Bifurcation study of a tumor-immune system with chemotherapy

Understanding the dynamics of cancer cell growth, the interplay between tumor and immune cells, and the efficacy of chemotherapy are pivotal areas of focus in cancer research. In this regard, mathematical modeling can provide significant insights. This study re-examines a classical two-dimensional model of tumor-immune cell interactions where the tumor's growth rate is assumed to adhere to von Bertalanffy's model instead of the logistic model. We investigate the model both without chemotherapy and with treatment. The equilibrium points are identified, classified, and their stability analyzed. Our results reveal that the model can demonstrate a broad spectrum of behaviors, including bi-stability and multi-stability as well as regions of stable periodic behavior. We establish analytical conditions for the existence of Hopf points. Furthermore, we assess the impact of model parameters on the various behavior predicted by the model. This mathematical investigation can provide general guidance on treatment strategies.