Modeling And Study Of Strategies Of Insulin Therapy Based On Control Effect Of Plasma Glucose

Diabetes is a chronic disease in which hyperglycemia or hypoglycemia can cause a variety of complications.Currently,there is no effective drug or means to cure this disease,but only drugs can be used to control the patient's blood glucose concentration in a certain range to reduce and delay the occurrence of complications,and the main means of control is the input of exogenous insulin and its analogues.Based on the mechanism of glycemic-insulin interaction,this paper considers how to administer insulin drugs more economically and effectively in the course of treatment.According to the different research objectives,this paper constructs two kinds of mathematical dynamics models: Firstly,Filippov equation is used to study a kind of switching system with glucose control threshold and discuss the influence of the reasonable setting of the control threshold on the effect of insulin treatment.Then the finite time control of plasma glucose with insulin impulse injection is studied by using the optimization theory of impulsive differential equation.This paper is divided into three parts:In the first chapter,this paper introduces the background of diabetes and the research status of glycemic-insulin interaction at home and abroad briefly,gives the basic knowledge of Filippov system and optimal impulsive control theory applied in this paper.In the second chapter,a nonsmooth Filippov glycemia-insulin model with threshold control strategy is constructed,and the strategy of insulin injections in a large time scale are studied.Firstly,for the two subsystems of Filippov system,the existence and stability of the equilibrium point are obtained through qualitative and stability analysis.Then the real equilibrium?virtual equilibrium and pseudoequilibrium of Filippov system are obtained by using the theory of non-smooth dynamical systems.The numerical simulations show that appropriate selection of control threshold and effective regulation of the dosage of insulin can improve the control level of plasma glucose and reduce the economic input of plasma glucose control.In the third chapter,we study the injection strategies of insulin for the therapy of diabetes in a limited time.According to whether considering the risk of hypo glycemia or not,two different control objectives which take into account both of the fluctuations of plasma glucose level and the amount of insulin injected are set.For each control objective,three different injection strategies are investigated.To solve technical problems in optimal impulsive control,a time rescaling method is applied and the gradient formulas of cost functions with respect to injection doses and injection timings are obtained.In addition,with the aid of numerical simulations,we get the optimal injection doses and injection timings for each injection strategy.Our study indicates that for the control objective without considering the risk of hypoglycemia,the optimal injection timing control is superior to the optimal injection dose control,while the mixed control achieves almost the same effect as the optimal injection timing control.While for the control objective considering the risk of hypoglycemia,the optimal injection timing control is superior to the optimal injection dose control in avoiding problems with hypoglycemia,and the mixed control makes the most in avoiding problems with hyperglycemia.In the fourth chapter,the main contents of the paper are summarized.