Dynamics of a stochastic tumor-immune interaction system with an Ornstein-Uhlenbeck process

This paper investigates the dynamical behavior of a class of stochastic tumor-immune interaction systems incorporating Ornstein-Uhlenbeck (OU) processes. We first establish a four-dimensional deterministic ordinary differential equation (ODE) model describing the interactions among tumor cells, natural killer (NK) cells, CD8⁺T cells, and dendritic cells, then extend this model by incorporating OU processes to account for intrinsic stochastic perturbations with memory effects in the tumor microenvironment. Subsequently, the existence and local stability of equilibrium points in the deterministic system were analyzed, identifying multiple steady states, including death equilibrium, tumor-free equilibrium, and tumor-present equilibrium, along with their stability conditions. Further, the existence and uniqueness of global solutions for the stochastic system were proven, and an existence theorem for steady distributions was established using the Lyapunov function method. Moreover, through decoupling time scales, the system was simplified into a coupled model of tumor cells and OU noise. The stochastic basic reproduction number R₀^s was defined, revealing the system's threshold behavior: when R₀^s<1, tumors almost certainly become extinct; when R₀^s>1, tumors persist with positive probability. The study also identifies a critical noise intensity σ_c. When σ > σ_c, the system may undergo a noise-induced phase transition from extinction to persistence, even if the deterministic reproduction number R₀<1. This research provides theoretical foundations for understanding the nonlinear dynamics of tumor-immune systems and may provide ideas for the future development of immunotherapies.