SIR模型原理優(yōu)缺點中英混排

1、SIR模型是傳染病模型中最經(jīng)典的模型，其中S表示易感者。模型中把傳染病流行范圍 內(nèi)的人群分成三類：S類，易感者(Susceptible),指未得病者，但缺乏免疫能力，與感病者 接觸后容易受到感染；I類，感病者(Infective),指染上傳染病的人，它可以傳播給S類成 員；R類，移出者(Removal),指被隔離，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.假設總人數(shù)N不變，易感者、感病者、移出者三者的比例分別為s(t)、i(t)、r(t),并設

3、 病人的口接觸率(每個病人每天有效接觸的平均人數(shù))為常數(shù)入，口治愈率(每天被治愈的 病人占總病人數(shù)的比例)為常數(shù)H，則傳染期接觸數(shù)則有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不妨設初始時刻的易感染者，染

4、病者，恢復者的比例分別為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基礎模型用微分方程組表示如下：Using differential equations, we describeBasal SIR model as follows:di .=xsi- /Zidtds 與.=-xsidtdr而通常情況下，r(O)=ro都很小，可近似看作roO,祐+，。知1,以上方程可化簡為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)的求解十分困難，可利用相軌線分析討論解i(t)、s(t)的性質(zhì)，其中箭頭表示 了隨著時間t的增加s(t)和i(t)的變化趨向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.分析圖像可以得到以下結論：Analyzingthe figure, we come to the conclusions downside.為保證傳染病不蔓延，需要滿足So V 1/0。為了達到這個目的，一方面，可以提高閾值 lg，需降低6即減小口接觸率；I,可通過提高衛(wèi)生水平的方式；增大日治愈率H，可以通 過提高醫(yī)療水平的方式。另一方面，也可以通過群體免疫來提高從而降低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.模型優(yōu)缺點：Advantage and disadvantage:基于微分方程組求解的SIR模型可以根據(jù)己有數(shù)據(jù)比較準確地擬合曲線，并利用相軌線 分析得出使傳染病不蔓延的措施，理論依據(jù)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.但是應注意到，模型對人群的分類不夠細致，沒有明確考慮隔離的因素。而現(xiàn)實中對疑 似病人的隔離是控制疫情傳播的有效手段。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.模型沒有引入反饋機制，在預測過程中，單純依據(jù)己有數(shù)據(jù)預測未來較長一段時間的數(shù) 據(jù)，必然會使準確度降低。尤其是題目中藥物的介入和衛(wèi)生條件的改善在過去的數(shù)據(jù)中是無 法體現(xiàn)出來的，采用己有數(shù)據(jù)無法體現(xiàn)出這些因素對疫情控制的影響，這是模型致命的漏洞。 為此必須引入反饋機制達到自我調(diào)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，微分方程組求解較為困難，旦對初值比較敏感，這對模型的穩(wěn)健性是一個很大的 影響。Last but not the least, differential equations are sensitive to initial value. These disadvantages will gravely reduce the stability基于以上考慮，我們引入了反饋機制。但是這對原有的連續(xù)模型提出了一個挑戰(zhàn)，我們 無法做到實時反饋，事實上，我們只需要將連續(xù)的時間劃分為等距的時間段，然后按照時間 段反饋，這和每口統(tǒng)計疫情數(shù)據(jù)比較相似。于是，連續(xù)模型就改為離散模型。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