具有年龄结构的传染病模型研究
发布时间：2018-10-15 19:11
【摘要】： 本论文对具有年龄结构的传染病模型进行了研究。 建立传染病模型的主要目的是利用模型对影响疾病传播的生物学和社会学机理作清晰描述,然后通过模型的研究来揭示疾病流行规律,预测流行趋势,为发现、预防和控制疾病的流行提供根据和策略。 本文针对不同的传染病的传播特点,建立了三类具有年龄结构的传染病模型,分别是：具有年龄和隔离措施的SEIQR模型,具有年龄结构的手足口病模型和具有年龄结构和常数迁移率的SIR模型。 这三类模型都是偏微分方程组,本文使用传染病动力学中的有关方法和实变函数论中的有关理论对模型进行了分析,得到了模型无病平衡态和地方病平衡态的存在性和稳定性条件,并且证明了在基本再生数小于1时,无病平衡态是局部渐近稳定(或全局渐近稳定)的,当基本再生数大于1时,地方病平衡态在一定条件下局部渐近稳定(或模型在地方病平衡态处的线性化系统的特征方程无非负实根)。 最后,在上述理论结果的基础上,对这些偏微分方程组的解进行数值模拟,给出了模型的差分格式,然后对模型的差分格式进行了Matlab编程。最后,对模型中易感者和染病者的发展趋势进行了模拟。在进行计算机试验的过程中,通过改变各个参数的数值,寻找对传染病发展影响较大的因素,为以后的生活实践提供了很好的依据。 试验结果表明,传染病模型的两个平衡态是和基本再生数息息相关的。而基本再生数又依赖于模型中的参数,于是,我们可以通过对某些参数的控制,来控制传染病的传播。 本文的创新点体现在将传染病动力学中的有关方法成功地应用于具有年龄结构的传染病模型中,而且对模型的地方病平衡态进行了比较深入的分析。
[Abstract]:In this paper, the infectious disease model with age structure is studied. The main purpose of establishing infectious disease model is to clearly describe the biological and sociological mechanism of disease transmission by using the model, and then to reveal the epidemic law and predict the epidemic trend through the study of the model. The prevention and control of disease prevalence provides the basis and strategy. In this paper, according to the transmission characteristics of different infectious diseases, three kinds of infectious disease models with age structure are established, which are SEIQR model with age and isolation measures. Hand-foot-mouth disease model with age structure and SIR model with age structure and constant mobility. These three models are all partial differential equations. In this paper, the models are analyzed by using the relevant methods in infectious disease dynamics and the theory of real variable function theory. The existence and stability conditions of disease-free equilibrium state and endemic equilibrium state are obtained, and it is proved that the disease-free equilibrium state is locally asymptotically stable (or globally asymptotically stable) when the basic regeneration number is less than 1. When the basic regeneration number is greater than 1, the endemic equilibrium state is locally asymptotically stable under certain conditions (or the characteristic equations of the linearized system of the model at the endemic equilibrium state are all negative real roots). Finally, on the basis of the above theoretical results, the solutions of these partial differential equations are numerically simulated, the difference scheme of the model is given, and the difference scheme of the model is programmed by Matlab. Finally, the development trend of susceptible and infected people in the model was simulated. In the course of computer experiment, by changing the values of each parameter, the factors which have great influence on the development of infectious diseases are found, which provides a good basis for the later life practice. The experimental results show that the two equilibrium states of the infectious disease model are closely related to the basic regeneration number. The basic reproduction number depends on the parameters in the model, so we can control the spread of infectious diseases by controlling some parameters. The innovation of this paper lies in the successful application of the relevant methods in the dynamics of infectious diseases to the epidemic model with age structure and the analysis of the endemic equilibrium state of the model.
【学位授予单位】：北京林业大学
【学位级别】：硕士
【学位授予年份】：2010
【分类号】：R181.3
本文编号：2273551
[Abstract]:In this paper, the infectious disease model with age structure is studied. The main purpose of establishing infectious disease model is to clearly describe the biological and sociological mechanism of disease transmission by using the model, and then to reveal the epidemic law and predict the epidemic trend through the study of the model. The prevention and control of disease prevalence provides the basis and strategy. In this paper, according to the transmission characteristics of different infectious diseases, three kinds of infectious disease models with age structure are established, which are SEIQR model with age and isolation measures. Hand-foot-mouth disease model with age structure and SIR model with age structure and constant mobility. These three models are all partial differential equations. In this paper, the models are analyzed by using the relevant methods in infectious disease dynamics and the theory of real variable function theory. The existence and stability conditions of disease-free equilibrium state and endemic equilibrium state are obtained, and it is proved that the disease-free equilibrium state is locally asymptotically stable (or globally asymptotically stable) when the basic regeneration number is less than 1. When the basic regeneration number is greater than 1, the endemic equilibrium state is locally asymptotically stable under certain conditions (or the characteristic equations of the linearized system of the model at the endemic equilibrium state are all negative real roots). Finally, on the basis of the above theoretical results, the solutions of these partial differential equations are numerically simulated, the difference scheme of the model is given, and the difference scheme of the model is programmed by Matlab. Finally, the development trend of susceptible and infected people in the model was simulated. In the course of computer experiment, by changing the values of each parameter, the factors which have great influence on the development of infectious diseases are found, which provides a good basis for the later life practice. The experimental results show that the two equilibrium states of the infectious disease model are closely related to the basic regeneration number. The basic reproduction number depends on the parameters in the model, so we can control the spread of infectious diseases by controlling some parameters. The innovation of this paper lies in the successful application of the relevant methods in the dynamics of infectious diseases to the epidemic model with age structure and the analysis of the endemic equilibrium state of the model.
【学位授予单位】：北京林业大学
【学位级别】：硕士
【学位授予年份】：2010
【分类号】：R181.3
【引证文献】
相关硕士学位论文 前1条
1 彭华勤;多个斑块间传播的传染病模型的研究[D];广州大学;2012年
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