具有双线性发病率病毒变异的SEIR模型系统的最优控制问题

发布时间：2018-05-04 02:26

本文选题：传染病模型 + 全局稳定性　；参考：《哈尔滨师范大学》2015年硕士论文

【摘要】：在当今时代,传染病的爆发将会给人类带来毁灭性的灾难。由于其具有变异性的特点,出现了许多新型的病毒。长期以来人们与各种疾病做着不屈不挠的斗争。通过建立适当的数学模型来研究传染病的传播过程和预测传染病发展的最终趋势,已成为数学知识应用的一个重要领域。随着传染病动力学的迅速发展,研究传染病模型成为当下研究的主要趋势并且取得了许多成果。目前研究传染病流行规律的方法主要有：描述性研究,分析性研究,实验性研究和理论性研究。除了一些典型的研究方法外,出现了一些分盆、混沛等系统动力学方法,为理论研究提供了新的参考。其中最为丰富的一类是用微分系统描述传染病模型.国内外学者借助极限系统理论和构造适当的Liapunov函数的方法,对一类具有双线性发生率的病毒变异SEIR模型进行讨论,当无染病者输入时,找到地方病平衡点存在的阈值,找到无病平衡点和地方病平衡点全局渐进稳定的充要条件。判别疾病能否长期存在的重要依据是稳定性。文中主要研究了具有病毒变异情况的传染病模型,建立了具有双线性发生率的病毒变异的SEIR传染病模型,给出无病平衡点及地方病平衡点的存在条件；借助Liapunov函数,证明无病平衡点和地方病平衡点的全局渐近稳定性；最后将病毒变异速率作为控制变量,用范数指标函数作为衡量控制变量的标准得出该模型最优控制元的存在性。
[Abstract]:In the present age, the outbreak of infectious diseases will bring devastating disasters to mankind. Because of its variability, many new viruses have emerged. For a long time people have been fighting indefatigably with various diseases. It has become an important field of mathematical knowledge application to establish appropriate mathematical models to study the spreading process of infectious diseases and to predict the final trend of infectious diseases. With the rapid development of infectious disease dynamics, the research of infectious disease model has become the main trend and has made many achievements. At present, the methods of studying epidemic law of infectious diseases are descriptive, analytical, experimental and theoretical. In addition to some typical research methods, there are some system dynamics methods, such as basin division, mixing and so on, which provide a new reference for theoretical research. The most abundant one is to describe infectious disease model by differential system. With the help of the limit system theory and the method of constructing appropriate Liapunov function, a class of virus mutation SEIR model with bilinear incidence is discussed. The threshold of the existence of endemic equilibrium point is found when there is no infection. A sufficient and necessary condition for global asymptotic stability of disease-free equilibrium and endemic equilibrium is found. Stability is an important basis for judging the long-term existence of disease. In this paper, the infectious disease model with virus variation is studied, the SEIR infectious disease model with bilinear incidence of virus variation is established, the conditions for the existence of disease-free equilibrium and endemic equilibrium are given, and the Liapunov function is used. The global asymptotic stability of disease-free equilibrium point and endemic equilibrium point is proved, and the existence of optimal control element of the model is obtained by taking virus mutation rate as control variable and norm index function as criterion to measure control variable.
【学位授予单位】：哈尔滨师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175;O232

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