【作者】 徐芳；

【导师】 栗永安；

【作者基本信息】 兰州交通大学 ， 运筹学与控制论， 2009， 硕士

【摘要】 研究传染病的传播和预测传染病的发展趋势,是研究传染病的一个重要方面,它是政府部门和卫生医疗机构制定相应措施的基础。本文仍然采用建立数学模型的方法来研究传染病的传播机理。全文共分五章:第一章介绍了传染病的研究概况和一些关于研究流行病模型的基本方法;第二章介绍了一些理论知识;第三章建立并分析了一类非线性高维自治微分系统SEIQR流行病传播数学模型。该模型采用的非线性发生率包括了传统的双线性发生率。通过对模型的全局性态的分析,我们对疾病的传播规律有一个透彻的了解,疾病的灭绝与否由基本再生数R₀决定。这在理论上表明,增大隔离强度和提高疾病的治愈率可以控制疾病的蔓延;第四章主要研究了具有常数输入和非线性发生率的SEIS传染病模型,利用LaSalle不变集原理、Liapunov函数、Hurwith判据证明了无病平衡点的全局渐近稳定性。同时利用第二加性复合矩阵原理证明了惟一地方病平衡点的全局渐近稳定性。本章结果表明,给予病人及时的治疗,提高治疗的有效性是控制疾病的有效方法;第五章主要研究了一类潜伏期和染病期均具有传染力的SEIS传染病模型。该模型的全局性态的分析显的更加复杂,通过理论分析和数值模拟我们发现给予病人及时有效的治疗和减少染病系数是控制疾病蔓延的有效方法。更多还原

【Abstract】 Exploring epidemic spreading and predicting its development trend are important aspects for epidemic study,and they are the basis of control policy adopted by the government and medical department.The aim of this work is to construct several mathematical models in epidemiology and to analyze the asymptotic behavior of these models.The full text is divided into five chapters:The first chapter introduced the study of infectious diseases and some methods about epidemiological models.The second chapter introduced some theoretical knowledge.The third chapter not only establish but also analyze a nonlinear high dimensional autonomous differential system SEIQR model in epidemiology.That the nonlinear incidence rate is used in the model,has included the traditional bilinear incidence rate.By analyzing global behavior of the model,we get a incisive comprehension about the disease propagation law and the disease extinction is determined by the basic reproduction number R₀.It showed that incresing isolation and improving the cure rate of the disease can control the spread of diaease;An SEIS epidemic model with constant input and nonlinear incidence rate is maily studied in the fourth chapter.The disease-free equilibrium is globally asymptotical stable by the means of Lasalle’s invariant set theorem,Liapunov function and Hurwitze criterion.Using second additive compound matrix,it is proved that the unique endemic equilibrium is globally asymptotical stable.The results in this chapter suggest that giving the patients timely treatment and improving the cure rate are an effective way to control the disease.A kind of SEIS model infections in both latent and infected period is studied in the fifth chapter. The global analysis of this model becomes more complex.Through mathematical studied and simulation we find that giving the patients timely treatment,improving the cure efficiency and decreasing the infective coefficient are effective methods for the control of the disease.更多还原