Abstract
It is important to get good dietary advice such as a daily value of exogenous glucose intake to keep blood glucose within reasonable range for everyone, especially for those with diabetes mellitus or with borderline diabetes mellitus. In this paper, two novel glucose-insulin models with impulsive glucose intake are considered to investigate the significance of diet to the glucose-insulin system. For the first model, we prove the positiveness, boundedness, and permanence of the solutions and then the existence and stability of the periodic solutions are demonstrated. For the second model, we consider the insulin secretion delay based on the first model. The sufficient delay-dependent conditions which determine the stability of the periodic solution are also obtained. In addition, the theoretical results are verified by numerical simulation. The results can give people some dietary advice how to prevent diabetes, and also, the intake amount and frequency of the exogenous glucose from diet are recommended for diabetic patients to assist in the treatment of diabetes.
Highlights
During recent decades, diabetes mellitus has become an epidemic disease in the sense of life style [1]
Many mathematical models have been developed to better understand the mechanisms of the glucose-insulin regulatory system ([3,4,5,6,7,8,9,10,11,12]). e most well-known clinical model is the Minimal Model, which was proposed by Bergman et al in 1979 [6] and 1980 [7]. e Minimal Model is widely used in the intravenous glucose tolerance test (IVGTT)
Two glucose-insulin models with impulsive intake of glucose are proposed and studied. e impulsive models that we presented can intuitively show the intake amount and intake period of exogenous glucose, which is convenient to implement numerical simulation and discuss
Summary
Diabetes mellitus has become an epidemic disease in the sense of life style [1]. In 2017, Shi et al [5] proposed a novel approach to model the insulin secretion time delay, which uses two parameters to simulate both discrete time delay and distributed time delay. Ere are many mathematical models modeling the insulin pump control of glucose level by impulsive insulin injections ([13,14,15,16,17]), and a number of impulsive differential equation models have been developed for the artificial pancreas ([18, 19]). Erefore, in this paper, we propose novel impulsive differential equation model (1) to simulate the glucose intake from the daily diet with periodic impulses.
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