Mathematical analysis and simulations of a cancer model with interleukins and delayed immunotherapy

A new system of delayed differential equations for tumor-immune system interactions is proposed and studied. The system describes the interactions between tumor cells and the immune system, at the most aggressive phase of cancer, where tumor cells have developed mechanisms from earlier stages to evade immune responses. The model incorporates adoptive cellular immunotherapy as the only effective treatment, and accounts for the complex influence of interleukins. To render the model more biologically realistic, it also includes a time delay representing the lag between the interaction of tumor cells and activated immune cells and the subsequent effectiveness of the immune response. These aspects, captured together, reflect the biological novelty of our approach. We first establish the well-posedness of the system, proving the global existence of the solution and its positivity. Then, we focus on the equilibrium states of the system, their asymptotic stability, and how the delay affects the stability of the equilibrium states and the behaviour of the solution. We investigate the global stability of a tumor equilibrium state with biological relevance, and estimate the length of the delay that preserves the stability of a tumor-free equilibrium. These results ensure the mathematical consistency of the model and provide a solid theoretical basis for analysing how delayed immune responses can influence the long-term behaviour of the tumor progression. Finally, we conduct some numerical simulations of the delayed system in order to illustrate the theoretical analysis and investigate the impact of the delay. We interpret the results and explore their biological implications in the context of cancer progression.