The Finial Taskveronica.docx
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The Finial Taskveronica.docx
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TheFinialTaskveronica
TheFinialTask:
SIRModel
Veronica
1、Effectivenessofimmunization
(1)Assumingthevaccineis100%effective
Thecontactnumberforpoliois7.Inordertomakesurethattheepidemicwillnotoccur,theinfectionrateshouldbelessthantherecoveryrate,thatis,cid(1-ve)<1,7·(1-v)<1,v>
.Sothefractionofthepopulationthatshouldbevaccinatedwouldbemorethan6/7.
Thecontactnumberformeaslesandpertussisis18.Asaresult,thefractionofthepopulationthatmustbevaccinatedwouldbemorethan17/18.
(2)Assumingthevaccineis90%effective
Thecontactnumberforpoliois7.Ifwedonotwanttheepidemictooccur,theinfectionrateshouldbelessthantherecoveryrate,whichmeansthatcid(1-ve)<1,7·(1-0.9v)<1,v>60/63.Sothefractionofthepopulationthatmustbevaccinatedshouldbemorethan60/63.
Thecontactnumberformeaslesandpertussisis18.Sothefractionofthepopulationthatmustbevaccinatedwillbemorethan85/81,thatistosay,althoughwecanhaveallthepeoplevaccinated,theepidemicofthesediseasescannotstopspreading.That’swhymeaslesandpertussispersistwhilepoliohasbeeneffectivelyeliminated.
(3)Assumingthathalfofthepopulationhasbeenimmunized,theinitialrecoveredpopulationwillchangefromzeroto5000,andthecurvewillbemuchflatter,whichmeansitwilltakemoretimetooccur,andwilllastforalongtime.
Accordingtotheformula:
cid(1-ve)<1,
<1,v>2/3,so2/3ofthepopulationshouldbeimmunizedtopreventanepidemic.
2、Effectivenessofquarantine
(1)Assumingthequarantinetimeishalfofaday,andthequarantinecontactrateiszero.Quarantinefractionforcurrent1,2,3,and4are0.2,0.4,0.5,and0.6respectively.Fromthefollowinggraph,wecanseethatwhenthequarantinefractionislessthan0.5,therewillbenoepidemic.
(2)Assumingthatthecontactrateofquarantinedindividualsis3
Bytestingvariousquarantinefractionslike0.2,0.4,0.6,0.8,1(theyarethenumbersofthefraction0,1,2,3,4),wegetthatthecontactratechangesfrom0to3,andevenperfectquarantinecannotpreventanepidemicfromhappening.
Thequarantinefractionis1.Bytestingvariousquarantinetimelike0.5,0.3,0.1,0.07(theyarethenumbersofthetime1,2,3,4inthegraph),wegetthatthefastertheinfectiouspeoplearemovingtoquarantine,thehigherthesusceptiblepopulationwillbe.Whenthetimeislessthan0.07,thereissomethingwrongwiththemodel.Butitstillcannotpreventtheepidemic.
Sowegettoknowthatifthecontactrateofthequarantinedindividualscannotreducetoacertainnumber,anepidemicwouldstillhappennomatterhowfasttheinfectiouspeoplearemovedtoquarantineandnomatterhowhighthequarantinefractionis.
Finally,quarantineandvaccinearebotheffective,butitisconvenienttoimplementquarantine.Therefore,inanemergency,just-in-timequarantineispossible,andvaccinewouldbeimpossibletopreventthedisease,becausethelatterneedsalotoftimeandmoneytoproduce.
3、Lossofimmunity
Lossofimmunitymakesrecoveredpeoplebecomethesusceptiblepopulationafteraperiodoftime.Bychangingtheaveragedurationofimmunity,wehavethefollowinggraph.Itcanbeconcludedthatwhentheaveragedurationofimmunitybecomesshorter,thecontinuousoccurrenceoftheepidemicwouldbemorelikelytohappen.Sowhentheimmunityisnotpermanent,itisdifficulttoeradicateadisease.
4、ExtendingtheSIRmodel
Assumingthevaccinatedfractioniszero,theaveragedurationofimmunityis10,000.Thenrunthemodel,wehavethefollowinggraghs.
Obviouslyanincubationperiodmakestheepidemicoccurmoreeasily.Fromthecharts,thereisnearly79%ofthesusceptiblepopulationbeinguninfectedwhile1%ofthepopulationwassymptomaticinabout8days;andabout5%ofthesusceptiblepopulationisuninfectedwhile10%ofthepopulationbecomessickin13days.
(1)Thecontactrateofthesymptomaticpopulationis0.Theinfectivityis0.25.
Settingdifferentincubationtimes,wegetthatthelongertheincubationtime,thequickerandbiggerofthediseasetospread.Theasymptomaticinfectivecurvewillreachahighertop,andtakealongertimetodecline.Thesymptomaticpopulationissmallandconstant.Therecoveredpopulationincreasesinasteadyway.Sotheincubationperiodresultsininefficiencyofthequarantine.Becausewhentheincubationperiodislong,therewillbemanypeopleintheasymptomaticstagewithoutbeingquarantined,buttheyareasactiveashealthypeople,alotofotherpeoplewillbeinfectedbythemgradually.Sowithlesssymptomaticpeoplebeingquarantinedandalotofasymptomaticpeople,thequarantinewillbecomeineffectiveandtheabilityofthepopulationtoenjoyherdimmunitywillbeworse.
(2)Thecontactrateofthesymptomaticpopulationis0.Incubationtimeis18days
Thefollowinggraphsshowthatthehighertheinfectivity,thequickertheepidemictospread,andthefasterthedeclinetrendwillbe.
5、Just-in-TimeImmunization
(1)Thevaccineeffectivenessis100%,thefractionvaccinatedis100%,thetimetodeployvaccineis2days,andtheaveragedurationofthediseaseis4days.Thethresholdforimmunizationis5%.Accordingtothegraphbelow,about77%ofthepopulationeventuallygetsthedisease.ComparedwiththecasewithoutJITvaccination,itcanreducethefractionoftheinfectedpeople.
Whenthethresholdis1%,theasymptomaticinfectivewillbehalflessthantheprevioussituations,whichmeansitismoreeffectivetostoptheepidemicfromthebeginning.
(2)Thevaccineeffectivenessis95%,thefractionvaccinatedis80%,thetimetodeployvaccineis7days,andtheaveragedurationofthediseaseis4days.Comparingthepreviousgraphandthefollowingone,wecanknowthatwhentheeffectivenessislow,thefractionvaccinatedissmall,thetimetodeployvaccineislong.Asaconsequence,theJITimmunizationprogramwillbelesseffectivenomatterhowearlyyouimplementit.
Inconclusion,itisprobablymoreeffectivetouseJITvaccinationintheconditionthatwhendiseasesmostlyoccurinurbanareaswherethepopulationisdenseandthecontactrateishigh,andwhendiseasesarespreadingthroughdailycontactorinhalationlikeinfluenza.Theineffectivesituationmayhappenwhendiseasesoccurinruralareasordiseasescannotspreadviapeople’sdailycontact,whichmeanstheinfectivityislow,likeHIV.
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