The Mean First-Passage Time and Stochastic Resonance of a Stochastic Tumor-immune Model via BPNN Method

Abstract

The Mean First-Passage Time (MFPT) and Stochastic Resonance (SR) of a stochastic tumor-immune model with noises 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 the follows, the long-time evolution of the tumor characterized by the Stationary Probability Density (SPD) and MFPT are 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 higher efficient algorithm of Back-Propagation Neural Network (BPNN) is applied in order to testify the correction of the theoretic SPD and MFPT. With the existence of weak signal, the functional relationship between Signal-to-Noise Ratio (SNR), noise intensities and correlation time is also studied. These results show that both multiplicative Gaussian colored noise and additive Gaussian white noise can promote the extinction of tumors, and the multiplicative Gaussian colored noise will lead to the resonance-like peak on MFPT curves, while the increasing intensity of additive Gaussian white noise will cause the minimum of MFPT. In addition, the correlation times are negatively correlated with MFPT. As for the SNR, we find the intensities of Gaussian white noise and Gaussian colored noise, as well as their correlation intensity will induce SR. Especially, SNR is monotonously increased in the case of Gaussian white noise with the change of the correlation time, however, SNR displays the point of inflection in other cases. 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.