An Atangana-Baleanu-Caputo Fractal-Fractional Model for Alcohol-Addiction Dynamics: Qualitative Analysis, Numerical Simulation and a Physics-Informed Neural Network Surrogate
DOI:
https://doi.org/10.66972/iscis202619Keywords:
Physics-Informed Neural Network, Alcohol-Addiction Model, Atangana-Baleanu-Caputo Fractal-Fractional Derivative, Mittag-Leffler Kernel, Basic Reproduction Number, Global StabilityAbstract
Alcohol dependence is a persistent public-health problem shaped by social contact and by long behavioural memory that local integer-order models cannot capture. This paper proposes a four-compartment alcohol-addiction model with susceptible, occasional-user, regular-user, and recovered classes in the Atangana-Baleanu-Caputo fractal-fractional framework, whose Mittag-Leffler kernel encodes nonlocal memory and whose fractal order encodes irregular temporal scaling. We prove existence, uniqueness, positivity, and boundedness, as well as derive the equilibria and the basic reproduction number. An Atangana-Toufik product-integration scheme and a Levenberg-Marquardt-trained physics-informed neural network confirm the analysis with sub-percent error, giving a consistent and flexible tool for studying memory-driven addiction dynamics and for informing intervention policy.
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[1] Brauer, F., Castillo-Chavez, C., & Feng, Z. (2012). Mathematical models in epidemiology. Springer.
[2] Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM Review, 42(4), 599–653. https://doi.org/10.1137/S0036144500371907
[3] Diekmann, O., Heesterbeek, J. A. P., & Metz, J. A. J. (1990). On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology, 28(4), 365–382. https://doi.org/10.1007/BF00178324
[4] Van den Driessche, P., & Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180(1–2), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6
[5] Li, S., Zhang, Y., Wang, H., & Chen, X. (2024). Global dynamics and computational modeling approach for alcohol addiction with treatment intervention. Mathematical Biosciences and Engineering, 21(8), 14231–14258. https://doi.org/10.1038/s41598-024-54578-9
[6] Bhosale, S., Lande, C., Jafari, H., & Raut, D. K. (2026). Mathematical analysis and numerical simulation of a fractional-order alcohol consumption model incorporating binge drinkers. Boundary Value Problems, 2026, 69. https://doi.org/10.1186/s13661-026-02269-x
[7] Wanassi, O. K., & Baleanu, D. (2024). Modeling blood alcohol concentration using fractional differential equations based on the -caputo derivative. Mathematical Methods in the Applied Sciences, 47(12), 9451–9470. https://doi.org/10.1002/mma.10002
[8] Prakash, A., Kumar, S., & Singh, R. (2024). Analysis and numerical simulation of fractional-order blood alcohol models. Computational and Mathematical Biophysics, 12(1), 1–20. https://doi.org/10.1515/cmb-2024-0001
[9] Podlubny, I. (1999). Fractional differential equations. Academic Press.
[10] Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and applications of fractional differential equations. Elsevier.
[11] Diethelm, K. (2010). The analysis of fractional differential equations. Springer.
[12] Li, C., & Zeng, F. (2020). Numerical methods for fractional calculus. CRC Press.
[13] Garrappa, R. (2018). Numerical solution of fractional differential equations: A survey and soft-ware tutorial. Mathematics, 6(2), 16. https://doi.org/10.3390/math6020016
[14] Tarasov, V. E. (2019). Fractional dynamics: Applications of fractional calculus to dynamics of particles, fields and media. Springer. https://doi.org/10.1007/978-3-642-14003-7
[15] Machado, J. A. T., Kiryakova, V., & Mainardi, F. (2022). Recent history of fractional calculus. Communications in Nonlinear Science and Numerical Simulation, 102, 105939. https://doi.org/10.1016/j.cnsns.2010.05.027
[16] Chen, Y. Q., Ding, H., & Li, C. (2021). Fractional-order systems and controls: Fundamentals and applications. Annual Reviews in Control, 51, 1–15. https://doi.org/10.1007/978-1-84996-335-0
[17] Atangana, A., & Baleanu, D. (2016). New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. Thermal Science, 20(2), 763–769. https://doi.org/10.2298/TSCI160111018A
[18] Abdeljawad, T. (2020). Fractional operators with generalized mittag-leffler kernels and their applications. Advances in Difference Equations, 2020, 468. https://doi.org/10.1063/1.5085726
[19] Djida, J. D., Area, I., & Atangana, A. (2017). New numerical scheme of atangana-baleanu fractional integral: Application to groundwater flow within leaky aquifers. European Physical Journal Plus, 132(5), 1–17.
[20] Algahtani, O. J. J. (2021). Comparing the atangana-baleanu and caputo-fabrizio derivatives with fractional order. Chaos, Solitons & Fractals, 89, 552–559. https://doi.org/10.1016/j.chaos.2016.03.026
[21] Martınez-Fuentes, O., & Delf´ın-Prieto, S. M. (2020). Stability of fractional nonlinear systems with mittag-leffler kernels and design of state observers. Chaos, Solitons & Fractals, 140, 110220.
[22] Sabatier, J. (2020). Fractional-order derivatives defined by continuous kernels: Are they really too restrictive? Fractal and Fractional, 4(3), 40. https://doi.org/10.3390/fractalfract4030040
[23] Khan, F. S., Khan, A., & Baleanu, D. (2022). Freelance model with atangana-baleanu-caputo fractional derivatives. Symmetry, 14(11), 2424. https://doi.org/10.3390/sym14112424
[24] Kebede, S. G., & Lakoud, A. G. (2023). Analysis of mathematical models involving nonlinear systems of caputo-fabrizio fractional differential equations. Boundary Value Problems, 2023(1), 44. https://doi.org/10.1186/s13661-023-01730-5
[25] Kumar, P., Kumar, S., Alkahtani, B. S. T., & Alzaid, S. S. (2024). A mathematical model for simulating the spread of infectious disease using the caputo-fabrizio fractional-order operator. AIMS Mathematics, 9(11), 30864–30897. https://doi.org/10.3934/math.20241490
[26] Chavada, A., & Pathak, N. (2024). Transmission dynamics of breast cancer through caputo-fabrizio fractional derivative operator with real data. Mathematical Modelling and Control, 4(1), 119–132. https://doi.org/10.3934/mmc.2024011
[27] Atangana, A. (2021). Fractal-fractional differentiation and integration: Connecting fractal geometry with memory effects. Chaos, Solitons & Fractals, 102, 396–406. https://doi.org/10.1016/j.chaos.2017.04.027
[28] Rahman, M. (2022). Generalized fractal-fractional order problems under non-singular mittag-leffler kernel. Results in Physics, 35, 105346. https://doi.org/10.1016/j.rinp.2022.105346
[29] Chithra, S. M., Kumar, R., & Devi, M. (2025). Fractal-fractional modelling of social media addiction with atangana-baleanu operators. European Journal of Pure and Applied Mathematics, 18(1), 1–25. https://doi.org/10.29020/nybg.ejpam.v18i4.6737
[30] Ahmed, S., Jahan, S., & Nisar, K. S. (2025). A fractional model for the dynamics of covid-19 using atangana-baleanu fractional operators. Journal of Mathematics and Computer Science, 39(2), 233–248.
[31] Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686–707. https://doi.org/10.1016/j.jcp.2018.10.045
[32] Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., & Yang, L. (2021). Physics-informed machine learning. Nature Reviews Physics, 3(6), 422–440. https://doi.org/10.1038/s42254-021-00314-5
[33] Pang, G., Lu, L., & Karniadakis, G. E. (2019). fPINNs: Fractional physics-informed neural networks. SIAM Journal on Scientific Computing, 41(4), A2603–A2626. https://doi.org/10.1137/18M1229845
[34] [34] Banach, S. (1922). Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fundamenta Mathematicae, 3(1), 133–181.
[35] Matignon, D. (1996). Stability results for fractional differential equations with applications to control processing. Proceedings of the Computational Engineering in Systems Applications Conference, 963–968.
[36] LaSalle, J. P. (1976). The stability of dynamical systems. SIAM.
[37] Khalil, H. K. (2002). Nonlinear systems (3rd ed.). Prentice Hall.
[38] Lakshmikantham, V., Leela, S., & Devi, J. V. (2009). Theory of fractional dynamic systems. Cambridge Scientific Publishers.
[39] Toufik, M., & Atangana, A. (2017). New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models. European Physical Journal Plus, 132(10), 444. https://doi.org/10.1140/epjp/i2017-11717-0
[40] Hagan, M. T., & Menhaj, M. B. (1994). Training feedforward networks with the marquardt algorithm. IEEE Transactions on Neural Networks, 5(6), 989–993. https://doi.org/10.1109/72.329697
[41] Marquardt, D. W. (1963). An algorithm for least-squares estimation of nonlinear parameters. Journal of the Society for Industrial and Applied Mathematics, 11(2), 431–441. https://doi.org/10.1137/0111030
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