Analysis Of Some Types Glucose-insulin Interaction Models

Diabetes mellitus is a chronic,lifelong non-communicable disease,which is an important public health problem concerned by many countries.Diabetes is a group of metabolic diseases characterized by high blood sugar.Insulin can accelerate the exploitation of glucose and inhibit the production of glucose,that is,to increase the way of blood sugar and reduce the source,so that blood sugar is reduced.The glucose-insulin regulation system can maintain the stability of blood glucose in the body,but once this regulation system is damaged,it may cause hyperglycemia or hypoglycemia.Therefore,in order to prevent and treat diabetes,we must deeply study the mechanism of the regulation system in human body firstly.In this dissertation,impulsive differential equation,time delay and stochastic differential equation are used to describe and investigate several types of glucose-insulin models.The main work of the dissertation is as follows.In the first chapter,the background knowledge of diabetes is introduced,including the growing trend of diabetes patients,the harm of diabetes to human body,and the main types and causes of diabetes.Next,the mechanism of glucose-insulin endocrine regulation system and the corresponding treatment methods are presented.Then,the research status of glucose-insulin model is showed.At last,some preliminary knowledge needed in this dissertation is given.In the second chapter,two glucose-insulin models are introduced and investigated respectively.Firstly,a glucose-insulin model with periodic pulse injection of glucose is established,and the positiveness,boundedness,and persistence of the system are analyzed.And the existence and global asymptotic stability of positive periodic solutions are proved by Krasnoselskii fixed point theorem and Lyapunov function.Then,on the basis of the first model a new model is constructed with time delay,and Lyapunov function is used to obtain the conditions for the global asymptotic stability of the periodic solution.Finally,the theoretical results are verified by numerical simulation,and some practical suggestions are given.In the third chapter,the glucose-insulin model which is interfered by white noise is introduced and invested.The deterministic model corresponding to the stochastic model has boundary equilibrium point and positive equilibrium point,but after adding stochastic terms,the boundary point and the positive equilibrium point are vanished.First of all,the existence of a unique global positive solution of the system is proved.Then,the asymptotic behavior of the global positive solution around the positive equilibrium point of the corresponding deterministic system is discussed.By applying It?o theorem,the solution of the random system is perturbed around the positive equilibrium point of the deterministic system and the vibration amplitude is related to the interference intensity of white noise under certain conditions.Finally,the theoretical results are validated by numerical simulation,and it is pointed out that the stochastic model can fit the IVGTT data better than the previous deterministic models,and further improve the diagnostic accuracy of diabetes mellitus by controlling the intensity of noise interference.In the last chapter,the whole dissertation are summarized and the problems that need to be studied further are pointed out.