SIR模型原理优缺点中英混排

1、SIR模型是传染病模型中最经典的模型，其中S表示易感者。模型中把传染病流行范围 内的人群分成三类：S类，易感者(Susceptible),指未得病者，但缺乏免疫能力，与感病者 接触后容易受到感染；I类，感病者(Infective),指染上传染病的人，它可以传播给S类成 员；R类，移出者(Removal),指被隔离，或因病愈而具有免疫力的人。SIR model is the most classic model in epidemic models. This model classify people as three groups follows：A Group S (Susceptible

2、): these healthy people have no immunity.They are easily infected when contacting with infected people.A Group I (Infected people): Theviruses have already infected them. They can spread virus to Group S.A Group R (Removal): people who are cured and died.假设总人数N不变，易感者、感病者、移出者三者的比例分别为s(t)、i(t)、r(t),并设

3、 病人的口接触率(每个病人每天有效接触的平均人数)为常数入，口治愈率(每天被治愈的 病人占总病人数的比例)为常数H，则传染期接触数则有Now we assume that the total number of people (N) is fixed, thus the proportion of each groups are s(t),i(t) and r(t). Every infected people contacts with A people every day,|J people are cured. Sos(t) + i(t) + r(t) = 1不妨设初始时刻的易感染者，染

4、病者，恢复者的比例分别为So、io、，即At the very beginning, the proportion of each groups are sQ. iQ. rQ, so, s(0) = s()(so 0) i(0) = io Uo 0) r(0) = r0(r0 0)SIR基础模型用微分方程组表示如下：Using differential equations, we describeBasal SIR model as follows:di .=xsi- /Zidtds 与.=-xsidtdr而通常情况下，r(O)=ro都很小，可近似看作roO,祐+，。知1,以上方程可化简为In

5、 general, r) = % are smallso it can be considered as% 幻 0 , i()+ sQ 1. Then, the equations can be simplified to(di瓦adsdi=-Asis(0) = SoI i(0) = iQ但s(t)、i(t)的求解十分困难，可利用相轨线分析讨论解i(t)、s(t)的性质，其中箭头表示 了随着时间t的增加s(t)和i(t)的变化趋向However; s(t) and i(t) aredifficult to solve. We can usetrajectorytoanalyze and obt

6、ain the characters of i(t)、s(t). The arrows stand for the tendencies of i(t)、s(t) with time going by.分析图像可以得到以下结论：Analyzingthe figure, we come to the conclusions downside.为保证传染病不蔓延，需要满足So V 1/0。为了达到这个目的，一方面，可以提高阈值 lg，需降低6即减小口接触率；I,可通过提高卫生水平的方式；增大日治愈率H，可以通 过提高医疗水平的方式。另一方面，也可以通过群体免疫来提高从而降低so,使病情不 蔓延。W

7、hens0 1/a, the contagion will not spread. To achieve this condition there are two ways. On one hand, by improving hygiene levels, we can lower A and lessen 山 namely raisethe threshold valuel /a. On the other hand, by promoting herd immunity, we can imp rover o, thereby reduces。 In these measures, th

8、e state of the illness will not rise.模型优缺点：Advantage and disadvantage:基于微分方程组求解的SIR模型可以根据己有数据比较准确地拟合曲线，并利用相轨线 分析得出使传染病不蔓延的措施，理论依据充分。The solutions of SIR model based on differential equation can fit to the realistic curve approximately. Meanwhile, by analyzing with trajectory, we conclude ways to con

9、trol the illness from spreading. The results show that the theoretical basis is practicable.但是应注意到，模型对人群的分类不够细致，没有明确考虑隔离的因素。而现实中对疑 似病人的隔离是控制疫情传播的有效手段。But we should realize that this model classifies people in a very simple way and considers nothingabout isolation. Howevec in reality, isolation makes

10、 a great difference in controlling the illness.模型没有引入反馈机制，在预测过程中，单纯依据己有数据预测未来较长一段时间的数 据，必然会使准确度降低。尤其是题目中药物的介入和卫生条件的改善在过去的数据中是无 法体现出来的，采用己有数据无法体现出这些因素对疫情控制的影响，这是模型致命的漏洞。 为此必须引入反馈机制达到自我调整的功能。There is no feedback mechanism in SIR model. In predictions, forecasting a long run data only using data we alr

11、eady have surely will let down the accuracy. But we should know that many factors will change in the future. Data collected before cannot reflect the changes especially those mentioned in the question such as improvements of medicine and hygiene level. There is a drawback here.To optimize this model

12、 and make a more accurate prediction, we add feedback mechanism in that.此外，微分方程组求解较为困难，旦对初值比较敏感，这对模型的稳健性是一个很大的 影响。Last but not the least, differential equations are sensitive to initial value. These disadvantages will gravely reduce the stability基于以上考虑，我们引入了反馈机制。但是这对原有的连续模型提出了一个挑战，我们 无法做到实时反馈，事实上，我们只需要将连续的时间划分为等距的时间段，然后按照时间 段反馈，这和每口统计疫情数据比较相似。于是，连续模型就改为离散模型。Considering all these we add feedback mechanism to optimize our model. However; our model cannot feedback instantly.ln fact, we need to divide time into