Transition and basin stability in a stochastic tumor growth model with immunization

Abstract

The phenomenon of noise-induced transition driven by the correlated Gaussian and non-Gaussian colored noises is investigated by the first escape probability (FEP) and the mean first exit time (MFET). To derive the Markovian approximation of the original tumor growth model and obtain the analytical expressions of the FEP and MFET, we reduce the non-Gaussian colored noise and then expand the unified colored noise approximation (UCNA). Additionally, the stochastic basin of attraction (SBA), a recent geometric concept based on the FEP and MFET, is introduced to provide further insight into the effects of noisy fluctuations on the basin stability of a certain domain. A higher FEP or shorter MFET in the high tumor population region B facilitates the transition from B to the low tumor population region Bc, which indicates the weaker stability of domain B. Our main results demonstrate that (i) the transitions from B to Bc can be induced by both the Gaussian and non-Gaussian noise sources; (ii) the stronger noise intensity, especially the non-Gaussian noise intensity, with a larger deviation parameter and immune coefficient improves the FEP, shortens the MFET, and hence benefits the transitions. However, the enlargement of the correlation between noises strengthens the basin stability of domain B and impedes the transitions; (iii) the size of SBA expands due to the larger cross-correlated intensity. In contrast, the enhancements of noise intensities with a larger departure degree reduce the size of SBA, which weakens the basin stability and is less in favor of tumor treatment. Furthermore, the Monte Carlo simulations of the original system are employed to verify the feasibility and accuracy of the analytical predictions.