Dynamics classification and threshold analysis of three-dimensional stochastic tumor-immune system

Understanding the dynamics of cancer-immune interactions is imperative for oncologists and applied mathematicians, particularly due to the unpredictable proliferation of tumor cells. Mathematical modeling serves as a paramount approach to elucidate this critical aspect of cancer biology. In response to the pressing needs for comprehending the cancer biology, this paper focuses on dynamical behaviors of a class of three-dimensional stochastic tumor-immune system perturbed by cellular microenvironment fluctuation. Firstly, the well-posedness, positivity, and Markov-Feller property of the solution for this system are proved. Then a nearly necessary and sufficient threshold-type criterion is provided, which shows the long-time behavior of the system can be classified by a real-valued parameter λ. Precisely, if λ 0, the system admits a unique invariant probability measure, and the transition probability of the solution process converges to this invariant measure. Finally, theoretical results are verified through numerical simulations, and trajectories of system behavior and the empirical distribution of the invariant probability measure are demonstrated.