具有饱和治疗率的SIR传染病模型的后向分支
发布时间：2018-11-14 15:12
【摘要】： 传染病模型的渐近行为已经被很多人研究。通常情况下,基本再生数是决定疾病流行与否的阈值。如果它小于1,无病平衡点是全局稳定的且疾病灭绝;如果它大于1,正平衡点是全局稳定的且发展为地方病。在这种情况下,从无病平衡点到正平衡点引起的分支是向前的。近年来,由于社群具有不同的感染性、非线性发生率和年龄结构等原因,许多关于传染病模型的论文发现了后向分支。在这种情况下,基本再生数不能完全描述疾病消除的效应,而这种效应能被转向点的关键参数描述,得到控制疾病的阈值对于确认后向分支是重要的。基于此,本文中我们研究了具有饱和治疗函数的传染病模型,通过数学分析和数值模拟主要得到以下结论: 1.当感染者治疗延滞的效应弱时,基本再生数是控制疾病的强阈值。当感染者治疗延滞的效应强时,后向分支将发生,对于消除疾病来说基本再生数小于1是不足的。 2.当后向分支发生时,转向点关键值是控制疾病的新阈值。 3.当基本再生数减少到一定程度时,无病平衡点是全局稳定的。 4.数学结果表明给病人及时的治疗、提高治疗率,和减少传染的协同因素对控制疾病是有效的。 最后,我们结合前人的一些工作,提出了今后努力的方向。
[Abstract]:The asymptotic behavior of infectious disease models has been studied by many people. In general, the basic number of reproduction is the threshold to determine the prevalence of the disease. If it is less than 1, the disease-free equilibrium is globally stable and the disease is extinct; if it is greater than 1, the positive equilibrium is globally stable and develops into endemic disease. In this case, the branches from the disease-free equilibrium to the positive equilibrium are forward. In recent years, due to the different infectivity, nonlinear incidence and age structure of community, many papers on infectious disease models have found backward branches. In this case, the basic regeneration number can not completely describe the effect of disease elimination, but this effect can be described by the key parameters of the turning point. It is important to obtain the threshold of disease control for confirming the backward branch. Based on this, we study the infectious disease model with saturation treatment function, and get the following conclusions by mathematical analysis and numerical simulation: 1. When the effect of treatment delay is weak, the number of basic regeneration is a strong threshold for disease control. When the effect of treatment delay is strong, the backward branch will occur, and the number of basic regeneration less than 1 is insufficient to eliminate the disease. 2. When the backward branch occurs, the critical value of the turning point is a new threshold for disease control. 3. When the number of basic regenerations is reduced to a certain extent, the disease-free equilibrium is globally stable. 4. The mathematical results show that timely treatment, improved treatment rate, and reduction of infectious co-factors are effective in disease control. Finally, combined with some previous work, we put forward the direction of future efforts.
【学位授予单位】：兰州大学
【学位级别】：硕士
【学位授予年份】：2009
【分类号】：R181.3
本文编号：2331530
[Abstract]:The asymptotic behavior of infectious disease models has been studied by many people. In general, the basic number of reproduction is the threshold to determine the prevalence of the disease. If it is less than 1, the disease-free equilibrium is globally stable and the disease is extinct; if it is greater than 1, the positive equilibrium is globally stable and develops into endemic disease. In this case, the branches from the disease-free equilibrium to the positive equilibrium are forward. In recent years, due to the different infectivity, nonlinear incidence and age structure of community, many papers on infectious disease models have found backward branches. In this case, the basic regeneration number can not completely describe the effect of disease elimination, but this effect can be described by the key parameters of the turning point. It is important to obtain the threshold of disease control for confirming the backward branch. Based on this, we study the infectious disease model with saturation treatment function, and get the following conclusions by mathematical analysis and numerical simulation: 1. When the effect of treatment delay is weak, the number of basic regeneration is a strong threshold for disease control. When the effect of treatment delay is strong, the backward branch will occur, and the number of basic regeneration less than 1 is insufficient to eliminate the disease. 2. When the backward branch occurs, the critical value of the turning point is a new threshold for disease control. 3. When the number of basic regenerations is reduced to a certain extent, the disease-free equilibrium is globally stable. 4. The mathematical results show that timely treatment, improved treatment rate, and reduction of infectious co-factors are effective in disease control. Finally, combined with some previous work, we put forward the direction of future efforts.
【学位授予单位】：兰州大学
【学位级别】：硕士
【学位授予年份】：2009
【分类号】：R181.3
【参考文献】
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1 李建全;马知恩;周义仓;;GLOBAL ANALYSIS OF SIS EPIDEMIC MODEL WITH A SIMPLE VACCINATION AND MULTIPLE ENDEMIC EQUILIBRIA[J];Acta Mathematica Scientia;2006年01期
，本文编号：2331530
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