Most probable dynamics of the tumor growth model with immune surveillance under cross-correlated noises

The noise is inherent and indispensable in the tumor growth system. Therefore, the behavior of the tumor system will display randomness. Different from the traditional analysis method (such as the mean value, variance etc.), the literature presents another deterministic tool, i.e. the most probable trajectories, which are defined by computing the spatial maximizers of the probability density function. Here we will investigate the tumor cell growth system with immune surveillance under correlated white noises from a deterministic point of view. Then the most probable extinction time is defined by the time when the most probable trajectories first escape to the extinction state from the tumor state. Afterward, the probability ratio of extinction state versus tumor state characterizes treatment effects. From the numerical simulation, we derive that for the increasing cross-correlation intensity of noises, the most probable extinction time is enhanced, and the therapeutic effect is weaken and conversely for the intensity of multiplicative noise. In contrast, there exists a critical intensity of additive noise at which the most probable extinction time is the smallest. Meanwhile treatment effects would be improved with the shrinking intensity of additive noise.