Dynamics of a time-delay differential model for tumour-immune interactions with random noise

This paper examines the dynamics of a time-delay differential model of the tumour immune system with random noise. The model describes the interactions between healthy tissue cells, tumour cells, and activated immune system cells. We discuss stability and Hopf bifurcation of the deterministic system. We then explore stochastic stability, and the dynamics of the system in view of environmental fluctuations. Criteria for persistence and sustainability are discussed. Using multiple Lyapunov functions, some sufficient criteria for tumour cell persistence and extinction are obtained. Under certain circumstances, stochastic noise can suppress tumour cell growth completely. In contrast to the deterministic model which shows no stable tumour-free state, the white noise can either lead to tumour dormancy or tumour elimination. Some numerical simulations, by using Milstein's scheme, are carried out to show the effectiveness of the obtained results.