A Fractional Model of an Effect of Awareness Programs by Media on Binge Drinking Model with Mittag-Leffler Kernel

Abstract

Alcoholism is one of the problems that societies have not been able to find a solution for a long time. Today, over 2 billions of people consume alcohol and it is estimated that approximately 76 million of these people are addicted. In this study, an alcoholism model is broadly researched with the help of AB derivative. The existence and uniqueness of the drinking model solutions together with the stability analysis is demonstrated through Banach fixed point theorem. The special solution of the model is investigated using the Sumudu transformation and following that, a set of numeric graphics are given for different fractional orders with the intention of showing the effectiveness of fractional derivative. Hence, it is obtained that while awareness programs driven by media increases, the number of heavy drinkers decreases.

Keywords

• Fractional differential equations,Atangana-Baleanu derivative,mathematical model

References

1. [1] Sanchez, F. Wang, X.H., Castillo-Chavez, C., Gorman, D.M., Gruenewald, P.J., Drinking as an epidemic simple mathematical model with recovery and relapse, Therapists Guide to Evidence-Based Relapse Prevention: Practical Resources for the Mental Health Professional, Academic Press, 2007, 353-368.
2. [2] Mulone, G., Straughan, B., Modeling binge drinking, Int. J. Biomath. Vol:5, (2012), 14pages.
3. [3] Gomez-Aguilar, J.F., Analytical and Numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations, Phys. A Vol:494, (2018), 52-75.
4. [4] Oldham, K.B., Spanier, J., The fractional calculus theory and applications of differentiation and integration to arbitrary order. Academic Press, 1974.
5. [5] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. Elsevier, 2006.
6. [6] Ozdemir, N., Karadeniz, D., Iskender, B.B., Fractional optimal control problem of a distributed system in cylindrical coordinates, Phys. Lett. A Vol:373, (2009), 221-226.
7. [7] Evirgen, F., Ozdemir, N., Multistage adomian decomposition method for solving NLP problems over a nonlinear fractional dynamical system. J. Comput. Nonlinear Dynam. Vol:6, (2010), 6pages.
8. [8] Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J., Fractional calculus models and numerical methods. World Scientific, 2012.

9. [9] Evirgen, F., Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM. Int. J. Optim. Control Theor. Appl. IJOCTA Vol:6, (2016), 75-83.
10. [10] Hammouch, Z., Mekkaoui, T., Circuit design and simulation for the fractional-order chaotic behavior in a new dynamical system. Complex Intell. Syst. Vol:4, (2018), 251-260.
11. [11] Ucar, E., Ozdemir, N., Altun, E., Fractional order model of immune cells influenced by cancer cells, Math. Model. Nat. Phenom. Vol:14, (2019), 12pages.
12. [12] Ozdemir, N., Yavuz, M., Numerical Solution of fractional Black-Scholes equation by using the multivariate Pade approximation, Acta Phys. Pol. A Vol:132, (2017), 1050-1053.
13. [13] Inc, M., Yusuf, A., Aliyu, A.I., Baleanu, D., Investigation of the logarithmic-KdV equation involving Mittag-Leffler type kernel with Atangana-Baleanu derivative, Phys. A Vol:506, (2018), 520-531.
14. [14] Khan, M.A., Ullah, S., Farooq, M., A new fractional model for tuberculosis with relapse via Atangana-Baleanu derivative, Chaos Solitons Fractals Vol:116, (2018), 227-238.
15. [15] Kumar, D., Singh, J., Baleanu, D., Sushila, Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel, Phys. A Vol:492, (2018), 155-167.
16. [16] Yavuz, M., Ozdemir, N,, Baskonus, H.M., Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel. Eur. Phys. J. Plus Vol:133, (2018), 11pages.
17. [17] Baleanu, D., Fernandez, A., On some new properties of fractional derivatives with Mittag-Leffler kernel, Commun. Nonlinear Sci. Numer. Simulat. Vol:59,(2018), 444-462.
18. [18] Fernandez, A., Baleanu, D., Srivastava, H.M., Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions, Commun. Nonlinear Sci. Numer. Simulat. Vol:67, (2019), 517-527.
19. [19] Uc¸ar, S., Uc¸ar, E., O¨ zdemir, N., Hammouch, Z., Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative, Chaos Solitons Fractals Vol:118, (2019), 300-306.
20. [20] Koca, I., Analysis of rubella disease model with non-local and non-singular fractional derivatives. Int. J. Optim. Control Theor. Appl. IJOCTA Vol:8, (2018), 17-25.
21. [21] Avcı D., Yetim, A., Analytical solutions to the advection-diffusion equation with the Atangana-Baleanu derivative over a finite domain. J. BAUN Inst. Sci. Technol. Vol:20, (2018), 382-395.
22. [22] Owolabi, K.M., Hammouch, Z., Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order derivative, Phys. A Vol:523, (2019), 1072-1090.
23. [23] Yavuz, M., Bonyah, E., New approaches to the fractional dynamics of schistosomiasis disease model. Phys. A Vol:525, (2019), 373-393.
24. [24] Yavuz, M., Ozdemir, N., Analysis of an Epidemic Spreading Model with Exponential Decay Law, Math. Sci. Appl. E-Notes, Vol:8, (2020).
25. [25] Yavuz, M., Characterizations of two different fractional operators without singular kernel, Math. Model. Nat. Phenom. Vol: 14, (2019), 13pages.
26. [26] Huo, H.F., Wang, Q., Modelling the influence of awareness programs by media on the drinking dynamics, Abstr. Appl. Anal. Vol:2014, (2014), 8pages.
27. [27] Atangana, A., Koca, I., Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fractals Vol:89, (2016), 447-454.
28. [28] Odibat, Z.M., Momani, S., Application of variational iteration method to nonlinear differential equation of fractional order. Int. J. Nonlinear Sci. Vol:7, (2006), 27-34.
29. [29] Atangana, A., Baleanu, D., New fractional derivatives with non-local and non-singular kernel: theory and applications to heat transfer model. Therm. Sci. Vol:20, (2016), 763-769.
30. [30] Baleanu, D., Jajarmi, A., Hajipour, M., On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag-Leffler kernel. Nonlinear Dyn., Vol:94, (2018), 397-414.