多因素HIV模型的研究

发布时间：2018-05-31 21:13

本文选题：双线性 + 标准型　；参考：《北方工业大学》2011年硕士论文

【摘要】：我国1985年首次发现艾滋病后,艾滋病以传播迅速、无法治疗而震惊全国.当前,人类正面临着艾滋病长期而严峻的威胁.因此,建立能反映艾滋病特性的数学模型.通过对模型的分析,揭示其流行规律,预测其发展趋势,分析疾病流行的原因,寻求预防和控制的策略是非常必要的. 本文建立了一类多因素HIV传染病动力学模型,建立了三种模型：双线性模型；标准型模型；SP模型.并进行了理论性研究.在建立数学模型时,主要参考了马知恩,James M. Hyman等人的文章.在证明过程中,主要利用了M矩阵理论,LaSalle不变原理,Liapunov函数进行证明,得到了无病平衡点的稳定性定理.所得结果推广、丰富了已有文献中的相关结论. 首先,介绍了研究传染病的重要意义、HIV/AIDS的研究现状、预备知识、本文的研究内容与结构安排. 其次,建立了双线性HIV模型.求出了基本再生数,证明了无病平衡点的局部渐近稳定性与全局渐近稳定性. 再次,建立了标准型HIV模型.求出了基本再生数,证明了无病平衡点的局部渐近稳定性与全局渐近稳定性；给出了一些持续生存的概念和结论,证明了系统的一致持续生存性. 最后,建立了SP模型.求出了基本再生数R0,证明了当R01时,无病平衡点是局部渐近稳定的和全局渐近稳定的；当R01时,无病平衡点是不稳定的.
[Abstract]:After our country first discovered AIDS in 1985, AIDS shocked the whole country because of its rapid spread and untreatable. At present, human beings are facing the long-term and severe threat of AIDS. Therefore, the establishment of mathematical models can reflect the characteristics of AIDS. Through the analysis of the model, it is very necessary to reveal its epidemic law, forecast its developing trend, analyze the cause of disease epidemic, and seek the strategy of prevention and control. In this paper, a class of multi-factor HIV epidemic dynamics model is established, and three models are established: bilinear model, standard model and SP model. And carried on the theoretical research. In establishing the mathematical model, we mainly refer to the paper by James M. Hyman et al. In the process of proving, we mainly use M-matrix theory and LaSalle invariant principle to prove the stability theorem of disease-free equilibrium. The generalization of the results enriches the relevant conclusions in the previous literature. Firstly, this paper introduces the significance of HIV / AIDS research, the preparatory knowledge, the content and structure of this study. Secondly, a bilinear HIV model is established. The basic reproducing numbers are obtained and the local asymptotic stability and global asymptotic stability of disease-free equilibrium are proved. Thirdly, the standard HIV model is established. The local asymptotic stability and global asymptotic stability of disease-free equilibrium are proved, and some concepts and conclusions of persistent existence are given, and the uniform persistence of the system is proved. Finally, the SP model is established. The basic reproducing number R _ 0 is obtained and it is proved that the disease-free equilibrium is locally asymptotically stable and globally asymptotically stable when R _ 01, and that the disease-free equilibrium is unstable at R _ 01.
【学位授予单位】：北方工业大学
【学位级别】：硕士
【学位授予年份】：2011
【分类号】：R311;O242.1

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