Stochastic Invasion and Immune Feedback in a Tumor–Immune–IL-4 Model

Deterministic and stochastic models describing tumor–immune–cytokine interactions regulated by IL-4 are developed and analyzed. The deterministic system provides baseline tumor–immune dynamics, while a continuous-time Markov chain (CTMC) formulation is introduced to capture intrinsic demographic fluctuations and exact absorbing states corresponding to tumor extinction. A diffusion approximation is derived, leading to a stochastic differential equation (SDE) model that describes fluctuations near positive equilibria when population sizes are sufficiently large. Invasion probabilities obtained from the full CTMC model are compared with those predicted by a branching process approximation under a frozen immune environment. Numerical simulations show that immune–cytokine feedback strongly influences tumor outcomes and can generate stochastic invasion behavior that is not predicted by the deterministic model.