Fractional order dynamics in breast cancer control: a Caputo perspective

Anil Chavada; Nimisha Pathak; Anil Chavada; Nimisha Pathak

doi:10.3934/mmc.2026005

Mathematical Modelling and Control

2026, Volume 6, Issue 1: 57-71. doi: 10.3934/mmc.2026005

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Research article

Fractional order dynamics in breast cancer control: a Caputo perspective

Anil Chavada ^,,
Nimisha Pathak

Department of Applied Mathematics, Faculty of Technology and Engineering, The Maharaja Sayajirao University of Baroda, Vadodara, Gujarat, India

Received: 08 April 2024 Revised: 21 May 2025 Accepted: 09 June 2025 Published: 03 March 2026

In this study, we present a novel fractional-order framework to model the dynamics of breast cancer, incorporating the Liouville–Caputo fractional derivative to capture memory effects inherent in biological systems. The model describes tumor progression and its regulation through chemotherapy within a fractional calculus setting, introducing three control variables-monoclonal antibody drugs, a ketogenic diet, and z-control to influence system behavior. The existence and uniqueness of solutions are rigorously established via Sadovskii's fixed-point theorem, while global stability is examined using Hyers–Ulam stability criteria. Numerical validation is carried out using a predictor–corrector method, and graphical simulations demonstrate the improved accuracy and realism of the fractional-order model compared to its integer-order counterpart. This framework offers a robust theoretical basis for improving breast cancer treatment strategies and has the potential to inform future clinical decision making.
- breast cancer dynamics,
- chemotherapy treatment,
- monoclonal antibody drug,
- Keto diet,
- z-control
Citation: Anil Chavada, Nimisha Pathak. Fractional order dynamics in breast cancer control: a Caputo perspective[J]. Mathematical Modelling and Control, 2026, 6(1): 57-71. doi: 10.3934/mmc.2026005

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Abstract

In this study, we present a novel fractional-order framework to model the dynamics of breast cancer, incorporating the Liouville–Caputo fractional derivative to capture memory effects inherent in biological systems. The model describes tumor progression and its regulation through chemotherapy within a fractional calculus setting, introducing three control variables-monoclonal antibody drugs, a ketogenic diet, and z-control to influence system behavior. The existence and uniqueness of solutions are rigorously established via Sadovskii's fixed-point theorem, while global stability is examined using Hyers–Ulam stability criteria. Numerical validation is carried out using a predictor–corrector method, and graphical simulations demonstrate the improved accuracy and realism of the fractional-order model compared to its integer-order counterpart. This framework offers a robust theoretical basis for improving breast cancer treatment strategies and has the potential to inform future clinical decision making.

References

[1]	C. W. Evans, The invasion and metastatic behaviour of malignant cells, In: The metastatic cell: behavior and biochemistry, Chapman and Hall: London, UK, 1991,137–214.
[2]	M. Patel, S. Nagl, The role of model integration in complex systems modelling: an example from cancer biology, Springer Berlin, Heidelberg, 2010. https://doi.org/10.1007/978-3-642-15603-8
[3]	D. A. Botesteanu, S. Lipkowitz, J. M. Lee, D. Levy, Mathematical models of breast and ovarian cancers, WIREs Syst. Biol. Med., 8 (2016), 337–362. https://doi.org/10.1002/wsbm.1343 doi: 10.1002/wsbm.1343
[4]	S. K. Dey, S. Charlie Dey, Mathematical modeling of breast cancer treatment, In: S. Sarkar, U. Basu, S. De, Applied mathematics, Springer Proceedings in Mathematics & Statistics, New Delhi: Springer, 146 (2015), 149–160. https://doi.org/10.1007/978-81-322-2547-8_13
[5]	S. I. Oke, M. B. Matadi, S. S. Xulu, Optimal control analysis of a mathematical model for breast cancer, Math. Comput. Appl., 23 (2018), 21. https://doi.org/10.3390/mca23020021 doi: 10.3390/mca23020021
[6]	P. Shah, T. Shah, Adaptive neuro fuzzy inference system based classifier in diagnosis of breast cancer, Results Control Optim., 14 (2024), 100358. https://doi.org/10.1016/j.rico.2023.100358 doi: 10.1016/j.rico.2023.100358
[7]	P. J. Shah, T. P. Shah, The ultimate kernel machine for diagnosis of breast cancer, Int. J. Appl. Pattern Recognit., 7 (2022), 1–14. https://doi.org/10.1504/IJAPR.2022.122259 doi: 10.1504/IJAPR.2022.122259
[8]	P. Shah, T. Shah, Identification of breast tumor using hybrid approach of independent component analysis and deep neural network, Int. J. Intell. Syst. Appl. Eng., 9 (2021), 209–219.
[9]	P. J. Shah, T. P. Shah, Regularized deep neural network in identification of breast cancer, Solid State Technol., 63 (2020), 22704–22715.
[10]	H. C. Wei, Mathematical modeling of ER-positive breast cancer treatment with AZD9496 and palbociclib, AIMS Math., 5 (2020), 3446–3455. https://doi.org/10.3934/math.2020223 doi: 10.3934/math.2020223
[11]	I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, In: Mathematics in science and engineering, San Diego: Academic Press, 198 (1998).
[12]	T. Q. Tang, Z. Shah, E. Bonyah, R. Jan, M. Shutaywi, N. Alreshidi, Modeling and analysis of breast cancer with adverse reactions of chemotherapy treatment through fractional derivative, Comput. Math. Methods Med., 2022 (2022), 5636844. https://doi.org/10.1155/2022/5636844 doi: 10.1155/2022/5636844
[13]	E. Ucar, N. Özdemir, E. Altun, Fractional order model of immune cells influenced by cancer cells, Math. Model. Nat. Phenom., 14 (2019), 1–12. https://doi.org/10.1051/mmnp/2019002 doi: 10.1051/mmnp/2019002
[14]	W. Adel, H. M. Srivastava, M. Izadi, A. Elsonbaty, A. El-Mesady, Dynamics and numerical analysis of a fractional-order toxoplasmosis model incorporating human and cat populations, Bound. Value Probl., 2024 (2024), 152. https://doi.org/10.1186/s13661-024-01965-w doi: 10.1186/s13661-024-01965-w
[15]	C. Baishya, S. J. Achar, P. Veeresha, An application of the Caputo fractional domain in the analysis of a COVID-19 mathematical model, Contemp. Math., 5 (2024), 255–283. https://doi.org/10.37256/cm.5120242363 doi: 10.37256/cm.5120242363
[16]	C. Baishya, M. K. Naik, R. N. Premakumari, Design and implementation of a sliding mode controller and adaptive sliding mode controller for a novel fractional chaotic class of equations, Results Control Optim., 14 (2024), 100338. https://doi.org/10.1016/j.rico.2023.100338 doi: 10.1016/j.rico.2023.100338
[17]	M. Izadi, H. M. Srivastava, Robust QLM-SCFTK matrix approach applied to a biological population model of fractional order considering the carrying capacity, Discrete Contin. Dyn. Syst. Ser. S, 18 (2025), 1159–1181. https://doi.org/10.3934/dcdss.2023101 doi: 10.3934/dcdss.2023101
[18]	M. K. Naik, C. Baishya, R. N. Premakumari, M. E. Samei, Navigating climate complexity and its control via hyperchaotic dynamics in a 4D Caputo fractional model, Sci. Rep., 14 (2024), 18015. https://doi.org/10.1038/s41598-024-68769-x doi: 10.1038/s41598-024-68769-x
[19]	M. K. Naik, C. Baishya, R. N. Premakumari, P. Veeresha, Exploring ocean pH dynamics via a mathematical modeling with the Caputo fractional derivative, J. Umm Al-Qura Univ. Appl. Sci., 11 (2025), 421–437. https://doi.org/10.1007/s43994-024-00168-4 doi: 10.1007/s43994-024-00168-4
[20]	R. N. Premakumari, C. Baishya, S. Rezapour, M. K. Naik, Z. M. Yaseem, S. Etemad, Qualitative properties and optimal control strategy on a novel fractional three-species food chain model, Qual. Theory Dyn. Syst., 23 (2024), 252. https://doi.org/10.1007/s12346-024-01110-z doi: 10.1007/s12346-024-01110-z
[21]	A. Singh, A. Chavada, N. Pathak, Fractional calculus approach to Pancreatic cancer therapy: modeling tumor and immune interactions with siRNA treatment, Int. J. Appl. Comput. Math, 11 (2025), 61. https://doi.org/10.1007/s40819-025-01873-2 doi: 10.1007/s40819-025-01873-2
[22]	A. Chavada, N. Pathak, A note on Lyapunov-type inequalities for fractional boundary value problems with Sturm-Liouville boundary conditions, J. Math. Ext., 15 (2021), 1–12. https://doi.org/10.30495/JME.2021.1634 doi: 10.30495/JME.2021.1634
[23]	N. Pathak, A. Chavada, M. Thakkar, Qualitative analysis of Caputo fractional mathematical model of Pancreatic Cancer, Math. Applicanda, 52 (2024), 205-226. https://doi.org/10.14708/ma.v52i2.7272 doi: 10.14708/ma.v52i2.7272
[24]	S. J. Achar, C. Baishya, P. Veeresha, L. Akinyemi, Dynamics of fractional model of biological pest control in tea plants with Beddington–DeAngelis functional response, Fractal Fract., 6 (2021), 1. https://doi.org/10.3390/fractalfract6010001 doi: 10.3390/fractalfract6010001
[25]	E. Ilhan, P. Veeresha, H. M. Baskonus, Fractional approach for a mathematical model of atmospheric dynamics of $CO_2$ gas with an efficient method, Chaos, Soliton. Fract., 152 (2021), 111347. https://doi.org/10.1016/j.chaos.2021.111347 doi: 10.1016/j.chaos.2021.111347
[26]	P. Veeresha, The efficient fractional order based approach to analyze chemical reaction associated with pattern formation, Chaos, Soliton. Fract., 165 (2022), 112862. https://doi.org/10.1016/j.chaos.2022.112862 doi: 10.1016/j.chaos.2022.112862
[27]	P. Veeresha, E. Ilhan, D. G. Prakasha, H. M. Baskonus, W. Gao, , A new numerical investigation of fractional order susceptible-infected-recovered epidemic model of childhood disease, Alex. Eng. J., 61 (2022), 1747–1756. https://doi.org/10.1016/j.aej.2021.07.015 doi: 10.1016/j.aej.2021.07.015
[28]	P. Veeresha, D. G. Prakasha, D. Baleanu, Analysis of fractional Swift‐Hohenberg equation using a novel computational technique, Math. Methods Appl. Sci., 43 (2020), 1970–1987. https://doi.org/10.1002/mma.6022 doi: 10.1002/mma.6022
[29]	J. E. Solís-Pérez, J. F. Gómez-Aguilar, A. Atangana, A fractional mathematical model of breast cancer competition model, Chaos, Soliton. Fract., 127 (2019), 38–54. https://doi.org/10.1016/j.chaos.2019.06.027 doi: 10.1016/j.chaos.2019.06.027
[30]	A. Yousef, F. Bozkurt, T. Abdeljawad, Mathematical modeling of the immune-chemotherapeutic treatment of breast cancer under some control parameters, Adv. Differ. Equ., 2020 (2020), 696. https://doi.org/10.1186/s13662-020-03151-5 doi: 10.1186/s13662-020-03151-5
[31]	D. K. Dave, T. P. Shah, Stability analysis and Z-control of breast cancer dynamics, Adv. Appl. Math. Sci., 21 (2021), 343–363.
[32]	H. M. Srivastava, M. Izadi, Generalized shifted airfoil polynomials of the second kind to solve a class of singular electrohydrodynamic fluid model of fractional order, Fractal Fract., 7 (2023), 94. https://doi.org/10.3390/fractalfract7010094 doi: 10.3390/fractalfract7010094
[33]	L. Mahto, S. Abbas, Existence and uniqueness of solution of Caputo fractional differential equations, AIP Conf. Proc., 1479 (2012), 896–899. https://doi.org/10.1063/1.4756286 doi: 10.1063/1.4756286
[34]	K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29 (2002), 3–22. https://doi.org/10.1023/A:1016592219341 doi: 10.1023/A:1016592219341
[35]	K. Diethelm, N. J. Ford, A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31–52. https://doi.org/10.1023/B:NUMA.0000027736.85078.be doi: 10.1023/B:NUMA.0000027736.85078.be

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