Analysis of a delayed mathematical model for tumor-immune cell interactions with Holling type II functional response
DOI:
https://doi.org/10.17268/sel.mat.2023.02.03Keywords:
Cancer, holling functional response, Hopf bifurcation, Delay differential equationAbstract
In this study, we analyzed a three-dimensional nonlinear differential system considering Holling type II functional response that describes the dynamics of tumor cells, cytotoxic T lymphocytes, and helper T cells, with a single interaction delay. The linear stability of the internal equilibrium point and the presence of the Hopf bifurcation are examined, with the discrete time delay serving as the bifurcation parameter. To demonstrate the rich dynamic behavior of the model, we present numerical simulations with various values of the time delay τ and the attack rate of cytotoxic T lymphocytes on tumor cells (α1). These simulations exhibit the presence of periodic oscillations and tumor death with survival of the mentioned immune cells, for all α1 greater than a fixed threshold with or without delay.
References
Adam, J. A. and Bellomo N., A survey of models for tumor immune system dynamics, Springer Science and Business Media, 1997.
Bajzer Z, Marusíc M, Vuk-Pavlovíc S. Conceptual frameworks for mathematical modeling of tumor growth dynamics. Mathematical and computer modelling. 1996;23(6):31-46.
Bellouquid A, De Angelis E, Knopoff D. From the modeling of the immune hallmarks of cancer to a black swan in biology. Mathematical Models and Methods in Applied Sciences. 2013;23(05):949-978.
Chun-Biao G, Matjaz P, Qing-Yun W. Delay-aided stochastic multiresonances on scale-free FitzHugh–Nagumo neuronal networks. Chinese Physics B. 2010;19(4):040508.
Dullens HFJ, Van Der Tol MWM, De Weger RA, Den Otter W. A survey of some formal models in tumor immunology. Cancer Immunology, Immunotherapy. 1986;23:159-164.
Eftimie R, Bramson JL, Earn DJ. Interactions between the immune system and cancer: a brief review of non-spatial mathematical models. Bulletin of mathematical biology. 2011;73:2-32.
Erbe LH, Freedman HI, Rao VSH. Three-species food-chain models with mutual interference and time delays. Mathematical Biosciences. 1986;80(1):57-80.
Farc O, Cristea V. An overview of the tumor microenvironment, from cells to complex networks. Experimental and therapeutic medicine. 2021;21(1):1-1.
Galach M. Dynamics of the tumor-immune system competition the effect of time delay. Int J Appl Math Comput Sci. 2003;13:395–406.
Grivennikov SI, Greten FR, Karin M. Immunity, inflammation, and cancer. Cell. 2010;140(6):883-899.
Guerrini L, Gori L, Matsumoto A, Sodini M, Zhang Z, Bianca C. Time Delayed Equations as Models in Nature and Society. Discrete Dynamics in Nature and Society. 2016.
Khajanchi S, Nieto JJ. Mathematical modeling of tumor-immune competitive system, considering the role of time delay. Applied mathematics and computation. 2019;340:180-205.
Mahdipour-Shirayeh A, Kaveh K, Kohandel M, Sivaloganathan S. Phenotypic heterogeneity in modeling cancer evolution. PLoS One. 2017;12(10):e0187000.
Mahlbacher GE, Reihmer KC, Frieboes HB. Mathematical modeling of tumor-immune cell interactions. Journal of Theoretical Biology. 2019;469:47-60.
Makhlouf AM, El-Shennawy L, Elkaranshawy HA. Mathematical modelling for the role of CD4+ T cells in tumor-immune interactions. Computational and mathematical methods in medicine. 2020.
North J, Bakhsh I, Marden C, Pittman H, Addison E, Navarrete C, et al. Tumor-primed human natural killer cells lyse NKresistant tumor targets: evidence of a two-stage process in resting NK cell activation. The Journal of Immunology. 2007;178(1):85-94.
de Pillis LG, Radunskaya AE, Wiseman CL. A validated mathematical model of cell-mediated immune response to tumor growth. Cancer research. 2005;65(17):7950-7958.
Rihan FA, Rihan NF. Dynamics of cancer-immune system with external treatment and optimal control. J. Cancer Sci. Ther. 2016;8(10):257-261.
Sarkar RR, Banerjee S. Cancer self remission and tumor stability–a stochastic approach. Mathematical Biosciences. 2005;196(1):65-81.
Tenen DG. Disruption of differentiation in human cancer: AML shows the way. Nature reviews cancer. 2003;3(2):89-101.
Thomlinson RH. Measurement and management of carcinoma of the breast. Clin Radiol. 1982;33(5):481-93.
Vesely MD, Kershaw MH, Schreiber RD, Smyth MJ. Natural innate and adaptive immunity to cancer. Annual review of immunology. 2011;29:235-271.
Villasana M, Radunskaya A. A delay differential equation model for tumor growth. Journal of mathematical biology. 2003;47:270-294.
Vogelstein B, Kinzler KW. The multistep nature of cancer. Trends in genetics. 1993;9(4):138-141.
Wang Q, Perc M, Duan Z, Chen G. Delay-induced multiple stochastic resonances on scale-free neuronal networks. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2009;19(2).
Weinberg RA. How cancer arises. Scientific American. 1996;275:62–70.
Wilkie KP. A review of mathematical models of cancer–immune interactions in the context of tumor dormancy. Systems biology of tumor dormancy. 2013;201-234.
Yang X, Chen L, Chen J. Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models. Computers and mathematics with applications. 1996;32(4):109-116.
Zamarron BF, Chen W. Dual roles of immune cells and their factors in cancer development and progression. International journal of biological sciences. 2011;7(5):651.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Selecciones Matemáticas
This work is licensed under a Creative Commons Attribution 4.0 International License.
The authors who publish in this journal accept the following conditions:
1. The authors retain the copyright and assign to the journal the right of the first publication, with the work registered with the Creative Commons Attribution License,Atribución 4.0 Internacional (CC BY 4.0) which allows third parties to use what is published whenever they mention the authorship of the work And to the first publication in this magazine.
2. Authors may make other independent and additional contractual arrangements for non-exclusive distribution of the version of the article published in this journal (eg, include it in an institutional repository or publish it in a book) provided they clearly state that The paper was first published in this journal.
3. Authors are encouraged to publish their work on the Internet (for example, on institutional or personal pages) before and during the review and publication process, as it can lead to productive exchanges and to a greater and more rapid dissemination Of the published work.
