Analytical and Numerical Methods to Study the MFPT and SR of a Stochastic Tumor-Immune Model

Abstract

The Mean First-Passage Time (MFPT) and Stochastic Resonance (SR) of a stochastic tumor-immune model with noise perturbation are discussed in this paper. Firstly, considering environmental perturbation, Gaussian white noise and Gaussian colored noise are introduced into a tumor growth model under immune surveillance. As follows, the long-time evolution of the tumor characterized by the Stationary Probability Density (SPD) and MFPT is obtained in theory on the basis of the Approximated Fokker-Planck Equation (AFPE). Herein the recurrence of the tumor from the extinction state to the tumor-present state is more concerned in this paper. A more efficient algorithm of Back-Propagation Neural Network (BPNN) is utilized in order to testify the correction of the theoretical SPD and MFPT. With the existence of a weak signal, the functional relationship between Signal-to-Noise Ratio (SNR), noise intensities and correlation time is also studied. Numerical results show that both multiplicative Gaussian colored noise and additive Gaussian white noise can promote the extinction of the tumors, and the multiplicative Gaussian colored noise can lead to the resonance-like peak on MFPT curves, while the increasing intensity of the additive Gaussian white noise results in the minimum of MFPT. In addition, the correlation times are negatively correlated with MFPT. As for the SNR, we find the intensities of both the Gaussian white noise and the Gaussian colored noise, as well as their correlation intensity can induce SR. Especially, SNR is monotonously increased in the case of Gaussian white noise with the change of the correlation time. At last, the optimal parameters in BPNN structure are analyzed for MFPT from three aspects: the penalty factors, the number of neural network layers and the number of nodes in each layer.