Study on Long-Time Evolution of a Stochastic Tumor-Immune Model with Simulation by Ann Algorithm

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 color noises, respectively. Then the corresponding Fokker-Planck equation (FPE) and the approximated Fokker-Planck equation (AFPE) are obtained to explore the long-time evolution of the tumor cells. The innovation of this paper is that artificial neural network (ANN) algorithm is introduced to solve the stationary probability density (SPD) based on FPE or AFPE with 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. As for the case of Gaussian color noises, the existence of the cross-correlation time between multiplicative and additive Gaussian color noises lead the tumor to be a large concentration with nearly probability 1. All the theoretical results are examined by ANN algorithm and they are all in good agreement. 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