Most probable trajectory of a tumor model with immune response subjected to asymmetric Lévy noise

Abstract

This paper is devoted to making realistic prediction for the trajectory transition of tumor cells induced by stochastic disturbances. To maintain better consistency with reality, an asymmetric Lévy noise is incorporated in a tumor–immune model. The corresponding Fokker–Planck equation is derived by the adjoint operator method. The transition behaviors of tumor cells are measured by maximizing the probability density function of system trajectories, namely, the most probable trajectory. The numerical algorithm for obtaining the most probable trajectory is offered. Based on the numerical calculations, the effect of Lévy noise on the evolution trajectories of tumor cells is discussed to explore the optimal parameters controlling or even eradicating tumor. It is found that by adjusting noise parameters, Lévy noise can induce the transition of tumor from the high concentration stable state toward the low concentration stable state or even to the tumor extinction state. When the skewness parameter is negative, Lévy noise with the stability index less than 1.0 and larger noise intensity is in favor of inhibiting tumor proliferation. While when the skewness parameter is positive, Lévy noise with the stability index larger than 1.0 and smaller noise intensity promotes the decrease of tumor cells.