In this paper, we studied the dynamical behavior of various phases of breast cancer using the Caputo Fabrizio (CF) fractional order derivative operator. The Picard-Lindelof (PL) method was used to investigate the existence and uniqueness of the proposed system. Moreover, we investigated the stability of the system in the sense of Ulam Hyers (UH) criteria. In addition, the two-step Adams-Bashforth (AB) technique was employed to simulate our methodology. The fractional model was then simulated using real data, which includes reported breast cancer incidences among females of Saudi Arabia from 2004 to 2016. The real data was used to determine the values of the parameters that were fitted using the least squares method. Also, residuals were computed for the integer as well as fractional-order models. Based on the results obtained, the CF model's efficacy rates were greater than those of the existing classical model. Graphical representations were used to illustrate numerical results by examining different choices of fractional order parameters, then the dynamical behavior of several phases of breast cancer was quantified to show how fractional order affects breast cancer behavior and how chemotherapy rate affects breast cancer behavior. We provided graphical results for a breast cancer model with effective parameters, resulting in fewer future incidences in the population of phases Ⅲ and Ⅳ as well as the disease-free state. Chemotherapy often raises the risk of cardiotoxicity, and our proposed model output reflected this. The goal of this study was to reduce the incidence of cardiotoxicity in chemotherapy patients while also increasing the pace of patient recovery. This research has the potential to significantly improve outcomes of patients and provide information of treatment strategies for breast cancer patients.
Citation: Anil Chavada, Nimisha Pathak. Transmission dynamics of breast cancer through Caputo Fabrizio fractional derivative operator with real data[J]. Mathematical Modelling and Control, 2024, 4(1): 119-132. doi: 10.3934/mmc.2024011
In this paper, we studied the dynamical behavior of various phases of breast cancer using the Caputo Fabrizio (CF) fractional order derivative operator. The Picard-Lindelof (PL) method was used to investigate the existence and uniqueness of the proposed system. Moreover, we investigated the stability of the system in the sense of Ulam Hyers (UH) criteria. In addition, the two-step Adams-Bashforth (AB) technique was employed to simulate our methodology. The fractional model was then simulated using real data, which includes reported breast cancer incidences among females of Saudi Arabia from 2004 to 2016. The real data was used to determine the values of the parameters that were fitted using the least squares method. Also, residuals were computed for the integer as well as fractional-order models. Based on the results obtained, the CF model's efficacy rates were greater than those of the existing classical model. Graphical representations were used to illustrate numerical results by examining different choices of fractional order parameters, then the dynamical behavior of several phases of breast cancer was quantified to show how fractional order affects breast cancer behavior and how chemotherapy rate affects breast cancer behavior. We provided graphical results for a breast cancer model with effective parameters, resulting in fewer future incidences in the population of phases Ⅲ and Ⅳ as well as the disease-free state. Chemotherapy often raises the risk of cardiotoxicity, and our proposed model output reflected this. The goal of this study was to reduce the incidence of cardiotoxicity in chemotherapy patients while also increasing the pace of patient recovery. This research has the potential to significantly improve outcomes of patients and provide information of treatment strategies for breast cancer patients.
| [1] |
C. Fitzmaurice, D. Dicker, A. Pain, H. Hamavid, M. Moradi-Lakeh, M. F. MacIntyre, et al., The global burden of cancer 2013, JAMA Oncol., 1 (2015), 505–527. https://doi.org/10.1001/jamaoncol.2015.0735 doi: 10.1001/jamaoncol.2015.0735
|
| [2] |
I. Vasiliadis, G. Kolovou, D. P. Mikhailidis, Cardiotoxicity and cancer therapy, Angiology, 65 (2014), 369–371. https://doi.org/10.1177/0003319713498298 doi: 10.1177/0003319713498298
|
| [3] | I. Podlubny, Fractional differential equations, Academic Press, 1999. |
| [4] | K. M. Owolabi, A Atangana, Mathematical modelling and analysis of fractional epidemic models using derivative with exponential kernel, CRC Press, 2020. |
| [5] |
S. R. Khirsariya, J. P. Chauhan, G. S. Hathiwala, Study of fractional diabetes model with and without complication class, Results Control Optim., 12 (2023), 100283. https://doi.org/10.1016/j.rico.2023.100283 doi: 10.1016/j.rico.2023.100283
|
| [6] |
S. R. Khirsariya, S. B. Rao, G. S. Hathiwala, Investigation of fractional diabetes model involving glucose-insulin alliance scheme, Int. J. Dyn. Control, 12 (2023), 1–14. https://doi.org/10.1007/s40435-023-01293-4 doi: 10.1007/s40435-023-01293-4
|
| [7] |
J. E. Solís-Pérez, J. F. Gómez-Aguilar, A. Atangana, A fractional mathematical model of breast cancer competition model, Chaos Solitions Fract., 127 (2019), 38–54. https://doi.org/10.1016/j.chaos.2019.06.027 doi: 10.1016/j.chaos.2019.06.027
|
| [8] |
M. Farman, M. Batool, K. S. Nisar, A. S. Ghaffari, A. Ahmad, Controllability and analysis of sustainable approach for cancer treatment with chemotherapy by using the fractional operator, Results Phys., 51 (2023), 106630. https://doi.org/10.1016/j.rinp.2023.106630 doi: 10.1016/j.rinp.2023.106630
|
| [9] |
C. Xu, M. Farman, A. Akgül, K. S. Nisar, A. Ahmad, Modeling and analysis fractal order cancer model with effects of chemotherapy, Chaos Solitons Fract., 161 (2022), 112325. https://doi.org/10.1016/j.chaos.2022.112325 doi: 10.1016/j.chaos.2022.112325
|
| [10] |
S. Kumar, R. P. Chauhan, A. H. Abdel-Aty, M. R. Alharthi, A study on transmission dynamics of HIV/AIDS model through fractional operators, Results Phys., 22 (2021), 103855. https://doi.org/10.1016/j.rinp.2021.103855 doi: 10.1016/j.rinp.2021.103855
|
| [11] |
S. Kumar, R. P. Chauhan, M. S. Osman, S. A. Mohiuddine, A study on fractional HIV-AIDs transmission model with awareness effect, Math. Methods Appl. Sci., 46 (2023), 8334–8348. https://doi.org/10.1002/mma.7838 doi: 10.1002/mma.7838
|
| [12] |
Z. Munawar, F. Ahmad, S. A. Alanazi, K. S. Nisar, M. Khalid, M. Anwar, et al., Predicting the prevalence of lung cancer using feature transformation techniques, Egypt. Inf. J., 23 (2022), 109–120. https://doi.org/10.1016/j.eij.2022.08.002 doi: 10.1016/j.eij.2022.08.002
|
| [13] |
S. T. Thabet, M. S. Abdo, K. Shah, Theoretical and numerical analysis for transmission dynamics of Covid-19 mathematical model involving Caputo-Fabrizio derivative, Adv. Differ. Equations, 1 (2021), 184. https://doi.org/10.1186/s13662-021-03316-w doi: 10.1186/s13662-021-03316-w
|
| [14] |
S. T. Thabet, M. S. Abdo, K. Shah, T. Abdeljawad, Study of transmission dynamics of Covid-19 mathematical model under ABC fractional order derivative, Results Phys., 19 (2020), 103507. https://doi.org/10.1016/j.rinp.2020.103507 doi: 10.1016/j.rinp.2020.103507
|
| [15] |
M. Farman, H. Besbes, K. S. Nisar, M. Omri, Analysis and dynamical transmission of Covid-19 model by using Caputo-Fabrizio derivative, Alex. Eng. J., 66 (2023), 597–606. https://doi.org/10.1016/j.aej.2022.12.026 doi: 10.1016/j.aej.2022.12.026
|
| [16] |
W. Ou, C. Xu, Q. Cui, Z. Liu, Y. Pang, M. Farman, et al., Mathematical study on bifurcation dynamics and control mechanism of tri-neuron BAM neural networks including delay, Math. Methods Appl. Sci., 2023. https://doi.org/10.1002/mma.9347 doi: 10.1002/mma.9347
|
| [17] |
C. Xu, D. Mu, Y. Pan, C. Aouiti, L. Yao, Exploring bifurcation in a fractional-order predator-prey system with mixed delays, J. Appl. Anal. Comput., 13 (2023), 1119–1136. https://doi.org/10.11948/20210313 doi: 10.11948/20210313
|
| [18] |
C. Xu, X. Cui, P. Li, J. Yan, L. Yao, Exploration on dynamics in a discrete predator-prey competitive model involving feedback controls, J. Biol. Dyn., 17 (2023), 2220349. https://doi.org/10.1080/17513758.2023.2220349 doi: 10.1080/17513758.2023.2220349
|
| [19] |
C. Xu, Q. Cui, Z. Liu, Y. Pan, X. Cui, W. Ou, et al., Extended hybrid controller design of bifurcation in a delayed chemostat model, Match Commun. Math. Comput. Chem., 90 (2023), 609–648. https://doi.org/10.46793/match.90-3.609X doi: 10.46793/match.90-3.609X
|
| [20] |
C. Xu, D. Mu, Z. Liu, Y. Pang, C. Aouiti, O. Tunc, et al., Bifurcation dynamics and control mechanism of a fractional-order delayed Brusselator chemical reaction model, Match Commun. Math. Comput. Chem., 89 (2023), 73–106. https://doi.org/10.46793/match.89-1.073x doi: 10.46793/match.89-1.073x
|
| [21] |
D. Mu, C. Xu, Z. Liu, Y. Pang, Further insight into bifurcation and hybrid control tactics of a chlorine dioxide-iodine-malonic acid chemical reaction model incorporating delays, Match Commun. Math. Comput. Chem., 89 (2023), 529–566. https://doi.org/10.46793/match.89-3.529M doi: 10.46793/match.89-3.529M
|
| [22] |
M. I. Ayari, S. T. Thabet, Qualitative properties and approximate solutions of thermostat fractional dynamics system via a nonsingular kernel operator, Arab J. Math. Sci., 2023. https://doi.org/10.1108/AJMS-06-2022-0147 doi: 10.1108/AJMS-06-2022-0147
|
| [23] |
S. Djennadi, N. Shawagfeh, M. Inc, M. S. Osman, J. F. Gómez-Aguilar, O. A. Arqub, The Tikhonov regularization method for the inverse source problem of time fractional heat equation in the view of ABC-fractional technique, Phys. Scr., 96 (2021), 094006. https://doi.org/10.1088/1402-4896/ac0867 doi: 10.1088/1402-4896/ac0867
|
| [24] |
A. Khalid, A. Rehan, K. S. Nisar, M. S. Osman, Splines solutions of boundary value problems that arises in sculpturing electrical process of motors with two rotating mechanism circuit, Phys. Scr., 96 (2021), 104001. https://doi.org/10.1088/1402-4896/ac0bd0 doi: 10.1088/1402-4896/ac0bd0
|
| [25] |
S. W. Yao, O. A. Arqub, S. Tayebi, M. S. Osman, W. Mahmoud, M. Inc, et al., A novel collective algorithm using cubic uniform spline and finite difference approaches to solving fractional diffusion singular wave model through damping-reaction forces, Fractals, 31 (2023), 2340069. https://doi.org/10.1142/S0218348X23400698 doi: 10.1142/S0218348X23400698
|
| [26] |
Z. A. Khan, S. U. Haq, T. S. Khan, I. Khan, K. S. Nisar, Fractional Brinkman type fluid in channel under the effect of MHD with Caputo-Fabrizio fractional derivative, Alex. Eng. J., 59 (2020), 2901–2910. https://doi.org/10.1016/j.aej.2020.01.056 doi: 10.1016/j.aej.2020.01.056
|
| [27] |
J. P. Chauhan, S. R. Khirsariya, G. S. Hathiwala, M. B. Hathiwala, New analytical technique to solve fractional-order Sharma-Tasso-Olver differential equation using Caputo and Atangana-Baleanu derivative operators, J. Appl. Anal. Anal., 2023. https://doi.org/10.1515/jaa-2023-0043 doi: 10.1515/jaa-2023-0043
|
| [28] |
S. T. Thabet, M. Vivas-Cortez, I. Kedim, Analytical study of $\mathcal ABC $-fractional pantograph implicit differential equation with respect to another function, AIMS Math., 8 (2023), 23635–23654. https://doi.org/10.3934/math.20231202 doi: 10.3934/math.20231202
|
| [29] |
S. W. Yao, S. Behera, M. Inc, H. Rezazadeh, J. P. S. Virdi, W. Mahmoud, et al., Analytical solutions of conformable Drinfel'd-Sokolov-Wilson and Boiti Leon Pempinelli equations via sine-cosine method, Results Phys., 42 (2022), 105990. https://doi.org/10.1016/j.rinp.2022.105990 doi: 10.1016/j.rinp.2022.105990
|
| [30] |
J. P. Chauhan, S. R. Khirsariya, A semi-analytic method to solve nonlinear differential equations with arbitrary order, Results Control Optim., 13 (2023), 100267. https://doi.org/10.1016/j.rico.2023.100267 doi: 10.1016/j.rico.2023.100267
|
| [31] |
S. R. Khirsariya, S. B. Rao, J. P. Chauhan, Semi-analytic solution of time-fractional korteweg-de vries equation using fractional residual power series method, Results Nonlinear Anal., 5 (2022), 222–234. https://doi.org/10.53006/rna.1024308 doi: 10.53006/rna.1024308
|
| [32] |
S. R. Khirsariya, S. B. Rao, J. P. Chauhan, A novel hybrid technique to obtain the solution of generalized fractional-order differential equations, Math. Comput. Simul., 205 (2023), 272–290. https://doi.org/10.1016/j.matcom.2022.10.013 doi: 10.1016/j.matcom.2022.10.013
|
| [33] |
S. R. Khirsariya, S. B. Rao, On the semi-analytic technique to deal with nonlinear fractional differential equations, J. Appl. Math. Comput. Mech., 22 (2023), 17–30. https://doi.org/10.17512/jamcm.2023.1.02 doi: 10.17512/jamcm.2023.1.02
|
| [34] |
S. Rashid, K. T. Kubra, S. Sultana, P. Agarwal, M. S. Osman, An approximate analytical view of physical and biological models in the setting of Caputo operator via Elzaki transform decomposition method, J. Comput. Appl. Math., 413 (2022), 114378. https://doi.org/10.1016/j.cam.2022.114378 doi: 10.1016/j.cam.2022.114378
|
| [35] |
L. Shi, S. Rashid, S. Sultana, A. Khalid, P. Agarwal, M. S. Osman, Semi-analytical view of time-fractional PDES with proportional delays pertaining to index and Mittag-Leffler memory interacting with hybrid transforms, Fractals, 31 (2023), 2340071. https://doi.org/10.1142/S0218348X23400716 doi: 10.1142/S0218348X23400716
|
| [36] |
P. Li, Y. Lu, C. Xu, J. Ren, Insight into Hopf bifurcation and control methods in fractional order BAM neural networks incorporating symmetric structure and delay, Cogn. Comput., 15 (2023), 1825–1867. https://doi.org/10.1007/s12559-023-10155-2 doi: 10.1007/s12559-023-10155-2
|
| [37] |
A. Atangana, K. M. Owolabi, New numerical approach for fractional differential equations, Math. Modell. Natural Phenom., 13 (2018), 3. https://doi.org/10.1051/mmnp/2018010 doi: 10.1051/mmnp/2018010
|
| [38] |
K. K. Ali, M. A. A. E. Salam, E. M. Mohamed, B. Samet, S. Kumar, M. S. Osman, Numerical solution for generalized nonlinear fractional integro-differential equations with linear functional arguments using Chebyshev series, Adv. Differ. Equations, 2020 (2020), 494. https://doi.org/10.1186/s13662-020-02951-z doi: 10.1186/s13662-020-02951-z
|
| [39] |
O. A. Arqub, S. Tayebi, D. Baleanu, M. S. Osman, W. Mahmoud, H. Alsulami, A numerical combined algorithm in cubic B-spline method and finite difference technique for the time-fractional nonlinear diffusion wave equation with reaction and damping terms, Results Phys., 41 (2022), 105912. https://doi.org/10.1016/j.rinp.2022.105912 doi: 10.1016/j.rinp.2022.105912
|
| [40] |
L. Shi, S. Tayebi, O. A. Arqub, M. S. Osman, P. Agarwal, W. Mahamoud, et al., The novel cubic B-spline method for fractional Painleve and Bagley-Trovik equations in the Caputo, Caputo-Fabrizio, and conformable fractional sense, Alex. Eng. J., 65 (2023), 413–426. https://doi.org/10.1016/j.aej.2022.09.039 doi: 10.1016/j.aej.2022.09.039
|
| [41] |
S. Qureshi, M. A. Akanbi, A. A. Shaikh, A. S. Wusu, O. M. Ogunlaran, W. Mahmoud, et al., A new adaptive nonlinear numerical method for singular and stiff differential problems, Alexandria Eng. J., 74 (2023), 585–597. https://doi.org/10.1016/j.aej.2023.05.055 doi: 10.1016/j.aej.2023.05.055
|
| [42] |
S. R. Khirsariya, S. B. Rao, Solution of fractional Sawada-Kotera-Ito equation using Caputo and Atangana-Baleanu derivatives, Math. Methods Appl. Sci., 46 (2023), 16072–16091. https://doi.org/10.1002/mma.9438 doi: 10.1002/mma.9438
|
| [43] |
S. R. Khirsariya, J. P. Chauhan, S. B. Rao, A robust computational analysis of residual power series involving general transform to solve fractional differential equations, Math. Comput. Simul, 216 (2023), 168–186. https://doi.org/10.1016/j.matcom.2023.09.007 doi: 10.1016/j.matcom.2023.09.007
|
| [44] |
T. Abdeljawad, S. T. Thabet, I. Kedim, M. I. Ayari, A. Khan, A higher-order extension of Atangana-Baleanu fractional operators with respect to another function and a Gronwall-type inequality, Bound. Value Probl., 2023 (2023), 49. https://doi.org/10.1186/s13661-023-01736-z doi: 10.1186/s13661-023-01736-z
|
| [45] |
S. T. Thabet, M. M. Matar, M. A. Salman, M. E. Samei, M. Vivas-Cortez, I. Kedim, On coupled snap system with integral boundary conditions in the G-Caputo sense, AIMS Math., 8 (2023), 12576–12605. https://doi.org/10.3934/math.2023632 doi: 10.3934/math.2023632
|
| [46] |
S. Thabet, I. Kedim, Study of nonlocal multiorder implicit differential equation involving Hilfer fractional derivative on unbounded domains, J. Math., 2023 (2023), 8668325. https://doi.org/10.1155/2023/8668325 doi: 10.1155/2023/8668325
|
| [47] |
S.Qureshi, A. Soomro, E. Hincal, J. R. Lee, C. Park, M. S. Osman, An efficient variable stepsize rational method for stiff, singular and singularly perturbed problems, Alex. Eng. J., 61 (2022), 10953–10963. https://doi.org/10.1016/j.aej.2022.03.014 doi: 10.1016/j.aej.2022.03.014
|
| [48] |
O. A. Arqub, M. S. Osman, C. Park, J. R. Lee, H. Alsulami, M. Alhodaly, Development of the reproducing kernel Hilbert space algorithm for numerical pointwise solution of the time-fractional nonlocal reaction-diffusion equation, Alex. Eng. J., 61 (2022), 10539–10550. https://doi.org/10.1016/j.aej.2022.04.008 doi: 10.1016/j.aej.2022.04.008
|
| [49] | M. Caputo, M. Fabrizio, New numerical approach for fractional differential equations, Progr. Fract. Differ. Appl., 1 (2015), 73–85. |
| [50] |
D. Baleanu, A. Jajarmi, H. Mohammad, S. Rezapour, A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative, Chaos Solitons Fract., 134 (2020), 109705. https://doi.org/10.1016/j.chaos.2020.109705 doi: 10.1016/j.chaos.2020.109705
|
| [51] |
S. Ullah, M. A. Khan, M. Farooq, Z. Hammouch, D. Baleanu, A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative, Discrete Contin. Dyn. Syst., 13 (2020), 975–993. https://doi.org/10.3934/dcdss.2020057 doi: 10.3934/dcdss.2020057
|
| [52] |
M. A. Dokuyucu, E. Celik, H. Bulut, H. M. Baskonus, Cancer treatment model with the Caputo-Fabrizio fractional derivative, Eur. Phys. J. Plus, 133 (2018), 92. https://doi.org/10.1140/epjp/i2018-11950-y doi: 10.1140/epjp/i2018-11950-y
|
| [53] |
M. M. El-Dessoky, M. A. Khan, Application of Caputo- Fabrizio derivative to a cancer model with unknown parameters, Discrete Contin. Dyn. Syst., 14 (2021), 3557–3575. https://doi.org/10.3934/dcdss.2020429 doi: 10.3934/dcdss.2020429
|
| [54] |
M. Ngungu, E. Addai, A. Adeniji, U. M. Adam, K. Oshinubi, Mathematical epidemiological modeling and analysis of monkeypox dynamism with non-pharmaceutical intervention using real data from United Kingdom, Front. Public Health, 11 (2023), 1101436 https://doi.org/10.3389/fpubh.2023.1101436 doi: 10.3389/fpubh.2023.1101436
|
| [55] |
A. Yousef, F. Bozkurt, T. Abdeljawad, Mathematical modeling of the immune-chemotherapeutic treatment of breast cancer under some control parameters, Adv. Differ. Equations, 2020 (2020), 696. https://doi.org/10.1186/s13662-020-03151-5 doi: 10.1186/s13662-020-03151-5
|
| [56] |
E. Alzahrani, M. M. El-Dessoky, M. A. Khan, Mathematical model to understand the dynamics of cancer, prevention diagnosis and therapy, Mathematics, 11 (2023), 1975. https://doi.org/10.3390/math11091975 doi: 10.3390/math11091975
|
| [57] |
S. M. Albeshan, Y. I. Alashban, Incidence trends of breast cancer in Saudi Arabia: a joinpoint regression analysis (2004–2016), J. King Saud Univ. Sci., 33 (2021), 101578. https://doi.org/10.1016/j.jksus.2021.101578 doi: 10.1016/j.jksus.2021.101578
|
| [58] | J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 87–92. https://doi.org/12785/pfda/010202 |
| [59] | S. M. Ulam, Problems in modern mathematics, Dover Publications, 2004. |
| [60] | S. M. Ulam, A collection of mathematical problems, Interscience Publishers, 1960. |
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Anil Chavada, Nimisha Pathak. Transmission dynamics of breast cancer through Caputo Fabrizio fractional derivative operator with real data[J]. Mathematical Modelling and Control, 2024, 4(1): 119-132. doi: 10.3934/mmc.2024011
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