时滞血糖—胰岛素系统的动力学与糖尿病机理

发布时间：2018-01-03 08:29

本文关键词：时滞血糖—胰岛素系统的动力学与糖尿病机理　出处：《郑州大学》2012年硕士论文　论文类型：学位论文

【摘要】：糖尿病是一种发病率高、并发症多、治疗费用昂贵的疾病,并且该疾病的发病机理尚不明确.为了能够为预防和治疗提供有效的方案,我们需要探索该疾病的发病机理.而临床实验等方法费用高、结果差,因此我们想通过建立血糖-胰岛素模型这种途径去研究其发病机理.目前存在的血糖-胰岛素模型,大部分都忽略了高血糖时肾代谢的作用,为了与实际更加接近,我们在模型中考虑了高血糖的实际影响.本文讨论了肝糖原产生时滞和胰岛素分泌时滞对血糖-胰岛素系统的动力学行为的影响以及糖尿病发病机理.中心流形约化和规范形方法已经广泛地应用到发生非共振双Hopf分岔的时滞系统中,但该方法比较抽象、难懂,因此我们尝试寻找简便的方法.本文首先将多尺度方法应用到研究具有两时滞通有系统的非共振双Hopf分岔点附近动力学行为,得到了规范形.然后引入了时滞血糖-胰岛素系统,得到了其解的存在性和有界性,由此说明模型的可行性；我们又研究了系统平衡点的存在唯一性、局部稳定条件和局部不稳定条件；然后我们又分别用中心流形约化-规范形方法和多尺度方法得到其规范形并对非共振双Hopf分岔点附近的动力学行为进行开折、分类,结果表明多尺度方法在研究具有两时滞系统的非共振双Hopf分岔的适用性,并且该方法简单、具体.最后,我们用WinPP进行了数值模拟,验证了上述理论分析.本文可得两个重要结论：当糖分摄入较多时,会造成平衡点坐标的升高,而平衡点对应于实际血糖的稳态,稳态过高可能导致糖尿病及一些并发症,如失明等,所以糖尿病患者应当注意糖分的摄入；当肝糖原产生反应时滞足够大时,胰岛素分泌时滞即使在较小的范围内波动都会引起显著的血糖波动,而血糖较大波动是罹患糖尿病的一个重要特征,这可能也是糖尿病的发病原因之一.
[Abstract]:Diabetes is a disease with high incidence, many complications, expensive treatment, and the pathogenesis of the disease is not clear. In order to provide an effective program for prevention and treatment. We need to explore the pathogenesis of the disease, and clinical trials and other methods are costly, the results are poor. Therefore, we want to study the pathogenesis of hyperglycemic insulin by establishing a blood glucose insulin model. Most of the existing blood glucose insulin models ignore the role of renal metabolism in hyperglycemia. To get closer to reality. In this paper, we discuss the effects of time delay of hepatic glycogen production and insulin secretion on the dynamic behavior of the blood-insulin system and the pathogenesis of diabetes mellitus. Reduction and normal form methods have been widely used in time-delay systems with non-resonant double Hopf bifurcation. But this method is abstract and difficult to understand, so we try to find a simple method. In this paper, we first apply the multi-scale method to study the dynamic behavior near the non-resonant double Hopf bifurcation point of the system with two delays. The canonical form is obtained. Then the existence and boundedness of the solution are obtained by introducing the time-delay glycemic insulin system, which shows the feasibility of the model. We also study the existence and uniqueness of the equilibrium point, the local stability condition and the local instability condition of the system. Then we use the center manifold reduced-normal method and the multi-scale method to obtain the normal form and break the dynamic behavior near the non-resonant double Hopf bifurcation point. The results show that the multi-scale method is applicable to the study of the non-resonant double Hopf bifurcation of systems with two delays, and the method is simple and specific. Finally, we use WinPP to simulate. Verify the above theoretical analysis. This paper can draw two important conclusions: when the sugar intake is more, the equilibrium point will increase, and the equilibrium point corresponds to the steady state of the actual blood sugar. High homeostasis may lead to diabetes and some complications, such as blindness, so diabetic patients should pay attention to sugar intake; When the time lag of liver glycogen reaction is large enough, insulin secretion delay will cause significant fluctuation of blood glucose even in a small range, which is an important characteristic of diabetes mellitus. This may also be one of the causes of diabetes.
【学位授予单位】：郑州大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：R311;R587.1

【共引文献】