Mathematical Models And Qualitative Analysis Of Impulsive Injections Of Insulin In The Therapy

In recent years, the researches on mathematical biology which base on the biological dy-namical system have been developing rapidly. The researches on continuous biological dynamical systems are gradually completed and the discussions on impulsive dynamical systems also obtain huge achievements. For the need to describe more actual natural word, mathematical biological models have always evolved. The discussions of continuous biological dynamical systems the main research direction in the past decades, and these studies really provide theoretical guid-ance for practice in many areas. However, people also find that continuous biological dynamical systems can not represent some natural phenomena and control behavior of human accurately, for example, the seasonal migration of the animals, restocking and harvesting in aqnaculture immunizations in disease prevention and pest control in agriculture, and impulsive dynamical systems can represent well because the relatively instantaneous behavior mentioned above can be described well in impulsive differential equations. In this paper, based on impulsive differen-tial equation's theorem, we introduce two novel mathematical models with impulsive injection of insulin or its analogues for type1and type2diabetes mellitus, and numerical simulations are used to investigate dynamical behaviors including the existence and stability of periodic solutions, permanence and all kinds of complexities.In Chapter2. we firstly consider a model incorporates with periodic impulsive injection of insulin. We analytically showed the existence and uniqueness of a positive globally asymptot-ically stable periodic solution for type1diabetes, which implies that the perturbation due to insulin injection will not disturb the homeostasis of plasma glucose concentration. At the sam time, we also showed that the system is uniformly permanent for type2diabetes, that is. the glucose concentration level is uniformly bounded above and below. Besides, we also consider one other model has the feature that determines the insulin injection by closely monitoring the glucose level when the glucose level reaches or passes a predefined but adjustable threshold value LG. We analytically proved the existence and stability of the order one periodic solution. which ensures that the perturbation by the injection in such an automated way can make the blood glucose concentration under control. At last, basing on a mass of simulations, we giv some suggestions of injection strategies of insulin for diabetic treatment:under the open-loop environment, for the same daily total dose, the impulsive injection with smaller dose but. shorter period has more efficient effect on controlling plasma glucose level than the injection with larger dose but longer period; for artificial pancreas, quite the opposite, impulsive injection with larger dose but longer period has more efficient effect.