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.

In this work, we develop a new mathematical model to describe the tumor-immune system interactions in the presence of interleukins and a recently developed immunotherapy. We describe the tumor-immune interactions at the cellular level, using the approach of the Kinetic Theory of active particles, by considering four populations, namely tumor cells, activated immune cells, immune cells and interleukins. We also derive the macroscopic model relative to the kinetic system and study the well-posedness of the Cauchy problem for the corresponding system of equations. Then, we determine the equilibrium states of the system and investigate their asymptotic stability. Particular attention is devoted to the global stability of a biologically meaningful tumor-free equilibrium. Some numerical simulations are also performed in view of illustrating the behaviour of the solution predicted by the theoretical analysis.