Dynamical analysis of a Tumor-Macrophages interaction model governed by the Caputo-Fabrizio derivatives

A. E. Matouk; Eman Ali Ahmed Ziada; Ausif Padder; Monica Botros; Taher S. Hassan; A. E. Matouk; Eman Ali Ahmed Ziada; Ausif Padder; Monica Botros; Taher S. Hassan

doi:10.3934/math.2026547

AIMS Mathematics

2026, Volume 11, Issue 5: 13257-13286. doi: 10.3934/math.2026547

Previous Article Next Article

Research article Special Issues

Dynamical analysis of a Tumor-Macrophages interaction model governed by the Caputo-Fabrizio derivatives

1.
Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah, 11952, Saudi Arabia
2.
Basic Science Department, Nile Higher Institute for Engineering and Technology, Mansoura 35511, Egypt
3.
Symbiosis Institute of Technology, Hyderabad Campus, Symbiosis International (Deemed University), Pune, India
4.
Basic Science Department, Faculty of Engineering, Delta University for Science and Technology, Gamasa 11152, Egypt
5.
Faculty of Artificial Intelligence, Delta University for Science and Technology, Gamasa, 11152, Egypt
6.
Department of Mathematics, Faculty of Science, University of Ha'il, Ha'il 2440, Saudi Arabia

Received: 26 February 2026 Revised: 08 April 2026 Accepted: 22 April 2026 Published: 13 May 2026
MSC : 26A33, 45J05, 65M70, 65R20

The Caputo-Fabrizio derivative is considered to be one of the most successful tools of fractional modeling since it has non singular nuclei, which make it a better candidate for modeling some interdisciplinary models that rely on memory effects and hereditary features such as the mathematical models in biology. In this study, tumor-macrophages interactions are modeled via the Caputo-Fabrizio derivatives. The uniqueness of the model's solutions is shown. A convergence analysis based on the Adomian decomposition method (ADM) is proved. The error analysis using the ADM is discussed. The Picard method is applied to the considered model. According to the stability theory of fractional-order systems governed by the Caputo-Fabrizio derivatives, the stability conditions of the tumor-free, the tumor-dominant, and the co-axial or the existence equilibrium points are discussed. Numerical simulations are carried out to show the rich complex dynamics in the model, including the existence of chaotic attractors.
- Tumor-Macrophages interaction model,
- Caputo-Fabrizio derivatives,
- Picard method,
- stability conditions,
- chaos
Citation: A. E. Matouk, Eman Ali Ahmed Ziada, Ausif Padder, Monica Botros, Taher S. Hassan. Dynamical analysis of a Tumor-Macrophages interaction model governed by the Caputo-Fabrizio derivatives[J]. AIMS Mathematics, 2026, 11(5): 13257-13286. doi: 10.3934/math.2026547

Related Papers:

Abstract

The Caputo-Fabrizio derivative is considered to be one of the most successful tools of fractional modeling since it has non singular nuclei, which make it a better candidate for modeling some interdisciplinary models that rely on memory effects and hereditary features such as the mathematical models in biology. In this study, tumor-macrophages interactions are modeled via the Caputo-Fabrizio derivatives. The uniqueness of the model's solutions is shown. A convergence analysis based on the Adomian decomposition method (ADM) is proved. The error analysis using the ADM is discussed. The Picard method is applied to the considered model. According to the stability theory of fractional-order systems governed by the Caputo-Fabrizio derivatives, the stability conditions of the tumor-free, the tumor-dominant, and the co-axial or the existence equilibrium points are discussed. Numerical simulations are carried out to show the rich complex dynamics in the model, including the existence of chaotic attractors.

References

[1]	M. R. Owen, J. A. Sherratt, Modelling the macrophage invasion of tumors: Effects on growth and composition, IMA J. Math. Appl. Med., 15 (1998), 165–185. https://doi.org/10.1093/imammb/15.2.165 doi: 10.1093/imammb/15.2.165
[2]	R. Eftimie, C. Barelle, Mathematical investigation of innate immune responses to Lung cancer: The role of macrophages with mixed phenotypes, J. Theor. Biol., 524 (2021), 110739. https://doi.org/10.1016/j.jtbi.2021.110739 doi: 10.1016/j.jtbi.2021.110739
[3]	I. Podlubny, Fractional differential equations, 1999.
[4]	R. Hilfer, Applications of fractional calculus in physics, World Scientific Publishing Co. Pte. Ltd, 2000. https://doi.org/10.1142/3779
[5]	N. Laskin, Fractional market dynamics, Physica A, 287 (2000), 482–492. https://doi.org/10.1016/S0378-4371(00)00387-3
[6]	D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus: Models and numerical methods, World Scientific, 2012.
[7]	A. Al-Khedhairi, A. E. Matouk, I. Khan, Chaotic dynamics and chaos control for the fractional-order geomagnetic field model, Chaos Soliton. Fract., 128 (2019), 390–401. https://doi.org/10.1016/j.chaos.2019.07.019 doi: 10.1016/j.chaos.2019.07.019
[8]	A. E. Matouk, I. G. Ameen, Y. A. Gaber, Analyzing the dynamics of fractional spatio-temporal SEIR epidemic model, AIMS Math., 9 (2024), 30838–30863. https://doi.org/10.3934/math.20241489 doi: 10.3934/math.20241489
[9]	A. E. Matouk, Advanced applications of fractional differential operators to science and technology, 2020.
[10]	D. Baleanu, H. Mohammadi, S. Rezapour, A fractional differential equation model for the COVID-19 transmission by using the Caputo-Fabrizio derivative, Adv. Differ. Equ., 2020 (2020), 299. https://doi.org/10.1186/s13662-020-02762-2 doi: 10.1186/s13662-020-02762-2
[11]	Z. Zhang, S. Jain, Mathematical model of Ebola and Covid-19 with fractional differential operators: Non-Markovian process and class for virus pathogen in the environment, Chaos Soliton. Fract., 140 (2020), 110175. https://doi.org/10.1016/j.chaos.2020.110175 doi: 10.1016/j.chaos.2020.110175
[12]	M. Caputo, Linear models of dissipation whose Q is almost frequency independent–Ⅱ, Geophys. J. Int., 13 (1967), 529–539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x doi: 10.1111/j.1365-246X.1967.tb02303.x
[13]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 73–85. https://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
[14]	J. M. Cruz-Duarte, J. Rosales-Garcia, C. R. Correa-Cely, A. Garcia-Perez, J. G. Avina-Cervantes, A closed form expression for the Gaussian-based Caputo-Fabrizio fractional derivative for signal processing applications, Commun. Nonlinear Sci., 61 (2018), 138–148. https://doi.org/10.1016/j.cnsns.2018.01.020 doi: 10.1016/j.cnsns.2018.01.020
[15]	M. Ran, X. Liao, D. Lin, R. Yang, Analog realization of fractional-order capacitor and inductor via the Caputo-Fabrizio derivative, J. Adv. Comput. Intell. Intell. Inform., 25 (2021), 291–300. https://doi.org/10.20965/jaciii.2021.p0291 doi: 10.20965/jaciii.2021.p0291
[16]	A. M. Alqahtani, S. Sharma, A. Chaudhary, A. Sharma, Application of Caputo-Fabrizio derivative in circuit realization, AIMS Math., 10 (2025), 2415–2443. https://doi.org/10.3934/math.2025113 doi: 10.3934/math.2025113
[17]	D. Yu, X. Liao, Y. Wang, Modeling and analysis of Caputo-Fabrizio definition-based fractional-order boost converter with inductive loads, Fractal Fract., 8 (2024), 81. https://doi.org/10.3390/fractalfract8020081 doi: 10.3390/fractalfract8020081
[18]	M. M. El-Dessoky, M. A. Khan, Application of Caputo-Fabrizio derivative to a cancer model with unknown parameters, Discrete Cont. Dyn. S, 14 (2020), 3557. https://doi.org/10.3934/dcdss.2020429 doi: 10.3934/dcdss.2020429
[19]	E. A. A. Ziada, M. Botros, Solution of a fractional mathematical model of brain metabolite variations in the circadian rhythm containing the Caputo-Fabrizio derivative, J. Appl. Math. Comput., 71 (2025), 3353–3380. https://doi.org/10.1007/s12190-024-02327-6 doi: 10.1007/s12190-024-02327-6
[20]	Y. Chu, M. F. Khan, S. Ullah, S. A. A. Shah, M. Farooq, M. B. Mamat, Mathematical assessment of a fractional-order vector-host disease model with the Caputo-Fabrizio derivative, Math. Method. Appl. Sci., 46 (2022), 232–247. https://doi.org/10.1002/mma.8507 doi: 10.1002/mma.8507
[21]	M. A. Khan, Z. Hammouch, D. Baleanu, Modeling the dynamics of hepatitis E via the Caputo-Fabrizio derivative, Math. Model. Nat. Phenom., 14 (2019), 311. https://doi.org/10.1051/mmnp/2018074 doi: 10.1051/mmnp/2018074
[22]	E. J. Moore, S. Sirisubtawee, S. Koonprasert, A Caputo-Fabrizio fractional differential equation model for HIV/AIDS with treatment compartment, Adv. Differ. Equ., 2019 (2019), 200. https://doi.org/10.1186/s13662-019-2138-9 doi: 10.1186/s13662-019-2138-9
[23]	D. Baleanu, A. Jajarmi, H. Mohammadi, S. Rezapour, A new study on the mathematical modelling of human liver with Caputo-Fabrizio derivative, Chaos Soliton. Fract., 134 (2020), 109705. https://doi.org/10.1016/j.chaos.2020.109705 doi: 10.1016/j.chaos.2020.109705
[24]	O. J. Peter, Transmission dynamics of fractional order Brucellosis model using Caputo-Fabrizio operator, Int. J. Differ. Equ., 2020 (2020), 2791380. https://doi.org/10.1155/2020/2791380 doi: 10.1155/2020/2791380
[25]	E. Bonyah, M. Juga, Fractional dynamics of coronavirus with comorbidity via Caputo-Fabrizio derivative, Commun. Math. Biol. Neurosci., 2022 (2022), 12.
[26]	Z. Shah, N. Ullah, R. Jan, M. H. Alshehri, N. Vrinceanu, E. Antonescu, et al., Existence and sensitivity analysis of a Caputo-Fabrizio fractional order vector-borne disease model, Eur. J. Pure Appl. Math., 18 (2025), 5687. https://doi.org/10.29020/nybg.ejpam.v18i2.5687 doi: 10.29020/nybg.ejpam.v18i2.5687
[27]	R. Shafqat, A. Alsaadi, Bifurcation in a fractional SIR model with normalized Caputo-Fabrizio derivative, Discrete Dyn. Nat. Soc., 2025 (2025), 4368999. https://doi.org/10.1155/ddns/4368999 doi: 10.1155/ddns/4368999
[28]	R. Shafqat, A. Al-Quran, A. Alsaadi, A. M. Djaouti, Normalized Caputo-Fabrizio SVIR modeling and bifurcation analysis, Sci. Rep., 16 (2026), 8193. https://doi.org/10.1038/s41598-026-38301-4 doi: 10.1038/s41598-026-38301-4
[29]	T. Zhang, Y. Li, Global exponential stability of discrete-time almost automorphic Caputo-Fabrizio BAM fuzzy neural networks via exponential Euler technique, Knowl. Based Syst., 246 (2022), 108675. https://doi.org/10.1016/j.knosys.2022.108675 doi: 10.1016/j.knosys.2022.108675
[30]	A. E. Matouk, The impact of Caputo-Fabrizio operator on the complex dynamics of a multidrug resistance model, AIMS Math., 11 (2026), 8655–8676. https://doi.org/10.3934/math.2026356 doi: 10.3934/math.2026356
[31]	E. Ahmed, A. Hashish, F. A. Rihan, On fractional order cancer model, J. Fract. Calc. Appl. Anal., 3 (2012), 1–6.
[32]	F. A. Rihan, A. Hashish, F. Al-Maskari, M. Sheek-Hussein, E. Ahmed, M. B. Riaz, et al., Dynamics of tumor-immune system with fractional-order, J. Tumor Res., 2 (2016), 109–115. https://doi.org/10.35248/2684-1258.16.2.109 doi: 10.35248/2684-1258.16.2.109
[33]	A. M. Wazwaz, A comparison between Adomian decomposition method and Taylor series method in the series solutions, Appl. Math. Comput., 97 (1998), 37–44. https://doi.org/10.1016/S0096-3003(97)10127-8 doi: 10.1016/S0096-3003(97)10127-8
[34]	A. Sadighi, D. D. Ganji, Exact solutions of Laplace equation by homotopy-perturbation and Adomian decomposition methods, Phys. Lett. A, 367 (2007), 83–87. https://doi.org/10.1016/j.physleta.2007.02.082 doi: 10.1016/j.physleta.2007.02.082
[35]	R. Rach, On the Adomian (decomposition) method and comparisons with Picard's method, J. Math. Anal. Appl., 128 (1987), 480–483. https://doi.org/10.1016/0022-247X(87)90199-5 doi: 10.1016/0022-247X(87)90199-5
[36]	E. A. A. Ziada, H. Hashem, A. Al-Jaser, O. Moaaz, M. Botros, Numerical and analytical approach to the Chandrasekhar quadratic functional integral equation using Picard and Adomian decomposition methods, Electron. Res. Arch., 32 (2024), 5943–5965. https://doi.org/10.3934/era.2024275 doi: 10.3934/era.2024275
[37]	E. A. A. Ziada, M. Botros, Solution of a fractional mathematical model of brain metabolite variations in the circadian rhythm containing the Caputo-Fabrizio derivative, J. Appl. Math. Comput., 71 (2025), 3353–3380. https://doi.org/10.1007/s12190-024-02327-6 doi: 10.1007/s12190-024-02327-6
[38]	E. A. A. Ziada, S. El-Morsy, O. Moaaz, S. S. Askar, A. M. Alshamrani, M. Botros, Solution of the SIR epidemic model of arbitrary orders containing Caputo-Fabrizio, Atangana-Baleanu and Caputo derivatives, AIMS Math., 9 (2024), 18324–18355. https://doi.org/10.3934/math.2024894 doi: 10.3934/math.2024894
[39]	L. Wang, A new algorithm for solving classical Blasius equation, Appl. Math. Comput., 157 (2004), 1–9. https://doi.org/10.1016/j.amc.2003.06.011 doi: 10.1016/j.amc.2003.06.011
[40]	S. Momani, Z. Odibat, Analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method, Appl. Math. Comput., 177 (2006), 488–494. https://doi.org/10.1016/j.amc.2005.11.025 doi: 10.1016/j.amc.2005.11.025
[41]	G. A. M. Nchama, L. D. Lau-Alfonso, A. M. L. Mecias, M. R. Ricard, Properties of the Caputo-Fabrizio fractional derivative, Appl. Math. Inf. Sci., 14 (2020), 761–769. https://doi.org/10.18576/amis/140503 doi: 10.18576/amis/140503
[42]	H. Li, J. Cheng, H. B. Li, S. M. Zhong, Stability analysis of a fractional-order linear system described by the Caputo-Fabrizio derivative, Mathematics, 7 (2019), 200. https://doi.org/10.3390/math7020200 doi: 10.3390/math7020200
[43]	Y. Shu, J. Huang, Y. Dong, Y. Takeuchi, Mathematical modeling and bifurcation analysis of pro- and anti-tumor macrophages, Appl. Math. Model., 88 (2020), 758–773. https://doi.org/10.1016/j.apm.2020.06.042 doi: 10.1016/j.apm.2020.06.042
[44]	A. E. Matouk, Fractional Routh-Hurwitz conditions and nonlinear dynamics in some 3D and 4D dynamical systems modeled by Caputo-Fabrizio operators, Results Appl. Math., 26 (2025), 100588. https://doi.org/10.1016/j.rinam.2025.100588 doi: 10.1016/j.rinam.2025.100588
[45]	A. E. Matouk, M. Botros, Hidden chaotic attractors and self-excited chaotic attractors in a novel circuit system via Grünwald-Letnikov, Caputo-Fabrizio and Atangana-Baleanu fractional operators, Alex. Eng. J., 116 (2025), 525–534. https://doi.org/10.1016/j.aej.2024.12.064 doi: 10.1016/j.aej.2024.12.064

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)