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Researchers should cite this work as follows:

• Yafia, R. (2023). 2011-Radouane_Yafia-A Study of Differential Equations Modeling Malignant Tumor Cells in Competition with Immune System. SIMIODE, QUBES Educational Resources. doi:10.25334/H4MZ-QQ20

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Yafia, Radouane. 2011. A Study of Differential Equations Modeling Malignant Tumor Cells in Competition with Immune System.  International Journal of Biomathematics. 4(2):  185-206.

See https://www.worldscientific.com/doi/10.1142/S1793524511001404 . Accessed 29 March 2023.

Abstract: In this paper, we present a competition model of malignant tumor growth that includes the immune system response. The model considers two populations: immune system (effector cells) and population of tumor (tumor cells). Ordinary differential equations are used to model the system to take into account the delay of the immune response. The existence of positive solutions of the model (with/without delay) is showed. We analyze the stability of the possible steady states with respect to time delay and the existence of positive solutions of the model (with and without delay). We show theoretically and through numerical simulations that periodic oscillations may arise through Hopf bifurcation. An algorithm for determining the stability of bifurcating periodic solutions is proved.

Keywords: Immunology, tumor, modeling, differential equation, delay , stability, Hopf bifurcation