Study on stationary probability density of a stochastic tumor-immune model with simulation by ANN algorithm

Abstract

In this paper, the theoretic analysis by stochastic dynamics and numerical simulation by ANN algorithm for tumor-immune models driven by random noises are studied. Firstly, a mathematical model about the competition between tumor cells and immune system is established based on the Michaelis-Menten enzyme kinetics. Secondly, the microenvironmental fluctuations are modeled as Gaussian white noises and Gaussian coloured noises, respectively. Then the corresponding Fokker-Planck equation (FPE) and the approximated Fokker-Planck equation (AFPE) are obtained to explore the stationary probability density (SPD) of the tumor cells. The innovation of this paper is that artificial neural network (ANN) algorithm is introduced to solve the SPD based on FPE or AFPE, which has higher robustness and accuracy. The SPDs of tumor cells show that the greater the intensity of Gaussian white noises, the more beneficial the prevention of tumor growth. In other words, the microenvironmental fluctuations could accelerate the extinction of tumor to a certain extent. For the case of Gaussian coloured noises, the existence of the cross-correlation time between multiplicative and additive Gaussian coloured noises lead the tumor to be large concentrations with probability of almost 1. All the theoretical results are examined by ANN algorithm and they are all in good agreement. In addition, we discuss the mean first passage time (MFPT) from the metastable state to the state of extinction of tumor cells, and discover the phenomena of the noise-enhanced stability (NES) as well as stochastic resonant activation (SRA). At last, the best parameters including penalty factors, the number of layers and nodes in ANN algorithm are also discussed in order to get the optimal accuracy.