Communications in Information and Systems

Volume 22 (2022)

Number 2

Gaussian and non-Gaussian colored noise induced escape in a tumor-immune model

Pages: 169 – 186



Xi Chen (School of Statistics, Xi’an University of Finance and Economics, Xi’an, China)

Yanmei Kang (School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China)


We investigate the mean first passage time of a tumor-immune model with Gaussian colored noise by the two analytic approximation methods of singular perturbation analysis and small correlation time approximation. For the first time, it is shown that the singular perturbation analysis is accurate in the sense of retaining linear term of the small correlation time parameter, while the small correlation time approximation keeps all the even higher-order terms of the same small parameter, but it neglects the linear leading order term. This contrast suggests that the singular perturbation method has a better accuracy than the small correlation approximation method when the correlation time parameter is small. As a further application of the singular perturbation method, the mean first passage time in the case of non-Gaussian noise is also deduced and discussed. It is shown that as the strength of immunization or the non-Gaussian deviation parameter increases, the mean first passage time decreases, and thus both enhancing immunization and applying heavy-tailed random perturbation can accelerate the extinction of tumor cells.


tumor-immune model, mean first passage time, singular perturbation analysis, small correlation approximation, non-Gaussian noise

2010 Mathematics Subject Classification

Primary 37H10, 60H10. Secondary 62P10.

The work was financially supported by Chinese National Natural Science Foundation with Grant Nos. 11372233, 11972270, 12101473, and by the Basic Research Program of Natural Science in Shaanxi Province with Grant Nos. 2021JQ-764 and 2021JQ-766.

Received 31 October 2021

Published 19 May 2022