Tumor expansion and immune regulation in a mathematical model of cancer under variations in tumor cell proliferation rate and innate immune stimulation

Manuel Arturo Nova-Martínez; Héctor Andrés Granada-Díaz; Manuel Arturo Nova-Martínez; Héctor Andrés Granada-Díaz

doi:10.3934/mbe.2025104

Mathematical Biosciences and Engineering

2025, Volume 22, Issue 11: 2826-2851. doi: 10.3934/mbe.2025104

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Tumor expansion and immune regulation in a mathematical model of cancer under variations in tumor cell proliferation rate and innate immune stimulation

Manuel Arturo Nova-Martínez ^{1
,
,},
Héctor Andrés Granada-Díaz ²

1.
Facultad de Ingeniería, Grupo de Investigación Programas de Ingenierías (Grupo-GIPIS), Universidad Cooperativa de Colombia, Villavicencio 500001, Colombia
2.
Departamento de Matemáticas y Estadística, Grupo de Matemáticas del Tolima (Grupo-MaT), Universidad del Tolima, Ibagué 730001, Colombia

Received: 19 July 2025 Revised: 29 August 2025 Accepted: 09 September 2025 Published: 15 September 2025

In this article, we proposed a simplified mathematical model of primary tumor growth that involves four cell populations: Two types of cancer cells with different levels of immunogenicity, and the immune response in its two components, innate and adaptive. By varying the proliferation rate of non-immunogenic cancer cells and the innate immune stimulation parameter, and applying biparametric numerical continuation techniques, we identified distinct stability regions that revealed scenarios of tumor escape and latency. A closed curve of supercritical Hopf bifurcation points was also detected, delineating the parameter region in which limit cycles emerged. By examining the population maxima of each cell type at steady state, we identified parameter values at which both immunogenic and non-immunogenic tumor cell populations remain in stable equilibrium at modest levels, sustained by an immune response that does not escalate to intensities associated with immunological damage.
- Hopf bifurcation,
- immune regulation,
- limit cycles,
- mathematical model,
- non-immunogenic tumors
Citation: Manuel Arturo Nova-Martínez, Héctor Andrés Granada-Díaz. Tumor expansion and immune regulation in a mathematical model of cancer under variations in tumor cell proliferation rate and innate immune stimulation[J]. Mathematical Biosciences and Engineering, 2025, 22(11): 2826-2851. doi: 10.3934/mbe.2025104

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Abstract

In this article, we proposed a simplified mathematical model of primary tumor growth that involves four cell populations: Two types of cancer cells with different levels of immunogenicity, and the immune response in its two components, innate and adaptive. By varying the proliferation rate of non-immunogenic cancer cells and the innate immune stimulation parameter, and applying biparametric numerical continuation techniques, we identified distinct stability regions that revealed scenarios of tumor escape and latency. A closed curve of supercritical Hopf bifurcation points was also detected, delineating the parameter region in which limit cycles emerged. By examining the population maxima of each cell type at steady state, we identified parameter values at which both immunogenic and non-immunogenic tumor cell populations remain in stable equilibrium at modest levels, sustained by an immune response that does not escalate to intensities associated with immunological damage.

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