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A mathematical model for tumor-immune competitive system with multiple time delays

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  • Sardar, Mrinmoy
  • Khajanchi, Subhas
  • Biswas, Santosh
  • Ghosh, Sumana

Abstract

A mathematical model is proposed to study the complicated dynamics of tumor-immune interactions with three discrete time delays. The proposed model consists of nine coupled ordinary differential equations (ODEs) whose components are tumor cells, tumor-specific CD8+T cells, dendritic cells, macrophages, regulatory T-cells (Tregs), interleukin-10 (IL-10), IL-12, transforming growth factor-β (TGF-β) and interferon-γ (IFN-γ). By utilizing the quasi-steady-state approximation for cytokine concentrations, we obtain a system of four coupled ODEs. We introduce three discrete time delays into our deterministic system to better understand the tumor-immune dynamics. Basic properties of the system, including existence, uniqueness, positivity, boundedness, and uniform persistence are discussed. Stability analysis of both the delayed and non-delayed system has been performed and the conditions for stability and direction of Hopf bifurcation have been determined. Furthermore, we assess the length of time delay required to maintain the stability of period-1 limit cycle. Parameter estimation techniques are discussed and numerical simulations are provided to support our theoretical analysis. Notably, we observe that time delays do not significantly influence the system behavior for the existing set of parameters value.

Suggested Citation

  • Sardar, Mrinmoy & Khajanchi, Subhas & Biswas, Santosh & Ghosh, Sumana, 2024. "A mathematical model for tumor-immune competitive system with multiple time delays," Chaos, Solitons & Fractals, Elsevier, vol. 179(C).
    • Handle: RePEc:eee:chsofr:v:179:y:2024:i:c:s0960077923012997
    DOI: 10.1016/j.chaos.2023.114397
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    References listed on IDEAS

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    1. 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.
    2. Khajanchi, Subhas, 2015. "Bifurcation analysis of a delayed mathematical model for tumor growth," Chaos, Solitons & Fractals, Elsevier, vol. 77(C), pages 264-276.
    3. Sardar, Mrinmoy & Biswas, Santosh & Khajanchi, Subhas, 2021. "The impact of distributed time delay in a tumor-immune interaction system," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    4. Al-Hussein, Abdul-Basset A. & Rahma, Fadihl & Jafari, Sajad, 2020. "Hopf bifurcation and chaos in time-delay model of glucose-insulin regulatory system," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
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