Research On A Class Dynamic Model Of Insulin Delivery

In recent years, the study of biological dynamic systems have been developing rapidly, the research on continuous biological dynamic systems are gradually complete and the investigation of impulsive dynam-ical systems are making great progress also. Differential equation models evolved during its development, so that it can be reflected the objective reality more effectively, and can be used to solve real problems. Many scholars studied continuous differential equations systems. In nature and human life activities, however, some phenomenon cannot be described by continuous differential equations very accurately, while impulsive differential equations can be. In the past 30 years, some literatures about subcutaneous injection of insulin or its analogues have appeared, most of the models simulate the absorption of insulin analogues in the injection depot, and then calculate the plasma insulin concentration. These models have been used or can be used to insulin pump technology or future closed-loop system, namely the artificial pancreas.In this paper, we review some models selectively, and then sum them up, we also point out these models play a key role on insulin absorption and effect. Then we give continuous subcutaneous insulin injection model, and obtain that the positivity and boundedness of solutions of the model by qualitative analysis, we also get that there exists a unique equilibrium which is globally asymptotically stable. Next, we study the periodic impulsive injection model, using Floquet multipliers theory of liner periodic impulsive differential and the comparison theorem, we can obtain the positivity and persistence of the system, that is, we can adjust sugar concentration in plasma to the ideal level range by controlling the insulin injected dosage and cycle. At the same time, we also obtain that the system exists a unique global asymptotically stable periodic solution, that is, the small charges of insulin injection dosage will not affect the stability of concentration of sugar in blood.