Dynamical analysis of a kind of two-stage tumor-immune model under Gaussian white noises

In this paper, a two-stage tumor-immune system with Gaussian white noises is investigated in order to understand the interaction mechanism between tumor cells and the immune system. Based on Langevin theory and the Fokker–Planck (FP) equation, a set of differential equations that governing the second-order moments of cells are obtained and solved. Following this, the statistical properties including the volatility of cell density, Fano factor, sensitivity analysis are discussed, respectively. It is concluded that mature immune cells around the tumor-free state change in a bigger range with regard to the noise intensity. However, the tumor cells around the tumor-present state have remarkable volatility all the time no matter what size of the noise intensity. In addition, the relative size of fluctuation described by Fano factors illustrates that Michaelis saturation coefficient and killing rate can bring larger fluctuation to the tumor cells, but dead rate and proliferation rate can greatly affect the fluctuation size of the mature immune cells. The sensitivity coefficients of three kinds of cells proved that the density of tumor cells are sensitively influenced by all the parametric values, but the mature immune cells are almost independent of the differentiation rate during the process from the immature to the mature immune cells.