| Citation: |
Fang Jin, Zi-Shan Qian, Yu-Ming Chu, Mati ur Rahman. ON NONLINEAR EVOLUTION MODEL FOR DRINKING BEHAVIOR UNDER CAPUTO-FABRIZIO DERIVATIVE[J]. Journal of Applied Analysis & Computation, 2022, 12(2): 790-806. doi: 10.11948/20210357
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ON NONLINEAR EVOLUTION MODEL FOR DRINKING BEHAVIOR UNDER CAPUTO-FABRIZIO DERIVATIVE
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Abstract
The investigation of this research article is the development of studying the dynamical behavior of the drinking population through the fractional drinking model in the sense of Caputo-Fabrizio (CF) arbitrary order operator along with the special non-singular kernel. The proposed system is analyzed for existence result and uniqueness of solution by applying fixed point theory and Picard's technique. Also on utilizing Adams-Bashforth method (ABM) of numerical analysis to interpret the approximate results through plots to observe dynamical behavior corresponding to different fractional order. For the mentioned simulation some real initial and parameter data are used.
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References
[1] A. Atangana and D. Baleanu, Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer, J. of Eng. Mech., 2017, 143(5), D4016005. [2] M. ur Rahman, M. Arfan and S. Ahmad, On the analysis of semi-analytical solutions of Hepatitis B epidemic model under the Caputo-Fabrizio operator, Cha., Sol. & Frac., 2021, 146, 110892. [3] A. Atangana, On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation, Appl. Math. Comput., 2016 273, 948-956. [4] M. Al-Refai and T. Abdeljawad, Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel, Adv. Differ. Equ., 2017, 2017(1), 1-12. doi: 10.1186/s13662-016-1057-2 [5] D. Baleanu and T. Abdeljawad, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 2017, 80(1), 11-27. doi: 10.1016/S0034-4877(17)30059-9 [6] A. E. Hassanien, I. Arpaci, I. Mahariq, M. Al-Emran, M. Khasawneh, S. Alshehabi and T. Abdeljawad, Analysis of twitter data using evolutionary clustering during the COVID-19 pandemic, Comp., Mate. and Cont., 2020, 65(1), 193-204. [7] A. Atangana and R. T. Alqahtani, Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation, Adv. Differ. Equ., 2016, 2016(1), 1-13. doi: 10.1186/s13662-015-0739-5 [8] A. Jajarmi, D. Baleanu, H. Mohammadi and S. Rezapour, A new study on the mathematical modeling of human liverwith Caputo-Fabrizio fractional derivative, Cha. Sol. & Frac., 2020, 134, 109705. [9] A. Mousalou, D. Baleanu, and S. Rezapour, The extended fractional Caputo-Fabrizio derivative of order $0= s<1, 0\leq\sigma<1$ on $CR [0, 1] C_ {\mathcal {R}}[0, 1] $ and the existence of solutions for two higher-order series-type differential equations, Adv. Differ. Equ., 2018, 2018, 1-11. [10] A. Mousalou, D. Baleanu, and S. Rezapour, On the existence of solutions for some infinite coefficient-symmetric Caputo-Fabrizio fractional integro-differential equations, Bound. Value Probl., 2017, 2017, 145. doi: 10.1186/s13661-017-0867-9 [11] D. Baleanu, S. Rezapour and Z. Saberpour, On fractional integro-differential inclusions via the extended fractional Caputo-Fabrizio derivation, Bound. Value Probl., 2019, 2019, 79. doi: 10.1186/s13661-019-1194-0 [12] N. T. J. Bailey, The mathematical theory of infectious diseases and its applications, Charles Griffin & Company Ltd, 5a Crendon Street, High Wycombe, Bucks HP13 6LE., 1975. [13] N. Crokidakis, R. M. Brum, Dynamics of tax evasion through an epidemic-like model, Int. J. Mod. Phy. C, 2017, 28, 1750023. doi: 10.1142/S0129183117500231 [14] D. Baleanu, H. Mohammadi, R. P. Agarwal and S. Rezapour, Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces, Bound. Val. Probl., 2013, 2013, 112. doi: 10.1186/1687-2770-2013-112 [15] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 2015, 1(2), 73-85. [16] J. S. Sa Martins and N. Crokidakis, Can honesty survive in a corrupt parliament?, Int. J. Mod. Phy. C, 2018, 29(10), 1850094. doi: 10.1142/S0129183118500948 [17] J. Chodkiewicz, J. Miniszewska, M. Talarowska, N. Nawrocka and P. Bilin-ski, Alcohol consumption reported during the covid-19 pandemic: The initial stage, Int. J. Envir. Res. Pub. Hea., 2020, 17, 4677. doi: 10.3390/ijerph17134677 [18] J. M. Clay and M. O. Parker, Alcohol use and misuse during the covid-19 pandemic: a potential public health crisis?, The Lan. Pub. Hea., 2020, 5, e259. doi: 10.1016/S2468-2667(20)30088-8 [19] L. Sigaud and N. Crokidakis, Modeling the evolution of drinking behavior: A Statistical Physics perspective, Phy. A: Stat. Mec. and its Appl., 2021, 570, 125814. doi: 10.1016/j.physa.2021.125814 [20] M. Caputo and M. Fabrizio, On the singular kernels for fractional derivatives. some applications to partial differential equations, Progr. Fract. Differ. Appl., 2021, 7(2), 1-4. [21] D. J. Daley and D. G. Kendall, Epidemics and rumours, Nature, 1964, 204(4963), 1118-1118. doi: 10.1038/2041118a0 [22] H. Nishiura, K. Ejima, K. Aihara, Modeling the obesity epidemic: social contagion and its implications for control, Theo. Bio. and Med. Mode., 2013, 10(1), 17. doi: 10.1186/1742-4682-10-17 [23] M. A. Javarone, S. Galam, Modeling radicalization phenomena in heterogeneous populations, PloS One, 2016, 11(5), e0155407. doi: 10.1371/journal.pone.0155407 [24] C. Hall, G. A. Kaplan and S. Galea, Social epidemiology and complex system dynamic modelling as applied to health behaviour and drug use research, Int. J. Drug Pol., 2009, 20, 209-216. doi: 10.1016/j.drugpo.2008.08.005 [25] A. Jajarmi and D. Baleanu, On the fractional optimal control problems with a general derivative operator, Asi. Jour. of Cont., 2021, 23(2), 1062-1071. doi: 10.1002/asjc.2282 [26] E. A. Kojaba and S. Rezapour, Approximate solutions of a sum-type fractional integro-differential equation by using Chebyshev and Legendre polynomials, Adv. Differ. Equ., 2017, 2017, 1-18. doi: 10.1186/s13662-016-1057-2 [27] A. Atangana and I. Koca, Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, Therm. Scie., 2017, 21(6 Part A), 2299-2305. doi: 10.2298/TSCI160209103K [28] A. Seadawy, G. Zaman, K. Shah, P. Kumam, S. A. Khan and Z. Shah, Study of mathematical model of Hepatitis B under Caputo-Fabrizo derivative, AIMS Math., 2020, 6, 195. [29] D. Baleanu, M. A. Khan and Z. Hammouch, Modeling the dynamics of hepatitis E via the Caputo-Fabrizio derivative, Math. Model. Nat. Phenom., 2019, 14(3), 311. doi: 10.1051/mmnp/2018074 [30] J. Losada and J. Nieto, Fractional integral associted to fractional derivatives with nonsingular kernels, Progr. Fract. Differ. Appl., 2021, 7(3), 1-7. [31] D. Bui, I. Mahariq, M. Kavyanpoor, M. Ghalandari and M. A. Nazari, Identification of nonlinear model for rotary high aspect ratio flexible blade using free vibration response, Alex. Eng. J., 2020, 59(4), 2131-2139. doi: 10.1016/j.aej.2020.01.029 [32] A. Atangana and K. M. Owolabi, On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems, Cha. : An Int. J. of Non. Scie., 2019, 29(2), 023111. [33] A. Yusuf and S. Qureshi, Fractional derivatives applied to MSEIR problems: Comparative study with real world data, The Eur. Phy. J. Plu., 2019, 134(4), 171. doi: 10.1140/epjp/i2019-12661-7 [34] A. Ali, F. Rahman, and S. Saifullah, Analysis of Time-Fractional ϕ4-Equation with Singular and Non-Singular Kernels, Int. J. of App. and Comp. Math., 2021, 7(5), 1-17. [35] F. Jarad, I. Mahariq, K. Shah and T. Abdeljawad, Qualitative analysis of a mathematical model in the time of COVID-19, Bio. Res. Inte., 2020, 2020. [36] A. Ali, K. Shah, M. Irfan and S. Saifullah, Time-Fractional Klein–Gordon Equation with Solitary/Shock Waves Solutions, Math. Prob. in the Eng., 2021, 2021, Article ID: 6858592. . [37] D. Baleanu, M. Shoaib, M. A. Z. Raja and Z. Sabir, Design of stochastic numerical solver for the solution of singular three-point second-order boundary value problems, Neu. Comp. and Appl., 2021, 33(7), 2427-2443. [38] F. Jarad, K. Shah and T. Abdeljawad, On a nonlinear fractional order model of dengue fever disease under Caputo-Fabrizio derivative, Alex. Eng. J., 2020, 59(4), 2305-2313. doi: 10.1016/j.aej.2020.02.022 [39] E. Śanchez, F. J. Santonja, M. Rubio and R. J. Villanueva, Predicting cocaine consumption in spain: A mathematical modelling approach, Drugs: Edu. Prev. Pol., 2011, 18(2), 108-115. doi: 10.3109/09687630903443299 [40] D. A. Fiellin, L. E. Sullivan and P. G. O'Connor, The prevalence and impact of alcohol problems in major depression: a systematic review, The Ame. J. Med., 2005, 118, 330-341. doi: 10.1016/j.amjmed.2005.01.007 [41] E. Śanchez, F. J. Santonja, J. L. Morera and M. Rubio, Alcohol consumption in spain and its economic cost: a mathematical modeling approach, Math. Com. Mod., 2010, 52, 999-1003. doi: 10.1016/j.mcm.2010.02.029 [42] A. Gilani, M. Talaee, M. Shabibi and S. Rezapour, On the existence of solutions for a pointwise defined multi-singular integro-differential equation with integral boundary condition, Adv. Differ. Equ., 2020, 2020, 1-16. doi: 10.1186/s13662-019-2438-0 [43] M. ur Rahman, N. A. Alshehri, R. T. Matoog, S. Ahmad and T. Khan, Study on the mathematical modelling of COVID-19 with Caputo-Fabrizio operator, Cha., Sol. & Frac., 2021, 111121. [44] I. Ullah, M. ur Rahman, M. Arfan and S. Ahmad, Investigation of fractional order tuberculosis (TB) model via Caputo derivative, Cha., Sol. & Frac., 2020, 138, 110479. [45] WHO, World health statistics 2018: monitoring health for the sdgs sustainable development goals, Geneva: World Health Organization CC BY-NC-SA, 2018, 3.0 IGO. -
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Fang Jin, Zi-Shan Qian, Yu-Ming Chu, Mati ur Rahman. ON NONLINEAR EVOLUTION MODEL FOR DRINKING BEHAVIOR UNDER CAPUTO-FABRIZIO DERIVATIVE[J]. Journal of Applied Analysis & Computation, 2022, 12(2): 790-806. doi: 10.11948/20210357
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Figure 1. Plot of the approximate solution for
$ \mathcal{S}(t) $ of the considered system (1.2) for the non-integer order$ \gamma $ between 0 and 1 -
Figure 2. Plot of the approximate solution for
$ \mathcal{M}(t) $ of the considered system (1.2) for the non-integer order$ \gamma $ between 0 and 1 -
Figure 3. Plot of the approximate solution for
$ \mathcal{R}(t) $ of the considered system (1.2) for the non-integer order$ \gamma $ between 0 and 1 -
Figure 4. Combine Dynamical behavior of the approximate solution for all the three agents of the considered system (1.2) for the non-integer order
$ \gamma $ between 0 and 1 for data of Case-Ⅰ -
Figure 5. Dynamical representation of the approximate solution for all the three agents
$ \mathcal{S}(t), \mathcal{M}(t), \mathcal{R}(t) $ of the system (1.2) for the non-integer order$ \gamma $ between 0 and 1 having parameters values of Case-Ⅱ -
Figure 6. Dynamical behavior of the approximate solution for all the three compartments
$ \mathcal{S}(t), \mathcal{M}(t), \mathcal{R}(t) $ of the considered system (1.2) for the non-integer order$ \gamma $ between 0 and 1 having parameters values of Case-Ⅲ -
Figure 7. Dynamical behavior of the approximate solution for all the three agents
$ \mathcal{S}(t), \mathcal{M}(t), \mathcal{R}(t) $ of the considered system (1.2) for the non-integer order$ \gamma $ between 0 and 1 having parameters values of Case-Ⅳ