中国手足口病动力学模型与数据模拟
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摘要
手足口病(Hand,foot and mouth disease(HFMD))是由肠道病毒引起的传染病,多发生于5岁以下儿童,小于三岁的年龄组发病率最高,可引起手、足、口腔等部位的疱疹,少数患儿可引起心肌炎、肺水肿、无菌性脑膜脑炎等并发症。个别重症患儿病情发展快,导致死亡。引发手足口病的肠道病毒有20多种(型),其中以柯萨奇病毒A16型(Cox A16)和肠道病毒71型(EV71)最为常见。而中国每年感染手足口病的患者累计超过100万人。事实上,对于手足口病而言,目前还没有有效的疫苗,也没有特效的抗病毒治疗方案,因此我们如果能提出切实可行的预防控制措施将会是很有意义的。
本文分为四章,在第一章里介绍了传染病动力学模型的背景,手足口病病原学、流行病学、诊断鉴别及研究现状及预备知识。
在第二章中,研究了一个具有非单调发生率的手足口病模型并提出一些控制方案。通过对无病平衡点的全局渐近稳定分析,地方病平衡点的存在性与全局分析,我们得到了疾病流行的充分条件,同时,对后向分支、双稳态进行了探讨。数值模拟证实了理论的正确性,最后给出了手足口病的预防方针。我们认为双稳态的产生是手足口病仍然在中国大范围流行的一个重要的原因。
在第三章中,我们建立了具有非单调接触率的时滞SEIQRS传染病模型,首先研究了无病平衡点、地方病平衡点的存在性,分析了该模型的渐近行为,使用Lyapunov泛函方法得到了无病平衡点及地方病平衡点稳定的充分条件。
在第四章里,我们关注中国2008年到2012年间手足口病患病数据的研究。先后建立了两个动力学模型,借助数据,运用拟合优度的方法,分别得出了两个模型的最优参数。通过统计推断的卡方检验,证实了模型的参数值都是合理的。我们得到了每一年的基本再生数都大于1。因此,断定在当前条件下,手足口病仍会流行。此外,我们给出了详细的预防控制措施,如果运用文中的预防措施,手足口病将会得到迅速的控制,患者的数量会得到急剧的减少。文章的最后,我们运用2011年和2012年模拟出的参数对2013年患病人数做了预测,预测结果表明,与手足口病实际流行情形是吻合的。
Hand, foot and mouth disease (HFMD), an infectious disease caused by en-terovirus and Coxsackievirus, usually happens to children under age of five, with an especially high incidence being observed for those under three. It can result in herpes in such body parts as hands, feet and mouth and even other complications such as myocarditis, pulmonary edema, and aseptic meningoencephalitis in some children. Some severely affected patients may die due to the quick progress of the disease. There are over twenty types of enterovirus leading to HFMD, Coxsackievirus A16and EV71are the most common, and cause severe mortality in children. And there are more than one million children infected with HFMD every year in China. As a matter of fact, there is no effective vaccine or antiviral treatment specifically for HFMD, but if we can provide some preventive measures to control the HFMD, it will be very meaningful.
This thesis has been divided into four chapters, the first chapter introduces the background of epidemic dynamics, etiology, epidemiology, diagnostication, research status and pre-knowledge of HFMD.
In chapter two, an epidemic model with non-monotonic contact rate is devel-oped to explore the control of HFMD. By analyzing globally asymptotic stability of disease-free equilibrium, existence and global analysis of endemic equilibria, we establish the sufficient conditions for the extinction of the endemic disease. Mean-while, backward bifurcation, bistable states are investigated. Numerical simulations are presented to verify the theoretical result and then preventive measures of HFMD are proposed in the end. The huge infected population is the important reason that HFMD is still epidemic in China.
In chapter three, we formulate a time-delay SEIQRS epidemic model with non-monotonic contact rate. Firstly, the existence of the disease-free equilibrium and the endemic equilibria is discussed. We further analyze the asymptotic behaviors of the dynamical model. With the help of Lyapunov functional methods we derive the sufficient conditions for the stability of these two equilibria in the end.
In the fourth chapter, we concentrates on the HFMD data of China from2008to2012. We set up two mathematical models to fit those data with the goodness of fit and obtain the optimal parameter values of the model. By the Chi-square test of statistical inference, the optimal parameter values of the models are reasonable. We get the basic reproductive number of the disease for each year, and it is larger than1. Thus, we conclude that HFMD will persist in China under the current conditions. So we investigate the preventive measures to control the HFMD, the HFMD will be controlled quickly and the number of infections will decline rapidly in a period of time. Finally, we make use of the parameter values of the year2011and2012to simulate and forecast the number of patients of2013, and the predictive results inosculate well with the real-world situations.
本文分为四章,在第一章里介绍了传染病动力学模型的背景,手足口病病原学、流行病学、诊断鉴别及研究现状及预备知识。
在第二章中,研究了一个具有非单调发生率的手足口病模型并提出一些控制方案。通过对无病平衡点的全局渐近稳定分析,地方病平衡点的存在性与全局分析,我们得到了疾病流行的充分条件,同时,对后向分支、双稳态进行了探讨。数值模拟证实了理论的正确性,最后给出了手足口病的预防方针。我们认为双稳态的产生是手足口病仍然在中国大范围流行的一个重要的原因。
在第三章中,我们建立了具有非单调接触率的时滞SEIQRS传染病模型,首先研究了无病平衡点、地方病平衡点的存在性,分析了该模型的渐近行为,使用Lyapunov泛函方法得到了无病平衡点及地方病平衡点稳定的充分条件。
在第四章里,我们关注中国2008年到2012年间手足口病患病数据的研究。先后建立了两个动力学模型,借助数据,运用拟合优度的方法,分别得出了两个模型的最优参数。通过统计推断的卡方检验,证实了模型的参数值都是合理的。我们得到了每一年的基本再生数都大于1。因此,断定在当前条件下,手足口病仍会流行。此外,我们给出了详细的预防控制措施,如果运用文中的预防措施,手足口病将会得到迅速的控制,患者的数量会得到急剧的减少。文章的最后,我们运用2011年和2012年模拟出的参数对2013年患病人数做了预测,预测结果表明,与手足口病实际流行情形是吻合的。
Hand, foot and mouth disease (HFMD), an infectious disease caused by en-terovirus and Coxsackievirus, usually happens to children under age of five, with an especially high incidence being observed for those under three. It can result in herpes in such body parts as hands, feet and mouth and even other complications such as myocarditis, pulmonary edema, and aseptic meningoencephalitis in some children. Some severely affected patients may die due to the quick progress of the disease. There are over twenty types of enterovirus leading to HFMD, Coxsackievirus A16and EV71are the most common, and cause severe mortality in children. And there are more than one million children infected with HFMD every year in China. As a matter of fact, there is no effective vaccine or antiviral treatment specifically for HFMD, but if we can provide some preventive measures to control the HFMD, it will be very meaningful.
This thesis has been divided into four chapters, the first chapter introduces the background of epidemic dynamics, etiology, epidemiology, diagnostication, research status and pre-knowledge of HFMD.
In chapter two, an epidemic model with non-monotonic contact rate is devel-oped to explore the control of HFMD. By analyzing globally asymptotic stability of disease-free equilibrium, existence and global analysis of endemic equilibria, we establish the sufficient conditions for the extinction of the endemic disease. Mean-while, backward bifurcation, bistable states are investigated. Numerical simulations are presented to verify the theoretical result and then preventive measures of HFMD are proposed in the end. The huge infected population is the important reason that HFMD is still epidemic in China.
In chapter three, we formulate a time-delay SEIQRS epidemic model with non-monotonic contact rate. Firstly, the existence of the disease-free equilibrium and the endemic equilibria is discussed. We further analyze the asymptotic behaviors of the dynamical model. With the help of Lyapunov functional methods we derive the sufficient conditions for the stability of these two equilibria in the end.
In the fourth chapter, we concentrates on the HFMD data of China from2008to2012. We set up two mathematical models to fit those data with the goodness of fit and obtain the optimal parameter values of the model. By the Chi-square test of statistical inference, the optimal parameter values of the models are reasonable. We get the basic reproductive number of the disease for each year, and it is larger than1. Thus, we conclude that HFMD will persist in China under the current conditions. So we investigate the preventive measures to control the HFMD, the HFMD will be controlled quickly and the number of infections will decline rapidly in a period of time. Finally, we make use of the parameter values of the year2011and2012to simulate and forecast the number of patients of2013, and the predictive results inosculate well with the real-world situations.
引文
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