Fractional-order mathematical modelling of cancer cells-cancer stem cells-immune system interaction with chemotherapy
Abstract
In this paper, we present a mathematical model of stem cells and chemotherapy for cancer treatment, in which the model is represented by fractional-order differential equations. Local stability of equilibrium points is discussed. Then, the existence and uniqueness of the solution are studied. In addition, in order to point out the advantages of the fractional-order modeling, memory trace and hereditary traits are taken into consideration. Numerical simulations have been used to investigate how the fractional-order derivative and different parameters affect the population dynamics, the graphs have been illustrated according to different values of fractional order $\alpha$ and different parameter values. Moreover, we have examined the effect of chemotherapy on tumor cells and stem cells over time. Furthermore, we concluded that the memory effect occurs as the $\alpha$ decreases from 1 and the chemotherapy drug is quite effective on the populations. We hope that this work will contribute to helping medical scientists take the necessary measures during the screening process and treatment.
Keywords
Fractional-order differential equations, cancer stem cells, immune system, numerical solutions, memory effect, existence and uniqueness
References
- El-Gohary, A. Chaos and optimal control of cancer self-remission and tumor system steady states. Chaos, Solutions and Fractals, 37(5), 1305-1316, (2008).
- El-Gohary, A. The chaos and optimal control of cancer model with complete unknown parameters. Chaos, Solutions and Fractals, 42(5), 2865-2874, (2009).
- Kirschner, D., Panetta, J.C. Modeling immunotherapy of the tumor–immune interaction. Journal of mathematical biology, 37(3), 235-252, (1998).
- Öztürk, I. & Özköse, F. Stability analysis of fractional order mathematical model of tumor-immune system interaction. Chaos, Solitons & Fractals, 133, 109614, (2020).
- Hadamard, J. Essai sur l’étude des fonctions données par leur développement de Taylor (PDF). Journal de Mathématiques Pures et Appliquées, 4(8), 101–186, (1892).
- Li, C. & Tao, C. On the fractional Adams method. Computers & Mathematics with Applications, 58(8), 1573-1588, (2009).
- Yavuz, M. & Sene, N. Stability analysis and numerical computation of the fractional predator–prey model with the harvesting rate. Fractal and Fractional, 4(3), 35, (2020).
- Naik, P.A., Owolabi, K.M., Yavuz, M. & Zu, J. Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells. Chaos, Solitons & Fractals, 140, 110272, (2020).
- Özköse, F. & Yavuz, M. Investigation of interactions between COVID-19 and diabetes with hereditary traits using real data: A case study in Turkey. Computers in biology and medicine, 105044, (2021).
- Hammouch, Z., Yavuz, M. & Özdemir, N. Numerical solutions and synchronization of a variable-order fractional chaotic system. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 11-23, (2021).