Ponente
Descripción
In this work, we study a new three-dimensional mathematical model of cancer that describes the interaction between tumor cells, immune system effector cells, and the cytokine IL-2 in patients undergoing immunotherapy. The model accounts for both functional exhaustion and the depletion of effector cells, as well as their proliferation induced by tumor stimulation, thereby capturing relevant aspects of the nonlinear dynamics of tumors. Our objective is to analyze certain aspects of the dynamics and mechanisms associated with long term phenomena in cancer progression, such as tumor clearance, disease persistence, and recurrence under immunotherapeutic treatment. In particular, we demonstrate that the system’s trajectories are bounded and well-defined for all positive time. Furthermore, we characterize and study the stability of the equilibrium solutions and show that, under certain parameter conditions, transcritical and pitchfork bifurcations emerge from the tumor-free equilibrium. Moreover, through a local analysis around a perturbed equilibrium, we establish the existence of a Hopf bifurcation leading to the emergence of periodic orbits, corresponding to cyclic oscillations in the cell populations.Our analysis reveals that the model exhibits rich dynamical behavior strongly influenced by the parameters $\mu_1$ (death rate of effector cells) and $p_1$ (production rate of effector cells stimulated by IL-2). These results provide a deeper theoretical understanding of the complex interactions between tumors and the immune system, as well as the impact of immunotherapy on cancer progression. Furthermore, the bifurcation analysis reveals how small changes in immune response or therapeutic efficacy may drastically alter the long-term dynamics of the disease. In this context, mathematical modeling offers a valuable framework for exploring therapeutic scenarios while reducing the cost and complexity associated with identifying effective treatments, since experimental and clinical studies are often invasive, time-consuming, and expensive.