Examining of a tumor system with Caputo derivative

Abstract

Cancer is a disease that many people are exposed to, which results in the recovery of some and the death of others. For this reason, A system reflecting the relationship between immune system and tumor growth in this study is examined. This system is handled with the traditional Caputo fractional derivative. The stability analysis of equilibrium points and solution properties of this system is searched. Then, the conditions about the existence and uniqueness of the solution for this system are given. In conclusion, the fractional system is solved benefiting from Grünwald-Letnikov scheme.

Keywords

• Caputo derivative
• mathematical modeling
• numerical solution.

Tümör sisteminin Caputo türev ile incelenmesi

Abstract

Birçok insanın maruz kaldığı kanser, bazı hastaların iyileşmesi bazılarının ölmesi ile sonuçlanan bir hastalıktır. Bu nedenle bu çalışmada bağışıklık sistemi ile tümör büyümesi arasındaki ilişkiyi yansıtan bir sistem inceliyoruz. Söz konusu sistem, Caputo kesirli türevi ile ele alınacaktır. Bu sistemin denge noktalarının kararlılık analizini ve çözüm özelliklerini vereceğiz. Daha sonra bu sistem için çözüm özellikleri belirtilecektir. Son olarak, bu kesirli sistemi Grünwald-Letnikov nümerik metodunu kullanarak çözeceğiz.

Keywords

• Caputo derivative
• mathematical modeling
• numerical solution

References

1. Castiglione, F. and Piccoli, B., Cancer immunoteraphy, mathematical modeling and optimal control, Journal of Theoratical Biology, 247, 723-732, (2007).Pillis, L.G. and Radunskaya A., A mathematical tumor model with immune resistance and drug therapy: an optimal control approach, Journal of Theoratical Medicine, 3, 79-100, (2000).
2. Kirschner, D. and Panetta, J.C., Modelling immunoterapy of tumor-immune interaction, Journal of Mathematical Biology, 37, 235-252, (1998).
3. Arshad, S., Baleanu, D., Huang, J., Tang, Y. and Qurashi, M.M.A. Dynamical analysis of fractional order model immugonemic tumors, Advances in Mechanical Engineering, 8, 1-13, (2016).
4. Kilbas, A.A and Marzan, S.A., Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differential Equations, 41, 82-89, (2005).
5. Podlubny, I., Fractional Differential Equations, Academic Press, New York, (1999).
6. Fernandez, A., Uçar, S. and Özdemir, N., Solving a well-posed fractional initial value problem by a complex approach, Fixed Point Theory and Algorithms for Sciences and Engineering, 1, 1-13, (2021).
7. Uçar, E., Uçar, S, Evirgen, F. and Özdemir, N., A Fractional SAIDR Model in the Frame of Atangana–Baleanu Derivative, Fractal and Fractional, 5, 32, (2021).
8. Uçar, S., Özdemir, N., Koca, İ., and Altun, E., Novel analysis of the fractional glucose–insulin regulatory system with non-singular kernel derivative, The European Physical Journal Plus, 135, 1-18, (2020).

9. Koca, i., Analysis of rubella disease model with non-local and non-singular fractional derivatives, An International Journal of Optimization and Control Theories & Applications (IJOCTA), 8, 17-25, (2018).
10. Hristov, J., Magnetic field diffusion in ferromagnetic materials: fractional calculus approaches, An International Journal of Optimization and Control Theories & Applications (IJOCTA), 12, 20-38, (2022).
11. Hammouch, Z., Yavuz, M., and 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).
12. Veeresha, P., Yavuz, M., and Baishya, C., A computational approach for shallow water forced Korteweg–De Vries equation on critical flow over a hole with three fractional operators, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 11(3), 52-67, (2021).
13. Özköse, F., Şenel, M. T., and Habbireeh, R., Fractional-order mathematical modelling of cancer cells-cancer stem cells-immune system interaction with chemotherapy, Mathematical Modelling and Numerical Simulation with Applications, 1(2), 67-83, (2021).
14. Baleanu, D., Güvenç, Z. and Teenreriro Machado, J.A. New trends in nanotechnology and fractional calculus applications, Springer, (2010).
15. Pinto, C.M.A. and Carvalho, A. R. M., Fractional modeling of typical stages in HIV epidemics with drug-resistance, Progress in Fractional Differentiation and Applications an International Journal, 2, 111-122, (2015).
16. Momani, S. and Odibat, Z., Numerical comparison of methods for solving linear differential equations of fractional order. Chaos, Solitons Fractals, 131, 1248-1255 (2007).
17. Özdemir, N., Avcı, D. And İskender, B. B., The numerical solutions of a two-dimensional-space-time Riesz-Caputo factional diffusion equation, An International Journal of Optimization and Control Theories & Applications (IJOCTA), 1, 17-26, (2011).
18. Scherer, R., Kalla, S. L., Yang, Y., and Huang, J., The Grunwald-Letkinov method for fractional differential equations, Computers Mathematics with Applications, 62, 902-917, (2011).
19. Kumar, V., Abbas, A. and Aster, J., Robbins and cotran pathologic basis of disease, Elsevier, (2014).
20. Minelli, A., Topputo, F. and Bernelli F., Controlled drug delivery in cancer immunotherapy: stability, optimization and monte carlo analysis, SIAM Journal on Applied Mathematics, 71, 2229-2245, (2011).
21. Ahmed E., El-Sayed A. M. A., El-Saka H. A. A., Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models, Journal of Mathematical Analysis and Applications, 325, 542-553, (2007).
22. Ahmed, E., El-Sayed, A. M. A. and El-Saka, H. A. A., Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, Journal of Mathematical Analysis and Applications, 325, 542-553, (2007).
23. Bozkurt, F., Stability analysis of fractional-order differential equation system of a GBM-IS interaction depending on the density, Applied Mathematics and Information Sciences, 8, 1021-1028, (2014).
24. El-Sayed A. M. A., El-Mesiry, A. E. M. and El-Saka, H. A. A., On the fractional- order logistic equation, Applied Mathematics Letters, 20, 817-823, (2007).