The threshold of stochastic tumor-immune model with regime switching

Abstract

In response to the pressing needs for comprehending the cancer biology, this paper focuses on dynamical behaviors of a class of stochastic tumor-immune models in random environment modulated by Markov chains. A sufficient and nearly necessary threshold-type criterion is investigated, which shows the long-time behavior of the system can be classified by a real-value parameter λ. Precisely, if λ<0, tumor cells die out. If λ>0, the system exists a unique invariant probability measure, and the transition probability of the solution process converges to this invariant measure. Moreover, we also estimate the expectations with respect to the invariant measure under some conditions. Two numerical examples are provided to illustrate our results.