Analysis of Th1/Th2 response pattern with Treg cell inhibition and stochastic effect

T cells differentiate into Th1 or Th2 cells upon maturation to influence different patterns of the immune response. Th1 and Th2 cells regulate each other and their responses are inhibited by Treg cells. With noisy external stimulation, Th1/Th2 cell differentiation can be dynamically balanced. The underlying mechanisms of Th cell differentiation under Treg cell inhibition and the extrinsic noise effects are not yet completely understood. In this paper, a mathematical model of the interactions between Th1 and Th2 cells with Treg cell inhibition and stochastic effects is developed to study the preference of outcomes and the noise-induced hopping among different states. First, we provide the conditions for different asymptotic phases of Th1 and Th2 responses under Treg cell regulation. Numerical simulations are applied to calculate the switching probability and the mean residence time to study how the noise affects the attractiveness of different states. Our results support that due to the stronger inhibitory effect of Treg cells on Th1 cell development, the high-Th2-low-Th1 state is more attractive under small noise effects. Additionally, we show that the attractiveness of the states is affected mainly by the extrinsic noise in Th2 cell signaling.