Global stability of a delay model of glucose–insulin interaction

Recently, there were several differential delay models of the glucose–insulin system analyzed in the literature. One of the mostly used models is generally known as the m i n i m a l m o d e l which was first published in 1979 and modified in 1986. This minimal model has been challenged by De Gaetano and Arino in 2000 from both physiological and modeling aspects. Instead, they proposed a new and mathematically more reasonable model, called d y n a m i c m o d e l . Following dynamic model, Li, Kuang and Li proposed a more popular, general and realistic model. Panunzi, Palumbo and De Gaetano believed that it requires a reasonably simple model, with as few parameters to be estimated as practicable, and with a qualitative behavior consistent with physiology, and they proposed a discrete single delay model (SDM). These models were proved to have the reasonable qualitative behavior, such as uniform persistence and global stability of equilibrium. In this paper we investigate a more general model, which includes SDM and one of the models in Li et al. (2001) [7] as its special cases. The model is shown to admit globally stable equilibrium under certain conditions of the parameters. For the two special models (SDM and one in [7]), it is shown that these conditions are slightly better. It is also shown that the model admits oscillating behavior due to the existence of Hopf-bifurcation.