具有时滞的SEIR的肺结核模型研究

发布时间：2018-01-16 20:06

本文关键词：具有时滞的SEIR的肺结核模型研究　出处：《西安科技大学》2012年硕士论文　论文类型：学位论文

【摘要】：本文考虑肺结核在潜伏期和染病期都具有传染性，建立了几类具有不同传染率的SEIR肺结核传播时滞微分方程模型.综合运用微分方程理论等方法对模型的无病平衡点的全局渐近稳定性、地方平衡点的局部稳定性及疾病的持续性做了系统的研究.得到了决定疾病绝灭与否的基本再生数. 首先，建立了一类具有双线性传染率的SEIR肺结核模型.利用不动点理论、微分方程稳定性理论及LaSalle不变集原理研究了模型的平衡点的存在性、稳定性及系统的持续性.当时，无病平衡点全局渐近稳定，疾病将消除；当时无病平衡点不稳定，惟一的地方病平衡点全局稳定且系统是持续的. 再次，建立了一类具有型传染率的肺结核模型.通过线性化方法和Hurwitz判别法及LaSalle不变集原理研究了模型的无病平衡点的全局渐近稳定性及地方病平衡点的局部渐近稳定性，得到了依赖于时滞的基本再生数，，并讨论了模型在时的持续性. 最后，建立了一类具有垂直传染和预防接种的肺结核模型.讨论了地方病平衡点的存在惟一性、稳定性及模型的持续性.通过分析模型的基本再生数并提出了合理的预防和控制措施，最后通过数值模拟验证所得的理论结果.为肺结核传播的控制和预防提供理论依据和数量基础.
[Abstract]:In this paper, we consider that tuberculosis is infectious in both incubation period and infection stage. In this paper, several kinds of delay differential equation models of SEIR tuberculosis transmission with different infection rates are established, and the global asymptotic stability of the disease-free equilibrium point of the model is obtained by using the differential equation theory and other methods. The local stability of the local equilibrium point and the persistence of the disease are systematically studied, and the basic regeneration numbers which determine the extinction or extinction of the disease are obtained. Firstly, a class of SEIR pulmonary tuberculosis model with bilinear infection rate is established, and the fixed point theory is used. The stability theory of differential equations and the LaSalle invariant set principle are used to study the existence, stability and persistence of the equilibrium point of the model. At that time, the disease-free equilibrium point is globally asymptotically stable and the disease will be eliminated. At that time, the disease-free equilibrium was unstable, the only endemic equilibrium was globally stable and the system was persistent. Again. A class of pulmonary tuberculosis models with type infection rate is established. The global asymptotic stability and ground of disease-free equilibrium of the model are studied by means of linearization method, Hurwitz criterion and LaSalle invariant set principle. The local asymptotic stability of the equilibrium point of square disease. The basic reproducing numbers dependent on time delay are obtained, and the persistence of the model in time is discussed. Finally, a class of pulmonary tuberculosis models with vertical transmission and vaccination are established, and the existence and uniqueness of endemic equilibrium are discussed. Stability and sustainability of the model. By analyzing the basic regeneration number of the model, reasonable prevention and control measures are proposed. Finally, the theoretical results are verified by numerical simulation, which provide theoretical basis and quantitative basis for the control and prevention of tuberculosis transmission.
【学位授予单位】：西安科技大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：O175;R311

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