, Solitons Fractals, 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006.  Google Scholar

[18]

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

[19]

I. Koca, Modelling the spread of Ebola virus with Atangana-Baleanu fractional operators, Eur. Phys. J. Plus, 133 (2018), 100. doi: 10.1140/epjp/i2018-11949-4.  Google Scholar

[20]

Y. Koçak, M. A. Dokuyucu and E. Çelik, Well-posedness of optimal control problem for the Schrodinger equations with complex potential, Int. J. Math. Comput., 26 (2015), 11-16.   Google Scholar

[21] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.  doi: 10.1142/9781848163300.  Google Scholar
[22]

G. Mulone and B. Straughan, Modeling binge drinking, Int. J. Biomath., 5 (2012), 1250005. doi: 10.1142/S1793524511001453.  Google Scholar

[23]

Z. M. Odibat and S. Momani, Application of variational iteration method to nonlinear differential equation of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 7 (2006), 27-34.  doi: 10.1515/IJNSNS.2006.7.1.27.  Google Scholar

[24]

N. Özdemir, Y. Povstenko, D. Avci and B. B. İskender, Optimal boundary control of thermal stresses in a plate based on time-fractional heat conduction equation, Journal of Thermal Stresses, 37 (2014), 969-980.  doi: 10.1080/01495739.2014.912937.  Google Scholar

[25]

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.  doi: 10.12693/APhysPolA.132.1050.  Google Scholar

[26]

Y. Povstenko, D. Avci, B. B. İ. Eroǧlu and N. Özdemir, Control of thermal stresses in axissymmetric problems of fractional thermoelasticity for an infinite cylindrical domain, Thermal Science, 21 (2017), 19-28.  doi: 10.2298/TSCI160421236P.  Google Scholar

[27]

D. G. Prakasha, P. Veeresha and H. M. Baskonus, Analysis of the dynamics of hepatitis E virus using the Atangana-Baleanu fractional derivative, The European Physical Journal Plus, 134 (2019), 11p. doi: 10.1140/epjp/i2019-12590-5.  Google Scholar

[28]

G. ur Rahman, R. P. Agarwal, L. Liu and A. Khan, Threshold dynamics and optimal control of an age-structured giving up smoking model, Nonlinear Anal. Real World Appl., 43 (2018), 96-120.  doi: 10.1016/j.nonrwa.2018.02.006.  Google Scholar

[29]

J. Singh, D. Kumara, Z. 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

[30]

D. L. Suthar, S. D. Purohit and R. K. Parmar, Generalized fractional calculus of the multiindex Bessel function, Mathematics in Natural Science, 1 (2017), 26-32.  doi: 10.22436/mns.01.01.03.  Google Scholar

[31]

K. A. Touchent, Z. Hammouch, T. Mekkaoui and F. B. M. Belgacem, Implementation and convergence analysis of homotopy perturbation coupled with sumudu transform to construct solutions of local-fractional PDEs, Fractal Fract., 2 (2018), 22p. doi: 10.3390/fractalfract2030022.  Google Scholar

[32]

M. Toufik and A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, The European Physical Journal Plus, 132 (2017), 444. doi: 10.1140/epjp/i2017-11717-0.  Google Scholar

[33]

E. Ucar, N. Özdemir and E. Altun, Fractional order model of immune cells influenced by cancer cells, Math. Model. Nat. Phenom., 14 (2019), 12 pp. doi: 10.1051/mmnp/2019002.  Google Scholar

[34]

S. Uçar, E. Uçar, N. 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

[35]

H. Xiang, C.-C. Zhu and H.-F. Huo, Modelling the effect of immigration on drinking behaviour, J. Biol. Dyn., 11 (2017), 275-298.  doi: 10.1080/17513758.2017.1337243.  Google Scholar

[36]

M. Yavuz and N. Ozdemir, Numerical inverse Laplace homotopy technique for fractional heat equations, Thermal Science, 22 (2018), 185-194.  doi: 10.2298/TSCI170804285Y.  Google Scholar

show all references

References:
[1]

B. S. T. Alkahtani, I. Koca and A. Atangana, Analysis of a new model of H1N1 spread: Model obtained via Mittag-Leffler function, Advances in Mechanical Engineering, 9 (2017). doi: 10.1177/1687814017705566.  Google Scholar

[2]

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

[3]

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

[4]

A. Atangana and I. Koca, New direction in fractional differentiation, Mathematics in Natural Science, 1 (2017), 18-25.  doi: 10.22436/mns.01.01.02.  Google Scholar

[5]

D. Baleanu and A. Mendes Lopes, Handbook of Fractional Calculus with Applications, Volume 7-8, De Gruyter, 2019. doi: 10.3389/978-2-88945-958-2.  Google Scholar

[6]

F. B. M. Belgacem, R. Silambarasan, H. Zakia and T. Mekkaoui, New and extended applications of the natural and sumudu transforms: Fractional diffusion and stokes fluid flow realms, in Advances in Real and Complex Analysis with Applications (eds. M. Ruzhansky, Y. Cho, P. Agarwal and I. Area), Birkhäuser, (2017), 107–120. doi: 10.1007/978-981-10-4337-6_6.  Google Scholar

[7]

H. Bulut, D. Kumar, J. Singh, R. Swroop and H. M. Baskonus, Analytic study for a fractional model of HIV infection of CD4+T lymphocyte cells, Mathematics in Natural Science, 2 (2018), 33-43.  doi: 10.22436/mns.02.01.04.  Google Scholar

[8]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404.  doi: 10.3934/mbe.2004.1.361.  Google Scholar

[9]

M. A. Dokuyucu and E. Celik, Nonlinear diffusion for chemotaxis and birth-death process for Keller-Segel model, New Trends Math. Sci., 4 (2016), 204-211. doi: 10.20852/ntmsci.2016318931.  Google Scholar

[10]

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

[11]

F. Evirgen and N. Özdemir, Multistage adomian decomposition method for solving NLP problems over a nonlinear fractional dynamical system, J. Comput. Nonlinear Dynam., 6 (2011), 021003. doi: 10.1115/1.4002393.  Google Scholar

[12]

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

[13]

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

[14]

J. F. Gómez-Aguilar and A. Atangana, Fractional derivatives with the power-law and the Mittag-Leffler kernel applied to the nonlinear Baggs-Freedman model, Fractal and Fractional, 2 (2018), 10. doi: 10.3390/fractalfract2010010.  Google Scholar

[15]

J. F. Gómez-Aguilar, A. Atangana and V. F. Morales-Delgado, Electrical circuits RC, LC, and RL described by Atangana-Baleanu fractional derivatives, International Journal of Circuit Theory and Applications, 45 (2017), 1514-1533.   Google Scholar

[16]

H.-F. Huo and L.-X. Feng, Global stability for an HIV/AIDS epidemic model with different latent stages and treatment, Appl. Math. Model., 37 (2013), 1480-1489.  doi: 10.1016/j.apm.2012.04.013.  Google Scholar

[17]

F. Jarad, T. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos, Solitons Fractals, 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006.  Google Scholar

[18]

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

[19]

I. Koca, Modelling the spread of Ebola virus with Atangana-Baleanu fractional operators, Eur. Phys. J. Plus, 133 (2018), 100. doi: 10.1140/epjp/i2018-11949-4.  Google Scholar

[20]

Y. Koçak, M. A. Dokuyucu and E. Çelik, Well-posedness of optimal control problem for the Schrodinger equations with complex potential, Int. J. Math. Comput., 26 (2015), 11-16.   Google Scholar

[21] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.  doi: 10.1142/9781848163300.  Google Scholar
[22]

G. Mulone and B. Straughan, Modeling binge drinking, Int. J. Biomath., 5 (2012), 1250005. doi: 10.1142/S1793524511001453.  Google Scholar

[23]

Z. M. Odibat and S. Momani, Application of variational iteration method to nonlinear differential equation of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 7 (2006), 27-34.  doi: 10.1515/IJNSNS.2006.7.1.27.  Google Scholar

[24]

N. Özdemir, Y. Povstenko, D. Avci and B. B. İskender, Optimal boundary control of thermal stresses in a plate based on time-fractional heat conduction equation, Journal of Thermal Stresses, 37 (2014), 969-980.  doi: 10.1080/01495739.2014.912937.  Google Scholar

[25]

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.  doi: 10.12693/APhysPolA.132.1050.  Google Scholar

[26]

Y. Povstenko, D. Avci, B. B. İ. Eroǧlu and N. Özdemir, Control of thermal stresses in axissymmetric problems of fractional thermoelasticity for an infinite cylindrical domain, Thermal Science, 21 (2017), 19-28.  doi: 10.2298/TSCI160421236P.  Google Scholar

[27]

D. G. Prakasha, P. Veeresha and H. M. Baskonus, Analysis of the dynamics of hepatitis E virus using the Atangana-Baleanu fractional derivative, The European Physical Journal Plus, 134 (2019), 11p. doi: 10.1140/epjp/i2019-12590-5.  Google Scholar

[28]

G. ur Rahman, R. P. Agarwal, L. Liu and A. Khan, Threshold dynamics and optimal control of an age-structured giving up smoking model, Nonlinear Anal. Real World Appl., 43 (2018), 96-120.  doi: 10.1016/j.nonrwa.2018.02.006.  Google Scholar

[29]

J. Singh, D. Kumara, Z. 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

[30]

D. L. Suthar, S. D. Purohit and R. K. Parmar, Generalized fractional calculus of the multiindex Bessel function, Mathematics in Natural Science, 1 (2017), 26-32.  doi: 10.22436/mns.01.01.03.  Google Scholar

[31]

K. A. Touchent, Z. Hammouch, T. Mekkaoui and F. B. M. Belgacem, Implementation and convergence analysis of homotopy perturbation coupled with sumudu transform to construct solutions of local-fractional PDEs, Fractal Fract., 2 (2018), 22p. doi: 10.3390/fractalfract2030022.  Google Scholar

[32]

M. Toufik and A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, The European Physical Journal Plus, 132 (2017), 444. doi: 10.1140/epjp/i2017-11717-0.  Google Scholar

[33]

E. Ucar, N. Özdemir and E. Altun, Fractional order model of immune cells influenced by cancer cells, Math. Model. Nat. Phenom., 14 (2019), 12 pp. doi: 10.1051/mmnp/2019002.  Google Scholar

[34]

S. Uçar, E. Uçar, N. 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

[35]

H. Xiang, C.-C. Zhu and H.-F. Huo, Modelling the effect of immigration on drinking behaviour, J. Biol. Dyn., 11 (2017), 275-298.  doi: 10.1080/17513758.2017.1337243.  Google Scholar

[36]

M. Yavuz and N. Ozdemir, Numerical inverse Laplace homotopy technique for fractional heat equations, Thermal Science, 22 (2018), 185-194.  doi: 10.2298/TSCI170804285Y.  Google Scholar

Figure 1.  The transfer diagram of the alcoholism model under the effect of immigration
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Figure 2.  System behavior of the fractional drinking model (5) with order $ \eta = 0.3 $ in respect to time $ t = 1 $ and $ t = 40 $
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Figure 3.  System behavior of the fractional drinking model (5) with order $ \eta=0.5 $ in respect to time $ t=1 $ and $ t=40 $
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Figure 4.  System behavior of the fractional drinking model (5) with order $ \eta = 0.7 $ in respect to time $ t = 1 $ and $ t = 40 $
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Figure 5.  System behavior of the fractional drinking model (5) with order $ \eta = 0.9 $ in respect to time $ t = 1 $ and $ t = 40 $
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