SIR模型原理優(yōu)缺點(diǎn)中英混排(暫定版)

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闈內(nèi)的人群分成三類(lèi)：S類(lèi)，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(lèi)，感病者(Infective),指染上傳染病的人，它可以傳播給S類(lèi)成員；R類(lèi)，移出者(Removal),指被隔離，或因病愈而具有免疫力的人。SIRmodelisthemostclassicmodelinepidemicmodels?Thismodelclassifypeopleasthreegroupsfollows：GroupS(Susceptible):thesehealthypeoplehavenoimmunity.Theyareeasilyinfectedwhencontactingwithinfectedpeople?GroupI(Infectedpeople):Theviruseshavealreadyinfectedthem?TheycanspreadvirustoGroupS?GroupR(Removal):peoplewhoarecuredanddied?假設(shè)總?cè)藬?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(shè)病人的口接觸率(每個(gè)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)卩，則傳染期接觸數(shù)a=A/r,則有Nowweassumethatthetotalnumberofpeople(N)isfixed,thustheproportionofeachgroupsares(t),i(t)andr(t).Everyinfectedpeoplecontactswith入peopleeveryday,|Jpeoplearecured?Sos(t)+i(t)+r(t)=l不妨設(shè)初始時(shí)刻的易感染者，染病者，恢復(fù)者的比例分別為so、i0.r0,即Attheverybeginning,theproportionofeachgroupsaresQ.iQ.rQ,so,s(0)=s()(so>0)i(0)=io(io>O)

r(0)=r0(r0>0)SIR基礎(chǔ)模型用微分方程組表示如下：Usingdifferentialequations,wedescribeBasalSIRmodelasfollows:citdsctt一久citdsctt一久sidr通常情況下，r(0)=r0都很小，可近似看作ro^O,io+so^l,以上方程可化簡(jiǎn)為Ingeneral,=roaresmallzsoitcanbeconsideredasq°0,i()+sQ1?Then,theequationscanbesimplifiedto(dids'di=-Asi

s(0)=SoIi(0)=io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(shí)間t的增加s(t)和i(t)的變化趨向However;s(t)andi(t)aredifficulttosolve.Wecanusetrajectorytoanalyzeandobtainthecharactersofi(t)、s(t).Thearrowsstandforthetendenciesofi(t)、s(t)withtimegoingby.分析圖像可以得到以下結(jié)論：Analyzingthefigure,wecometotheconclusionsdownside?為保證傳染病不蔓延，需要滿(mǎn)足SoVl/"。為了達(dá)到這個(gè)目的，一方面，可以提高閾值1/6需降低6即減小口接觸率；I，可通過(guò)提高衛(wèi)生水平的方式；增人口治愈率“可以通過(guò)提高醫(yī)療水平的方式。另一方面，也可以通過(guò)群體免疫來(lái)提高r°,從而降低so,使病情不蔓延。Whens。<1/<7,thecontagionwillnotspread?Toachievethisconditiontherearetwoways.Ononehand,byimprovinghygienelevels,wecanlowerAandlessen[i,namelyraisethethresholdvaluel/cr.Ontheotherhand,bypromotingherdimmunity,wecanimprovero,therebyreduceso?Inthesemeasures,thestateoftheillnesswillnotrise?模型優(yōu)缺點(diǎn)：Advantageanddisadvantage:基于微分方程組求解的SIR模型可以根據(jù)已有數(shù)據(jù)比較準(zhǔn)確地?cái)M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。ThesolutionsofSIRmodelbasedondifferentialequationcanfittotherealisticcurveapproximately.Meanwhile,byanalyzingwithtrajectory,weconcludewaystocontroltheillnessfromspreading?Theresultsshowthatthetheoreticalbasisispracticable?但是應(yīng)注意到，模型對(duì)人群的分類(lèi)不夠細(xì)致，沒(méi)有明確考慮隔離的因素。而現(xiàn)實(shí)中對(duì)疑似病人的隔離是控制疫情傳播的有效手段。Butweshouldrealizethatthismodelclassifiespeopleinaverysimplewayandconsidersnothingaboutisolation.Howevecinreality,isolationmakesagreatdifferenceincontrollingtheillness.模型沒(méi)有引入反饋機(jī)制，在預(yù)測(cè)過(guò)程中，單純依據(jù)已有數(shù)據(jù)預(yù)測(cè)未來(lái)較長(zhǎng)一段時(shí)間的數(shù)據(jù)，必然會(huì)使準(zhǔn)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(guò)去的數(shù)據(jù)中是無(wú)法體現(xiàn)出來(lái)的，釆用已有數(shù)據(jù)無(wú)法體現(xiàn)出這些因素對(duì)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(jī)制達(dá)到自我調(diào)整的功能。ThereisnofeedbackmechanisminSIRmodel.Inpredictions,forecastingalongrundataonlyusingdatawealreadyhavesurelywillletdowntheaccuracy.Butweshouldknowthatmanyfactorswillchangeinthefuture?Datacollectedbeforecannotreflectthechangesespeciallythosementionedinthequestionsuchasimprovementsofmedicineandhygienelevel.Thereisadrawbackhere.Tooptimizethismodelandmakeamoreaccurateprediction,weaddfeedbackmechanisminthat.此外，微分方程組求解較為困難，且對(duì)初值比較敏感，這對(duì)模型的穩(wěn)健性是一個(gè)很犬的影響。Lastbutnottheleast,differentialequationsaresensitivetoinitialvalue?Thesedisadvantageswillgravelyreducethestability基于以上考慮，我們引入了反饋機(jī)制。但是這對(duì)原有的連續(xù)模型提出了一個(gè)挑戰(zhàn)，我們無(wú)法做到實(shí)時(shí)反饋，爭(zhēng)實(shí)上，我們只需要將連續(xù)的時(shí)間劃分為等距的時(shí)間段，然后按照時(shí)間段反饋，這和每口統(tǒng)計(jì)疫情數(shù)據(jù)比較相似。于是，連續(xù)模型就改為離散模型。Consideringalltheseweaddfeedbackmechanismtooptimizeourmodel.However;ourmodelcannotfeedbackinstantly.lnfact,weneedtodivide