Dynamics of a Symmetric Model of Competition Between Tumor and Immune Cells Under Chemotherapy

This paper studies a model for competition between natural killer (NK) cells, cytotoxic T lymphocytes (CTLs) and tumor cells, and evaluates the outcomes in the absence and presence of chemotherapy treatment. The growth rate of the tumor is presumed to follow the classical logistic law. The model particularly emphasizes the rate-limiting recruitment of NK cells and CTL cells, which is activated by the presence of the tumor. It additionally includes the activation of CTL cells through debris produced by the lysis of tumor cells by NK cells, alongside the regulatory effect that NK cells have on CTL cells. Additionally, the model incorporates the reciprocal decreases in cell populations resulting from the interactions between tumor cells and immune cells, along with the impact of chemotherapy on all three types of cells. We analyze the stability of the equilibrium points. Utilizing parameter values that have been experimentally confirmed in the literature and applying some elementary principles of singularity theory, we investigate the bistability regimes anticipated by the model in the absence of chemotherapy, and evaluate the impact of model parameters on this behavior. This mathematical analysis serves to evaluate the effectiveness of chemotherapy treatment. We demonstrate that the interplay between the biological parameters in the model and those associated with chemotherapy can result in a range of treatment outcomes. The proposed mathematical analysis may serve as a valuable tool in directing the development of strategies for treatment interventions.