Chaos in a Three-Dimensional Cancer Model with Piecewise Constant Arguments

Senol Kartal

doi:10.17776/csj.1239101

EN

Chaos in a Three-Dimensional Cancer Model with Piecewise Constant Arguments

Abstract

In this study, we analyze a cancer model which includes the interactions among tumor cells, healthy host cells and effector immune cells. The model with continuous case has been studied in the literature and it has been shown that it exhibits chaotic behavior. In this paper, we aim to build a better understanding of how both discrete and continuous times affect the dynamic behavior of the tumor growth model. So, we reconsider the model as a system of differential equations with piecewise constant argument. To analyze dynamical behavior of the model, we consider the solution of the system in a certain subinterval which leads to the system of difference equations. Some theoretical results are obtained for local behavior of the system. In addition, we study chaotic dynamic of the system through Neimark-Sacker bifurcation by using Lyapunov exponents

Keywords

References

[1] Costa O.S., Molina L.M., Perez D.R., Antoranz J.C., Reyes M.C., Behavior of tumors under nonstationary therapy, Physica D., 178 (2003) 242-253.
[2] Onofrio A.D., A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences, Physica D., 208 (2005) 220-235.
[3] Kuznetsov V.A., Makalkin I.A., Taylor M.A., Perelson A.S., Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994) 295-321.
[4] Kirschner D., Panetta J.C., Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998) 235-252.
[5] Itık M., Banks S.P., Chaos in a three-dimensional cancer model, Int. J. Bifurcat. Chaos., 20 (2010) 71-79.
[6] Galindo M.C., Nespoli C., Messias M., Hopf bifurcation, cascade of period-doubling, chaos, and the possibility of cure in a 3D cancer model, Abstr. Appl. Anal., (2015) Article ID:354918.
[7] De Pillis L.G., Radunskaya A., The dynamics of an optimally controlled tumor model: A case study, Math. Comput. Model., 37 (2003) 1221-1244.
[8] Sarkar R.R., Banerjee S., Cancer self remission and tumor stability- a stochastic approach, Math. Biosci., 196 (2005) 65-81.

[9] Cho H., Lewis A.L, Storey K.M., Byrne H.M., Designing experimental conditions to use the Lotka–Volterra model to infer tumor cell line interaction types, J. Theor. Biol., 559 (2023) 111377.
[10] Pham H., Mathematical Modeling the Time-Delay Interactions between Tumor Viruses and the Immune System with the Effects of Chemotherapy and Autoimmune Diseases, Mathematics, 10(5) (2022) 756.
[11] Das A., Dehingia K., Sarmah H.K., Hosseini K., Sadri K., Salahshour S., Analysis of a delay-induced mathematical model of cancer, Adv. Cont. Discr. Mod., 15 (2022).
[12] Abernathy Z., Abernathy K., Stevens J., A mathematical model for tumor growth and treatment using virotherapy, AIMS Math., 5(5) (2020) 4136-4150.
[13] Rihan F.A., Alsakaji H.J., Kundu S., Mohamed O., Dynamics of a time-delay differential model for tumour-immune interactions with random noise, Alex. Eng. J., 61(12) (2022) 11913-11923.
[14] Bekker R.A., Kim S., Thomas S.P., Enderling H., Mathematical modeling of radiotherapy and its impact on tumor interactions with the immune system, Neoplasia, 28 (2022) 100796.
[15] Hussain J., Bano Z., Ahmed W., Shahid S., Analysis of stochastic dynamics of tumor with drug interventions, Chaos Soliton Fract., 157 (2022) 111932.
[16] Hu X., Jang S.R.J., Dynamics of tumor–CD4–cytokine–host cells interactions with treatments, Appl. Math. Comput., 321, (2018) 700-720.
[17] Rivaz A., Azizian M., Soltani M., Various Mathematical Models of Tumor Growth with Reference to Cancer Stem Cells: A Review, Iran J. Sci. Technol. Trans. Sci., 4 (2019) 687–700.
[18] Kemwoue F.F., Deli V., Mendimi J.M., Gninzanlong C.L., Tagne J.F., Atangana J., Dynamics of cancerous tumors under the effect of delayed information: mathematical and electronic study, Int. J. Dynam. Control., (2022).
[19] Banerjee S., Sarkar R.R., Delay-induced model for tumor-immune interaction and control of malignant tumor growth, Biosystems., 91 (2008) 268-288.
[20]Busenberg, S., Cooke, K.L., Models of vertically transmitted diseases with sequential continuous dynamics, Nonlinear Phenomena in Mathematical Sciences, Academic Press, New York (1982).
[21] Cooke K.L., Györi I., Numerical approximation of the solutions of delay-differential equations on an infinite interval using piecewise constant argument, Comput. Math. Appl., 28 (1994) 81-92.
[22] Shah S.M., Wiener J., Advanced differential equations with piecewise constant argument deviations, Int. J. Math. Math. Sci., 6 (1983) 671–703.
[23]Akhmet M., Nonlinear hybrid continuous/discrete-time models, Atlantis Press (2011).
[24]Karakoç F., Oscillation of a first order linear impulsive delay differential equation with continuous and piecewise constant arguments, Hacet. J. Math. Stat., 47(3) (2018) 601- 613.
[25]Chiu, K.S. Jeng, J.C. Stability of oscillatory solutions of differential equations with general piecewise constant arguments of mixed type, Math. Nachrichten., 288(10) (2015) 1085-1097.
[26]Oztepe G.S., Existence and qualitative properties of solutions of a second order mixed type impulsive differential equation with piecewise constant arguments, Hacet. J. Math. Stat., 46(6) (2017) 1077 – 1091.
[27] Oztepe G.S., Karakoç F., Bereketoglu H., Oscillation and Periodicity of a Second Order Impulsive Delay Differential Equation with a Piecewise Constant Argument, Commun. Math., 25 (2017) 89–98. [28]Ozturk, I., Bozkurt, F., Gurcan, F.: Stability analysis of a mathematical model in a microcosm with piecewise constant arguments, Math. Biosci., 240 (2012) 85-91.
[29]Bozkurt F., Yousef A., Bilgil H., Baleanu D., A mathematical model with piecewise constant arguments of colorectal cancer with chemo-immunotherapy, Chaos Soliton Fract., 168 (2023) 113207.
[30]Gurcan, F., Kartal, S., Ozturk, I., Bozkurt, F.: Stability and bifurcation analysis of a mathematical model for tumor-immune interaction with piecewise constant arguments of delay, Chaos Soliton. Fract. 68 (2014) 169-179.
[31] Kartal, S., Gurcan, F.: Stability and bifurcations analysis of a competition model with piecewise constant arguments, Math. Methods. Appl. Sci. 38 (2015) 1855-1866.
[32]Li, X., Mou, C., Niu, W., Wang, D.: Stability analysis for discrete biological models using algebraic methods, Math. Comput. Sci. 5 (2011) 247-262.
[33]Khan, A.Q., Qureshi, S.M., Alotaibi, A.M.: Bifurcation analysis of a three species discrete-time predator-prey model, Alex. Eng. J. 61 (2022) 2853-7875.
[34]Sandri, M.: Numerical Calculation of Lyapunov Exponents, The Mathematica Journal., 6 (1996) 78-84.
[35]Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.M.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. Part 2: Numerical application, 15 (1980) 21-30.

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Senol Kartal ^*
0000-0003-1205-069X
Türkiye

Publication Date

June 30, 2023

Submission Date

January 19, 2023

Acceptance Date

May 1, 2023

Published in Issue

Year 2023 Volume: 44 Number: 2

DOI

https://doi.org/10.17776/csj.1239101

IZ

https://izlik.org/JA88SG69KR

Cite

RIS / Bibtex

APA

Kartal, S. (2023). Chaos in a Three-Dimensional Cancer Model with Piecewise Constant Arguments. Cumhuriyet Science Journal, 44(2), 345-355. https://doi.org/10.17776/csj.1239101

Chaos in a Three-Dimensional Cancer Model with Piecewise Constant Arguments

Öz

Anahtar Kelimeler

Chaos in a Three-Dimensional Cancer Model with Piecewise Constant Arguments

Abstract

Keywords

References

Details

Primary Language

Subjects

Journal Section

Authors

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

Cite