# Filipin III br Substituting the parameters described in into

Substituting the parameters described in (56) into the model

dt

D

dt
D

E

dt

F

dt

The numerical results given in Fig. 4a–d shows numerical simu-lations of the special solution of our model as function of time for different values of αn for n ∈ [1; 5].

5. Conclusions

The solution of the model (57) can be obtained applying the

Adams-Bashforth method [34]. The Numerical scheme is given by
In this paper, we study a fractional breast cancer model. The

mathematical model is built using a Liouville–Caputo and Caputo–

Fabrizio–Caputo fractional derivatives. We consider the integer or-

D

der malaria transmission model proposed in [4] and modify it Filipin III to

become a fractional order model. Special solutions using an iter-

ative scheme via Laplace transform were obtained and the fixed

point theorem is discussed to prove the existence and uniqueness

of the coupled-solutions. Furthermore, we obtain numerical solu-

tions considering the Atangana–Toufik numerical scheme. We can

observe that the results obtained by using the Caputo–Fabrizio–

D

Caputo fractional derivative are different to those obtained by the

derivative of Liouville–Caputo type, for the first fractional deriva-

tive the memory effect is more evident. The numerical solutions

showed that the dynamical behaviour of the breast cancer model

depends on the fractional derivative. Also new behaviors have been

T

n

D

E

Declaration of Competing Interest

None.

C

T

Acknowledgments

E

Jesús Emmanuel Solís Pérez acknowledges the support pro-

KTn−1 (tn−1 )En−1 (tn−1 )
,
vided by CONACyT through the assignment doctoral fellowship.

by CONACyT: Cátedras CONACyT para jóvenes investigadores 2014

and the support provided by SNI-CONACyT.

n
F n

References

[2] Mufudza C, Sorofa W, Chiyaka ET. Assessing the effects of estrogen on the dy-namics of breast cancer. Comput Math Methods Med 2012;1:1–14.

[4] Abernathy K, Abernathy Z, Baxter A, Stevens M. Global dynamics of a breast cancer competition model. Differ Equations Dyn Syst 2017;1:1–15.

[5] Chen C, Baumann WT, Xing J, Xu L, Clarke R, Tyson JJ. Mathematical models of the transitions between endocrine therapy responsive and resistant states in breast cancer. J R Soc Interface 2014;11(96):1–11.

[6] Wang Z, Butner JD, Kerketta R, Cristini V, Deisboeck TS. Simulating cancer growth with multiscale agent-based modeling. In: Seminars in cancer biology., vol. 30. Academic Press.; 2015. p. 70–8.

[7] Paine I, Chauviere A, Landua J, Sreekumar A, Cristini V, Rosen J, Lewis MT. A geometrically-constrained mathematical model of mammary gland ductal elongation reveals novel cellular dynamics within the terminal end bud. PLoS Comput Biol 2016;12(4):1–23.

[9] Jenner AL, Yun CO, Kim PS, Coster AC. Mathematical modelling of the interac-tion between cancer cells and an oncolytic virus: insights into the effects of treatment protocols. Bull Math Biol 2018;1:1–15.

[10] Weis JA, Miga MI, Yankeelov TE. Three-dimensional image-based mechanical modeling for predicting the response of breast cancer to neoadjuvant therapy. Comput Methods Appl MechEng 2017;314:494–512.

[11] Lee AJ, Cunningham AP, Kuchenbaecker KB, Mavaddat N, Easton DF, Anto-niou AC. BOADICEA breast cancer risk prediction model: updates to cancer in-cidences, tumour pathology and web interface. Br J Cancer 2014;110(2):1–11.

[12] Owolabi KM. Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems. Chaos Solitons Fractals 2016;93:89–98.