论文总字数：20242字

摘 要

2019年，湖北武汉出现了第一例新冠肺炎冠状病毒传染病后，中国疫情迅速发展。根据政府数据可知，1月21日湖北省现存确诊数为338例，当日新增确诊数为105例，从此时开始，患者每日新增病例数量以几百增加，显示出传染病大流行的苗头，2月4日，湖北省现存确诊数为15679例，当日新增确诊数为3156例，从此时开始，患者每日新增病例数达到几千，现存病例数也已破万；2月12日湖北省新增病例数为14840例，现存病例数为43455例，疫情发展已经非常严重。

预测疫情发展趋势，何时达到峰值，何时疫情能够得到控制，这既能为政府人员做出决策提供参考价值，也能让人们对疫情发展有个大致的认识。

预测传染病发展趋势非常困难，很多模型有很多局限性。新冠肺炎传染病因为其感染者接受治疗后不再会被二次传染，新冠肺炎传播非常适合SIR模型的假设条件。因此，本文主要使用常微分方程组形式确定的SIR模型。

使用SIR模型有些不足之处：感染者有一部分入院接受治疗，此时他们处于被隔离状态，相应的也就没有了感染易感者的能力，有一部分并没有入院治疗，他们保持正常的社交，有感染易感者的能力。因此本文将SIR模型进行了一些修改，将感染者分为自由感染者和治疗感染者。自由感染者有传染病毒的能力，治疗感染者没有。另外，模型中的自由感染者有转化为治疗感染者的可能，本文没有进行详细讨论，今后需要进一步研究。

在实证方面，本文结合真实数据，对参数进行估计，并将疫情分为三阶段，分别求出各阶段的基本再生数，群体免疫阈值和有效再生数。并用龙格库塔法处理模型，预测出疫情的发展趋势，并结合真实情况进行讨论。最后发现模型适配程度较好。

关键词： 新型冠状肺炎病毒 SIR模型 基本再生数 群体免疫阈值

有效再生数

Abstract

In 2019,the first covid-19 epidemic occurred in wuhan,hubei province ,and the epidemic in China grew rapidly.According to official data,there were 338 confirmed cases in hubei province on January 21,with 105 new cases confirmed on that day.Since then,covid-19 has increased by more than 100 cases a day,indicating a pandemic trend.On February 4,there were 15679 confirmed cases of covid-19 in hubei peovince,with 3156 new cases confirmed on that day.Since then,there had been more than 1000 new cases of covid-19 every day and more than 10000 existing cases.On February 12,the number of new cases in hubei was 14840,while the number of existing cases was 43455.

Predicting the trend of the epidemic,the peak time and the end time of the epidemic can not only provide reference value for the government officials to make decisions,but also give people a general understanding of the development of the epidemic.

It is very difficult to predict trends in infectious diseases,and many models have many limitations.Covid-19 infection because its infected persons will no longer be re-infected after treatment,the transmission of covid-19 is will suited to the assumptions of the SIR model.Therefore,the paper mainly studied the epidemic situation through the SIR model.

The SIR model has some drawbacks.Some of those infected are hospitalized for treatment,and then they are quarantined,and they do not have the capacity to infect others.The other infected people are not hoapitalized,and they remained socially normal and have the ability to infect others.Therefore,the SIR model is modified in this paper,and the infected person are divided into free infected person and treatment infected person.Free infected people have the ability to transmit the virus,but not to treat infected people.In addition,it is possible to convert the free infected person in the model into a treatment infected person.The paper is not discussed in detail and further research is needed in the future.

In the empirical aspect,this paper estimate the parameters based on real data,divide the epidemic into three stages,and work out the basic regeneration number,the population immunity threshold and the effective regeneration number in each stage.The development trend of epidemic is predicted by using runge kutta method,and the real situation is discussed.Finally,the model is found to be reasonable.

Keywords: COVID-19；SIR model；basic reproduction number；

herd immunity threshold；effective reproductive number

目 录

摘要 - I -

Abstract - II -

第一章 绪论 - 1 -

1.1 研究背景 - 1 -

1.2 本文主要工作 - 1 -

第二章 基于SIR的新冠肺炎病毒的微分方程模型 - 3 -

2.1 前期研究与启发 - 3 -

2.2 假设与模型建造 - 3 -

2.3 理论分析 - 5 -

第三章 模型实证和疫情分析 - 7 -

3.1 武汉疫情数据分析 - 7 -

3.2 第一阶段模型实证和疫情分析 - 8 -

3.3 第二阶段模型实证和疫情分析 - 10 -

3.4 第三阶段模型实证和疫情分析 - 13 -

第四章 根据实际情况对建立模型的讨论 - 16 -

参考文献 - 17 –

附录 - 19 -

致谢 - 22 -

第一章 绪论

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