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On the Dynamics of Immune-Tumor Conjugates in a Four-Dimensional Tumor Model

Author

Listed:
  • Konstantin E. Starkov

    (Instituto Politecnico Nacional, CITEDI, Av. IPN 1310, Nueva Tijuana, Tijuana 22435, Mexico
    These authors contributed equally to this work.)

  • Alexander P. Krishchenko

    (Department of Mathematical Modeling, Bauman Moscow State Technical University, 2-ya Baumanskaya, 5, Moscow 105005, Russia
    Federal Research Center “Computer Science and Control”, Russian Academy of Sciences, ul. Vavilova, 44, k.2, Moscow 119333, Russia
    These authors contributed equally to this work.)

Abstract

We examine the ultimate dynamics of the four-dimensional model describing interactions between host cells, immune cells, tumor cells, and immune-tumor conjugate cells proposed by Abernethy and Gooding in 2018. In our paper, the ultimate upper bounds for all variables of this model are obtained. Formulas for positively invariant sets are deduced. Using these results, we establish conditions for the existence of the global attractor, derive formulas for its location, and present conditions under which immune and immune-tumor conjugate cells asymptotically die out. Next, we study equilibrium points, including the stability property for most of the equilibrium points. We discuss the existence of very low cancer-burden equilibrium points. Next, parametric conditions are derived under which the derivative of the density of the immune-tumor conjugate cell population eventually tends to zero; this mathematically rigorously confirms the correctness of the application of model reduction for this model in studies of its ultimate dynamics. In the final section, we summarize the results of this work and outline how to continue this study.

Suggested Citation

  • Konstantin E. Starkov & Alexander P. Krishchenko, 2024. "On the Dynamics of Immune-Tumor Conjugates in a Four-Dimensional Tumor Model," Mathematics, MDPI, vol. 12(6), pages 1-16, March.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:6:p:843-:d:1356375
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    References listed on IDEAS

    as
    1. Das, Parthasakha & Mukherjee, Sayan & Das, Pritha, 2019. "An investigation on Michaelis - Menten kinetics based complex dynamics of tumor - immune interaction," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 297-305.
    2. Abernethy, Sam & Gooding, Robert J., 2018. "The importance of chaotic attractors in modelling tumour growth," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 507(C), pages 268-277.
    3. Marluci Cristina Galindo & Cristiane Nespoli & Marcelo Messias, 2015. "Hopf Bifurcation, Cascade of Period-Doubling, Chaos, and the Possibility of Cure in a 3D Cancer Model," Abstract and Applied Analysis, Hindawi, vol. 2015, pages 1-11, March.
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