Abstract:

In this paper, we investigate the global dynamics of a glucose-insulin model and its corresponding stochastic differential equation version. For the deterministic model, we show that there exists a unique equilibrium point, which is globally asymptotically stable for all parameter values. For the stochastic model, we show that the system admits unique positive global solution starting from the positive initial value and derive the stochastic permanence of the solutions of the stochastic system. In addition, by using Hasminskiis methods, we prove that there exists a unique stationary distribution and it has ergodicity. Finally, numerical simulations are carried out to support our theoretical results. It is found that: (ⅰ) the difficulty of the prediction of the peak size of the plasma glucose concentration always increases with the increase of the intensity of environmental fluctuations; (ⅱ) environmental fluctuations can result in the irregular oscillating of the plasma glucose concentration and plasma insulin concentration. Moreover, the volatility of the plasma glucose concentration and plasma insulin concentration always increase with the increase of the intensity of environmental fluctuations.

Key words: Type 2 diabetes mellitus, Global stability, Environmental fluctuations, Stationary distribution and ergodicity