预测传染病的英文论文

Models,Prediction,andEstimationofOutbreaksofInfectiousDiseasePeterJ.Costa,JamesP.Dunyak,[e-mail]EIR(Susceptible–Exposed–Infectious–Recovered)modelshavebeenutilizedbynumerousresearcherstostudyandpredictdiseaseoutbreak.Bycombiningthepredictivenatureofsuchmathematicalmodelsalongwiththemeasuredoccurrencesofdisease,amorerobustestimateofdiseaseprogressioncanbemade.TheKalmanfilteristhemethoddesignedtoincorporatemodelpredictionandmeasurementcorrection.Consequently,weproduceanSEIRmodelwhichgovernstheshorttermbehaviourofanepidemicoutbreak.ThemathematicalstructureforanassociatedKalmanfilterisdevelopedandestimatesofasimulatedoutbreakareprovided1.IntroductionMathematicalmodelshavebeenusedtostudytheoutbreakofanumberofinfectiousdiseases[1,2,6].Inparticular,differenceanddifferentialequationsarethemethodologiesinwhichsuchmodelsarewritten[4,5,6].Manyresearchhospitalsand/orpublichealthdepartmentsaremaintainingadatabaseofemergencyroomvisitsbypatientswithcategorizedcomplaints.ThecombinationofamathematicalmodelofanoutbreakwithdailymeasurementsbeckonstheapplicationofaKalmanfiltertoprovideanoptimalestimateofthenumberofinfections.ThispaperwillprovidethemathematicalinfrastructurerequiredtoimplementaKalmanfilteronsimulatedemergencyroomdata.Theprogramofthisdiscussionwillbetoprovideageneralmodel,discussmodelsimplification,anddemonstratetheefficacyofthefilteronsimulateddata.Inthisfirstsection,weestablishcommonnotationandageneralmodelfortheoutbreakofaspecific(butunknown)infectiousdiseasethroughageneralpopulation.1.1NotationS=S(t)=numberofpeopleinthepopulationsusceptibletothediseaseattimetE=E(t)=numberofpeopleinthepopulationexposed/infectedbythediseaseattimetI=I(t)=numberofpeopleinthepopulationwhoareinfectiousattimetR=R(t)=numberofpeopleinthepopulationwhohaverecoveredfromthediseaseattimetThereareanumberofparameterswhichwillneedtobeeithermodeledorestimatedfromthedata.Itisassumedthattheseparametersaretimeinvariantthoughmoresophisticatedeffortsandinformationcouldproducetime–varyingmodels.Adescriptionoftheseparametersislistedbelow.1.2Parametersβ=probabilityofdiseasetransmissionv=rateofseroconversion(i.e.,fromexposedtoinfecti–ous)µI=deathrateofinfectiousduetothediseaseα=recoverydelayrateρ(I)=βI(t)=conversionratefromsusceptibletoexposed/infected(alsocalledtheforceofinfection)Infigure1below,aschematicdiagramexpressesthegraphicalrepresentationofthespreadofaninfectiousdiseasethroughapopulation.Implicitinthisfigureistheassumptionthateveryoneinthepopulationissusceptibletothedisease.ThefirstboxesillustratethemigrationofthepopulationofsusceptiblesS(t)tothoseexposedandinfectedE(t).TherateatwhichthesusceptiblesareinfectedisproportionaltothenumberofcontactscwiththeinfectiouspopulationI(t)timestheprobabilityofdiseasetransmissionpercontactβtimestheproportionofthepopulationwhichisinfectious:ρ(I)=βI(t).Sincetheinfectedleavethepopulationofsusceptiblesanegativesignisattachedtothisquantity.Consequently,dS(t)/dt=0–ρ(I)S(t)≡βI(t)S(t).Inasimilarmanner,thediseasedynamicsofequation(1.1)areformed.0-7803-8865-8/05/$20.00©2005IEEE.174Figure1.1.Diseasedynamics1.3DiseaseDynamics()()()()()()()()()()()()(1)()()(1)()(1)()IIIdSIStItStdtdEIStvEtItStvEtdtdIvEtItItdtvEtItdRItdtρβρβµµααµα=−≡−=−≡−=−−−−≡−−=−−(1.1)This“full”modelexpressedin(1.1)operatesunderthesimplifyingassumptionofasufficientlyshorttime–scalesuchthatnosignificantpopulationentersthesusceptiblepopulationandthattheparametersβ,ν,µI,andαdonotvarywithrespecttotime.Theeffortsbehindthisworkaretopresentamodelforashorttime–scalewithintheepidemiccycle(i.e.,ontheorderof2–3weeks).Consequently,aseriesofsimplifyingassumptionscanbemadewhicharelistedbelow.Assumptions(i)Shorttime–scale:t∈[to,to+∆t]wherethechangeintime∆tislessthanthreeweeks.(ii)Noimmigrationtooremigrationfromthesubpopulations(iii)InsufficienttimeforR(recovereds)toreturntothepopulationofsusceptibles(iv)Fort∈[to,to+∆t],S(t)=S(to)=So.From(iv),0dSdt=and()oStS=(constant).Setρ(I)=βSoI(t)≡ρoI(t),whereρo≡βSo,sothatthesecondandthirdequationsofthediseasedynamicsbecome()()()(1)()odEItvEtdtdIvEtItdtρα=−=−−(1.2)Observethatthefourthequationofthediseasedynamicsiscompletelydecoupledfromthemiddletwoequations.Consequently,thepopulationofrecoveredscanbecomputedas()()(1)()otoItRtRtIdµαττ=+−−³.(1.3)BysettingX=[E,I]T,thereducedsetofdiseasedynamicscanbewritteninthevector–matrixformdAdt=XX(1.4)where1oAνρνα−ªº=«»−¬¼.ExposedE(t)DeathbydiseaseSusceptiblesS(t)ρ(I)νµIInfectiousI(t)RecoveredR(t)1–µI–α175Themeasurementsofthissystemareaportionofthenumberofinfectiouswhichreporttoemergencyroomsonaday–to–daybasis.Moreprecisely,letTbetheprobabilitythatamemberoftheinfectiouspopulationappearsinareportingemergencyroom.Then,themeasurementsarem(t)=TI(t).(1.5)Themeasuredquantity,TI(t),ratherthanthemodeledpopulationofinfectiouspeopleI(t),iswhatemergencydepartmentsreported.Thus,makethefollowingchangeofvariables(1.6)totransformtheproblemtoa“non–dimensional”framework.l()()()()()()ItTItItEtTEtEt≡≡66(1.6)Since,ldEdETdtdt=anddIdITdtdt=,thenmultipl