Data‐driven modeling and simulation of Caputo–Fabrizio fractional order shingles disease model

Aziz Khan; Aiman Mukheimer; Thabet Abdeljawad; Rajermani Thinakaran; Aziz Khan; Aiman Mukheimer; Thabet Abdeljawad; Rajermani Thinakaran

doi:10.3934/math.2026248

AIMS Mathematics

2026, Volume 11, Issue 3: 5992-6018. doi: 10.3934/math.2026248

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Data‐driven modeling and simulation of Caputo–Fabrizio fractional order shingles disease model

1.
Department of Mathematics and Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia
2.
Department of Fundamental Sciences, Faculty of Engineering and Architecture, Istanbul Gelisim University, Avcilar, 34310, Istanbul, Turkey
3.
Department of Medical Research, China Medical University, Taichung, 40402, Taiwan
4.
Faculty of Data Science and Information Technology, INTI International University, Negeri Sembilan, Malaysia

Received: 23 December 2025 Revised: 12 February 2026 Accepted: 26 February 2026 Published: 10 March 2026
MSC : 26A33, 34A08, 34A12, 34A34

This study deals with Caputo–Fabrizio (CF) fractional-order shingles disease model to capture the intrinsic memory and nonlocal properties that govern the progression of herpes zoster within a population. The model is organized into interacting epidemiological compartments and its mathematical soundness is confirmed by establishing both the existence and stability of results through fixed-point theory and fractional theory. To approximate the model behavior, a high accuracy numerical method based on the Lagrangian interpolation technique is constructed, allowing smooth reconstruction of fractional trajectories and robust behavior of nonlocal operators. Complementing this numerical context, an artificial neural network technique is used and trained using the Levenberg–Marquardt optimization algorithm, enabling efficient learning of disease patterns and authentication of numerical fidelity. Performance indicators, involving regression, training test, error histogram, and convergence features, confirm the reliability of the ANN-supported evaluation. The combined modeling, numerical, and computational analysis offers a comprehensive estimation of the fractional dynamics governing shingles transmission, presenting deeper insights into disease evolution and displaying the capacity of fractional operators and intelligent systems to enhance health risk predictive epidemiological modeling.
- Caputo–Fabrizio operator,
- neural networks,
- shingles diseases,
- numerical method,
- training fit,
- Levenberg–Marquardt
Citation: Aziz Khan, Aiman Mukheimer, Thabet Abdeljawad, Rajermani Thinakaran. Data‐driven modeling and simulation of Caputo–Fabrizio fractional order shingles disease model[J]. AIMS Mathematics, 2026, 11(3): 5992-6018. doi: 10.3934/math.2026248

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Abstract

This study deals with Caputo–Fabrizio (CF) fractional-order shingles disease model to capture the intrinsic memory and nonlocal properties that govern the progression of herpes zoster within a population. The model is organized into interacting epidemiological compartments and its mathematical soundness is confirmed by establishing both the existence and stability of results through fixed-point theory and fractional theory. To approximate the model behavior, a high accuracy numerical method based on the Lagrangian interpolation technique is constructed, allowing smooth reconstruction of fractional trajectories and robust behavior of nonlocal operators. Complementing this numerical context, an artificial neural network technique is used and trained using the Levenberg–Marquardt optimization algorithm, enabling efficient learning of disease patterns and authentication of numerical fidelity. Performance indicators, involving regression, training test, error histogram, and convergence features, confirm the reliability of the ANN-supported evaluation. The combined modeling, numerical, and computational analysis offers a comprehensive estimation of the fractional dynamics governing shingles transmission, presenting deeper insights into disease evolution and displaying the capacity of fractional operators and intelligent systems to enhance health risk predictive epidemiological modeling.

References

[1]	C. Glynn, G. Crockford, D. Gavaghan, P. Cardno, D. Price, J. Miller, Epidemiology of shingles, J. Roy. Soc. Med., 83 (1990), 617–619. https://doi.org/10.1177/014107689008301007 doi: 10.1177/014107689008301007
[2]	N. R. Glassman, shingles, J. Cons. Hlth. Internet, 14 (2010), 423–433. https://doi.org/10.1080/15398285.2010.524126 doi: 10.1080/15398285.2010.524126
[3]	P. Sampathkumar, L. A. Drage, D. P. Martin, Herpes zoster (shingles) and postherpetic neuralgia, Mayo Clin. Proc., 84 (2009), 274–280. https://doi.org/10.4065/84.3.274 doi: 10.4065/84.3.274
[4]	M. H. Motswaledi, Herpes zoster (shingles), S. Afr. Fam. Pract., 60 (2018), 28–30.
[5]	E. Rafferty, W. McDonald, W. C. Qian, N. D. Osgood, A. Doroshenko, Evaluation of the effect of chickenpox vaccination on shingles epidemiology using agent-based modeling, PeerJ, 6 (2018), e5012. https://doi.org/10.7717/peerj.5012 doi: 10.7717/peerj.5012
[6]	B. C. Agbata, R. Dervishi, M. Gümüş, A. Smerat, G. C. E. Mbah, Analysis of fractional-order model for the transmission dynamics of malaria via Caputo–Fabrizio and Atangana–Baleanu operators, Sci. Rep., 15 (2025), 36359. https://doi.org/10.1038/s41598-025-20191-7 doi: 10.1038/s41598-025-20191-7
[7]	I. Ahmed, A. Akgül, F. Jarad, P. Kumam, K. Nonlaopon, A Caputo–Fabrizio fractional-order cholera model and its sensitivity analysis, Mathematical Modelling and Numerical Simulation with Applications, 3 (2023), 170–187.
[8]	R. Jan, Z. Shah, N. Vrinceanu, M. H. Alshehri, N. N. A. Razak, M. Racheriu, Fractional view analysis of the dynamics of a plant disease through Caputo–Fabrizio derivative, Fractals, 33 (2025), 2540079. https://doi.org/10.1142/S0218348X25400791 doi: 10.1142/S0218348X25400791
[9]	I. Ahmad, R. Jan, N. N. A. Razak, A. Khan, T. Abdeljawad, Numerical investigation of the dynamical behavior of hepatitis B virus via Caputo–Fabrizio fractional derivative, Eur. J. Pure Appl. Math., 18 (2025), 5509. https://doi.org/10.29020/nybg.ejpam.v18i1.5509 doi: 10.29020/nybg.ejpam.v18i1.5509
[10]	J. Z. Ndendya, J. A. Mwasunda, S. Edward, N. Shaban, A Caputo–Fabrizio fractional-order model with MCMC estimation for rabies transmission dynamics in a multi-host population, Scientific African, 29 (2025), e02885. https://doi.org/10.1016/j.sciaf.2025.e02885 doi: 10.1016/j.sciaf.2025.e02885
[11]	H. Khan, M. Abdel-Aty, D. K. Almutairi, J. F. Gómez-Aguilar, J. Alzabut, Artificial intelligence neural networking for data clustering of carbon dioxide model, Ain Shams Eng. J., 16 (2025), 103460. https://doi.org/10.1016/j.asej.2025.103460 doi: 10.1016/j.asej.2025.103460
[12]	H. Khan, J. Alzabut, M. Tounsi, D. K. Almutairi, AI-Based data analysis of contaminant transportation with regression of oxygen and nutrients measurement, Fractal Fract., 9 (2025), 125. https://doi.org/10.3390/fractalfract9020125 doi: 10.3390/fractalfract9020125
[13]	H. Khan, W. F. Alfwzan, J. Alzabut, D. K. Almutairi, M. A. Azim, R. Thinakaran, Artificial intelligence and neural networking for an analysis of fractal-fractional Zika virus model, Fractals, 33 (2025), 25401437. https://doi.org/10.1142/S0218348X25401437 doi: 10.1142/S0218348X25401437
[14]	H. Khan, J. Alzabut, D. K. Almutairi, W. K. Alqurashi, The use of artificial intelligence in data analysis with error recognitions in liver transplantation in HIV-AIDS patients using modified ABC fractional order operators, Fractal Fract., 9 (2024), 16. https://doi.org/10.3390/fractalfract9010016 doi: 10.3390/fractalfract9010016
[15]	H. Khan, M. Aslam, A. H. Rajpar, Y. M. Chu, S. Etemad, S. Rezapour, et al., A new fractal-fractional hybrid model for studying climate change on coastal ecosystems from the mathematical point of view, Fractals, 32 (2024), 2440015. https://doi.org/10.1142/S0218348X24400152 doi: 10.1142/S0218348X24400152
[16]	H. Khan, J. Alzabut, A. Shah, Z. Y. He, S. Etemad, S. Rezapour, et al., On fractal-fractional waterborne disease model: A study on theoretical and numerical aspects of solutions via simulations, Fractals, 31 (2023), 2340055. https://doi.org/10.1142/S0218348X23400558 doi: 10.1142/S0218348X23400558
[17]	D. Baleanu, A. Jajarmi, H. Mohammadi, S. Rezapour, A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative, Chaos Soliton. Fract., 134 (2020), 109705. https://doi.org/10.1016/j.chaos.2020.109705 doi: 10.1016/j.chaos.2020.109705
[18]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85.
[19]	S. Boulaaras, R. Jan, A. Khan, M. Ahsan, Dynamical analysis of the transmission of dengue fever via Caputo–Fabrizio fractional derivative, Chaos Soliton. Fract., 8 (2022), 100072. https://doi.org/10.1016/j.csfx.2022.100072 doi: 10.1016/j.csfx.2022.100072
[20]	H. Khan, A. Khan, T. Abdeljawad, A. Alkhazzan, Existence results in Banach space for a nonlinear impulsive system, Adv. Differ. Equ., 2019 (2019), 18. https://doi.org/10.1186/s13662-019-1965-z doi: 10.1186/s13662-019-1965-z
[21]	H. Khan, J. Alzabut, A. Shah, S. Etemad, S. Rezapour, C. Park, A study on the fractal-fractional tobacco smoking model, AIMS Mathematics, 7 (2022), 13887–13909. https://doi.org/10.3934/math.2022767 doi: 10.3934/math.2022767
[22]	H. Khan, J. Alzabut, J. F. Gómez-Aguilar, P. Agarwal, Piecewise mABC fractional derivative with an application, AIMS Mathematics, 8 (2023), 24345–24366. https://doi.org/10.3934/math.20231241 doi: 10.3934/math.20231241
[23]	Ç. Candan, An efficient filtering structure for Lagrange interpolation, IEEE Signal Proc. Let., 14 (2007), 17–19. https://doi.org/10.1109/LSP.2006.881528 doi: 10.1109/LSP.2006.881528
[24]	H. Ahmadi, A numerical scheme for advection dominated problems based on a Lagrange interpolation, Groundwater for Sustainable Development, 13 (2021), 100542. https://doi.org/10.1016/j.gsd.2020.100542 doi: 10.1016/j.gsd.2020.100542
[25]	N. Yogeesh, S. I. Mohammad, J. Divyashree, N. Raja, A. Vasudevan, H. L. Long, Pesticide residue induced hepatotoxicity: Determination for animal studies based on fuzzy logic, Appl. Math. Inform. Sci., 19 (2025), 365–378. http://doi.org/10.18576/amis/190212 doi: 10.18576/amis/190212
[26]	M. Farman, A. Talib, K. S. Nisar, A. Sambas, M. Bayram, M. Hafez, Investigation of ABPV predict dynamics infection in honeybee colony production: Soft patterns multiscale modeling with fractional approach, Ain Shams Eng. J., 16 (2025), 103626. https://doi.org/10.1016/j.asej.2025.103626 doi: 10.1016/j.asej.2025.103626
[27]	P. Giudici, A. Piergallini, M. C. Recchioni, E. Raffinetti, Explainable artificial intelligence methods for financial time series, Physica A, 655 (2024), 130176. https://doi.org/10.1016/j.physa.2024.130176 doi: 10.1016/j.physa.2024.130176
[28]	P. Giudici, V. Kolesnikov, SAFE AI metrics: An integrated approach, Machine Learning with Applications, 23 (2026), 100821. https://doi.org/10.1016/j.mlwa.2025.100821 doi: 10.1016/j.mlwa.2025.100821

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