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Mathematical modeling of tumor-immune competitive system, considering the role of time delay

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  • Khajanchi, Subhas
  • Nieto, Juan J.

Abstract

In this paper, we consider a three-dimensional nonlinear delay differential system (tumor cells, cytotoxic-T lymphocytes, T-helper cells) with single interaction delay. We perform linear stability of the equilibria and the existence of Hopf bifurcation in which the discrete time delay is used as a bifurcation parameter. We estimate the length of delay to preserve the stability of period-1 limit cycle. We also investigate the direction, period, and the stability of bifurcated periodic solutions by applying normal form method and center manifold theory. We observe that the discrete time delay plays an important role in stability switching. Numerical simulations are presented to illustrate the rich dynamical behavior of the model with different values for the time delay τ including the existence of periodic oscillations, which demonstrate the phenomena of long-term tumor relapse.

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  • Khajanchi, Subhas & Nieto, Juan J., 2019. "Mathematical modeling of tumor-immune competitive system, considering the role of time delay," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 180-205.
    • Handle: RePEc:eee:apmaco:v:340:y:2019:i:c:p:180-205
    DOI: 10.1016/j.amc.2018.08.018
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    References listed on IDEAS

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    1. Khajanchi, Subhas, 2018. "Modeling the dynamics of glioma-immune surveillance," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 108-118.
    2. El-Gohary, Awad, 2008. "Chaos and optimal control of cancer self-remission and tumor system steady states," Chaos, Solitons & Fractals, Elsevier, vol. 37(5), pages 1305-1316.
    3. Natalie Kronik & Yuri Kogan & Moran Elishmereni & Karin Halevi-Tobias & Stanimir Vuk-Pavlović & Zvia Agur, 2010. "Predicting Outcomes of Prostate Cancer Immunotherapy by Personalized Mathematical Models," PLOS ONE, Public Library of Science, vol. 5(12), pages 1-8, December.
    4. Khajanchi, Subhas, 2015. "Bifurcation analysis of a delayed mathematical model for tumor growth," Chaos, Solitons & Fractals, Elsevier, vol. 77(C), pages 264-276.
    5. Khajanchi, Subhas & Ghosh, Dibakar, 2015. "The combined effects of optimal control in cancer remission," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 375-388.
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    Cited by:

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