Research Article
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Year 2023, Volume: 44 Issue: 2, 345 - 355, 30.06.2023

Senol Kartal

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

Abstract

Keywords

Tumor model difference equation stability Neimark-Sacker bifurcation Lyapunov exponents

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.

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

Year 2023, Volume: 44 Issue: 2, 345 - 355, 30.06.2023

Senol Kartal

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

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

Tumor model difference equation stability Neimark-Sacker bifurcation Lyapunov exponents

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.
There are 34 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Senol Kartal 0000-0003-1205-069X

Publication Date June 30, 2023
Submission Date January 19, 2023
Acceptance Date May 1, 2023
Published in Issue Year 2023Volume: 44 Issue: 2

Cite

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