兰州大学机构库 >数学与统计学院
艾滋病及狂犬病的数学模型及其动力学分析
Alternative TitleMathematical Models of AIDS and Rabies and Their Kinetics
刘洪涛
Thesis Advisor李自珍
2008-05-23
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name硕士
KeywordHopf分支 数值模拟 狂犬病 捕杀 免疫 基本再生数
Abstract传染病的传播模型可追述到1760年Daniel Bernoulli对天花的分析。传染病动力学的建模与研究于二十世纪中叶开始蓬勃发展,作为标志性的著作是Bailey于1957年出版的专著《数理流行病学》。近年来,国际上传染病动力学的研究进展迅速,大量的数学模型被用于分析各种各样的传染病问题,也有部分是针对诸如麻疹、流感等诸多具体疾病的模型。从模型的角度来划分,主要可以分为四类:一类是研究以常微分方程描述的流行病模型;一类是研究以偏微分方程描述的流行病模型;第三类是研究同时含有常微分方程和偏微分方程的流行病 模型,这主要包括类年龄结构的流行病模型;另外一类模型为随机模型,可以在相应常微分方程的基础上增加随机考虑或利用Markov链进行Monte Carlo模拟。本文是针对艾滋病,狂犬病而建立了常微分方程模型。艾滋病是由于人体感染人类免疫缺陷病毒而引起的一种病死率极高的恶性传染病。狂犬病是由狂犬病毒引起的人兽共患烈性传染病,一旦发病病死率几乎100%。本文基于其相关的病理知识,分别对应建立了数学模型,并用动力学的知识对其进行分析,通过数值模拟,得到以下结论: 1.在病毒对健康细胞的感染率是非线性时,找出了决定系统周期解的稳定性,周期解的周期的因素。 2.通过对目前狂犬病流行现状的研究,在考虑中国实际情况的基础上,建立了新的数学模型。本模型不仅涉及到狂犬病在犬中传播,还考虑了暴露和患病犬以及携带病毒的”健康犬”使人感染狂犬病的现象。通过寻找”基本再生数”对比了捕杀、免疫、捕杀和免疫相结合三种不同策略在控制狂犬病传播中的有效性。分析和模拟结果表明三种控制狂犬病的方法中捕杀的效果最好,免疫的效果次之,捕杀和免疫相结合的效果最差。同时根据中国目前城市和农村发展不平衡的现状,提出了在城市以免疫为主,在农村采用捕杀和免疫相结合的控制狂犬病的措施,从而为中国目前控制狂犬病的流行提供了理论依据。
Other AbstractEpidemic models have a long history dating back to Daniel Bernoulli’s analysis of smallpox in 1760.In the middle twentieth century.There have a great development of the epidemic dynamics.The symbol work is the published of mathematical epidemiology written by Baily in 1957,and later republished.In recent years,the development is also very fast.Many mathematical models were used to analysis various disease.Meanwhile some mainly focused on measles flu.So we can divide the following four cases in the point of view mathematical models:one is described by ODE,the second is described by PED,the third is described by ODE and PDE. mainly include the age-structured epidemic models,and the fourth is described by stochastic models. we can add stochastic disturbance in ODE or use Monte Carlo simulation by markov chain. In this paper, we mainly established the ODE models of AIDS and rabies. AIDS is a malignant infections disease with high death rate that caused by Human immuno-deficiency virus .Rabies is an virulent contagious disease which transmit between people and animals.When infected,the deathrate is 100%. This paper based on some pathological facts.Mathematical models were established. We studied the dynamic behavior of the models through mathematical analysis and numerical simulation,we obtained the following results: 1.When the force of infection in assumed to be nonlinears.We find the main factor of which determining the stability of the system’s Periodic solutions and it’s Period. 2.we studied the outbreak of rabies and established the mathematical model based on the reality of our country. We not only focued on the rabies transmitted in the dogs but also considered people infected by exposed dogs, infected dogs and ”sound dogs” carrying virus. We found out the basic reproduction number and compared the efficiency of the three strategies for controlling the rabies: culling, vaccination, culling and vaccination. The results of analysis and simulation indicate that the strategy of culling is the most efficiency strategies, vaccination is the intermediate and culling and vaccination is the worse. Meanwhile, as the inbalance between city and country, we stated that vaccination in the city and culling and vaccination in the country are recommended for controlling the rabies. Our study provides a theoretical basis for controlling the rabies in our country.
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/225641
Collection数学与统计学院
Recommended Citation
GB/T 7714
刘洪涛. 艾滋病及狂犬病的数学模型及其动力学分析[D]. 兰州. 兰州大学,2008.
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