霍乱与媒体效应的传染病模型及研究

发布时间：2018-09-01 06:09

【摘要】：传染病动力学是对传染病进行理论性定量研究的一种重要方法.它是根据种群生长的特性,建立模型来分析疾病的发生及在种群内的传播、流行规律,并通过分析和数值模拟,来揭示疾病的流行规律,预测其变化发展趋势,寻找对其预防和控制的策略.本文用常微分方程组描述了两类传染病动力学模型,同时讨论了所提出模型的一些动力学行为,其中包括解的正不变区域,平衡点的存在性和稳定性,系统的持续性与灭绝性等动力学行为,并讨论其生物学意义.主要有以下两个方面的内容: 第二章研究具有接种效应的霍乱模型.提出了一类自治的五维SVIR-B模型,给出了决定霍乱消亡与否的控制再生数Rv,当Rv1时,利用Routh-Hurwitz条件证明了无病平衡点是局部渐近稳定的,又得到了无病平衡点是全局渐近稳定的,此时霍乱会消亡;当Rv1时,存在唯一一个地方病平衡点,利用Routh-Hurwitz条件证明了地方病平衡点是局部渐近稳定的,接着证明了系统的持续性,并使用复合矩阵理论讨论了地方病平衡点的全局渐近稳定性,得到了地方病平衡点是全局渐近稳定的充分条件,霍乱将持续存在.最后,对Rv的参数灵敏性分析并进行数值模拟,提出需要同时增加接种率和减小免疫失去率达到某个临界值后,霍乱才得以控制. 第三章研究了具有媒体效应的传染病模型.建立了一类具有一般性接触率SIRS模型,得到了基本再生数的表达式R0,当R01时,利用Routh-Hurwitz条件证明了无病平衡点是局部渐近稳定的,此时，传染病会消亡;当R01时,利用函数的零点定理证明了地方病平衡点的唯一存在性,使用Routh-Hurwitz条件证明了地方病平衡点是局部渐近稳定的,并使用Bendixson判据讨论了地方病平衡点的全局渐近稳定性,得到了地方病平衡点全局渐近稳定的充分条件,此时，传染病将一直存在.最后,进行了数值模拟,显示了媒体报道对传染病传播和控制的影响.
[Abstract]:Infectious disease dynamics is an important method for theoretical and quantitative study of infectious diseases. According to the characteristics of population growth, it establishes a model to analyze the occurrence, spread and prevalence of disease within the population, and by means of analysis and numerical simulation, it reveals the epidemic law of disease and predicts its changing trend. Look for strategies to prevent and control them. In this paper, two kinds of infectious disease dynamics models are described by ordinary differential equations, and some dynamic behaviors of the proposed model are discussed, including the positive invariant region of solution, the existence and stability of equilibrium point. The dynamics of the system such as persistence and extinction are discussed and its biological significance is discussed. The main contents are as follows: chapter 2 studies the cholera model with inoculation effect. In this paper, an autonomous five-dimensional SVIR-B model is proposed, and the control reproducing number Rv, is given to determine the extinction of cholera. When Rv1 is used, it is proved that the disease-free equilibrium is locally asymptotically stable and that the disease-free equilibrium is globally asymptotically stable. At this point, cholera will die out; when Rv1, there is only one endemic equilibrium, and the Routh-Hurwitz condition is used to prove that the endemic equilibrium is locally asymptotically stable, and then the persistence of the system is proved. The global asymptotic stability of endemic equilibrium is discussed by using the compound matrix theory. A sufficient condition is obtained that the endemic equilibrium is globally asymptotically stable, and cholera will continue to exist. Finally, the sensitivity of the parameters of Rv is analyzed and simulated. It is suggested that cholera can only be controlled if the immunization coverage rate and the immune loss rate reach a certain critical value at the same time. Chapter three studies the infectious disease model with media effect. In this paper, a class of SIRS model with general contact rate is established, and the expression R0 is obtained. When R01, we prove that the disease-free equilibrium is locally asymptotically stable by using the Routh-Hurwitz condition, and the infectious disease will die out when R01. The unique existence of endemic equilibrium is proved by using the zero point theorem of function, the local asymptotic stability of endemic equilibrium is proved by using Routh-Hurwitz condition, and the global asymptotic stability of endemic equilibrium is discussed by using Bendixson criterion. A sufficient condition for global asymptotic stability of endemic equilibrium is obtained. Finally, numerical simulation is carried out to show the influence of media reports on the transmission and control of infectious diseases.
【学位授予单位】：北京建筑大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：R181

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