Stochastic tumor immune dynamics with saturating response

We study a stochastic tumor-immune interaction model with a saturating response. The stochastic perturbation is driven by one Brownian motion acting through the tumor-immune interaction term, so random fluctuations modify effective immune contact rather than the two populations independently. For positive initial data, the system admits a unique global positive solution. If immune loss dominates baseline immune proliferation, [Formula: see text], then the total population is ultimately bounded in first moment. Tumor extinction requires sustained immune pressure. If [Formula: see text] for all sufficiently large [Formula: see text] almost surely, and if [Formula: see text], then [Formula: see text] almost surely and in mean. Persistence is governed by the saturated upper bound [Formula: see text]The resulting threshold is stochastic: weak persistence in mean follows when [Formula: see text]The term [Formula: see text] is the Itô correction produced by interaction noise, and the deterministic saturated threshold is recovered when [Formula: see text]. Krylov-Bogoliubov averaging gives existence of invariant probability measures on the closed state space under a Feller assumption. Uniqueness, ergodicity, and exponential convergence remain open for the present rank-one degenerate diffusion. Numerical experiments with dimensional parameters from tumor-immune calibrations support the persistence threshold, saturation sensitivity, bounded empirical occupation measures, time-average consistency, and step-size reliability.