Estimating the SARS-CoV-2 infected population fraction and the infection-to-fatality ratio: A data-driven case study based on Swedish time series data

aim: to estimate the covidMQY i¯fectio¯MtoM fatality ratio HifrIL i¯fectio¯MtoMcase ratio HicrIL a¯d i¯fectio¯MtoMicu admissio¯ ratio HiiarI i¯ swede¯[ to sugM gest methods for time series reco¯structio¯ a¯d predictio¯N

we optimize a set of simple fi¯ite impulse reM spo¯seHfirImodelscomprisi¯gofascali¯gfactora¯dtimeM delaybetwee¯officiallyreportedcasesLicuadmissio¯sa¯d deathstimeseriesusi¯gtheleastsquaresmethodNcombi¯ed with ra¯domized pcr study resultsL we utilize this simple model to estimate the total ¯umber of i¯fectio¯s i¯ swede¯L a¯d the correspo¯di¯g ifrN

the model class provides a good fit betwee¯ icu admissio¯s a¯d deaths throughout RPRPN cases fit co¯M siste¯tly from july RPRPL by whe¯ pcr tests had become broadlyavailableNweobserveadimi¯ishedifri¯latesumM mer as well as a stro¯g decli¯e duri¯g RPRQL followi¯g the lau¯chofa¯atio¯Mwidevacci¯atio¯programNthetotal¯umM ber of i¯fectio¯s duri¯g RPRP is estimated toQ.Smillio¯N

a fir model with a delta filter fu¯ctio¯ describes the evolutio¯ of epidemiological data i¯ swede¯ wellNthefactthatwefou¯difrLicra¯diiarco¯sta¯tover large parts of RPRP is i¯ co¯trast with claims of healthcare adaptatio¯ or mutated virus varia¯ts importa¯tly a;ecti¯g aN wacker mathematical physicsL lu¯d u¯iversityL swede¯ eMmailZ a¯dreasNwacker@fysikNluNse aN jöud occupatio¯al a¯d e¯viro¯me¯tal medici¯eL departme¯t of laboratory medici¯eL lu¯d u¯iversityL swede¯ bN ber¯hardsso¯ a¯d kN soltesz automatic co¯trolL lu¯d u¯iversityL swede¯ pN gerlee mathematical scie¯cesL chalmers u¯iversity of tech¯ology a¯d u¯iM versity of gothe¯burgL swede¯ fN gustafsso¯ electrical e¯gi¯eeri¯gL li¯köpi¯g u¯iversityL swede¯ theseratiosNthemodelallowsustoretrospectivelyestimate thecovidMQYepidemiologicaltrajectoryLa¯dco¯cludethat swede¯ was far from herd immu¯ity by the e¯d of RPRPN

sarsMcovMR· covidMQY· swede¯· herd immu¯ity· healthcare dema¯d predictio¯· dataMdrive¯ modelli¯g 1 Introduction the covidMQY pa¯demic has posed e¯ormous global chalM le¯ges to the healthcare sectorN to estimate the future ¯eed ofperso¯¯elLequipme¯ta¯dhospitalbedsLreliablestatistical a¯alysistoolsarerequiredNhistoricdataisa¯importa¯tasset i¯figuri¯gouthowtobestcombi¯eavailabletimeseriesdata to gai¯ predictive capability while reduci¯g the i¯flue¯ce of biasa¯dothersourcesofpredictio¯errora¯du¯certai¯tyNat the same timeL statistical a¯alysis of the historical epidemic evolutio¯ ca¯ provide i¯dicatio¯s for the success of medical treatme¯ts a¯d vacci¯atio¯ programsN it also allows estimaM tio¯ of the accumulated ¯umber of i¯fectio¯sN this ¯umber esse¯tially determi¯es the level of herd immu¯ityL a¯d thus receivedmuchatte¯tio¯i¯swede¯duri¯gthespri¯gofRPRPN itisadifficulttasktopredicthealthcareLa¯d6ofparticM ular i¯terest i¯ the covidMQY co¯text6icu dema¯dN this isespeciallytruei¯a¯earlyphaseofa¯epidemiccausedby a previously u¯k¯ow¯ pathoge¯L such as the sarsMcovMR virusthatcausescovidMQYNwhileitwaspossibletofalsify severalearlypredictio¯modelsbasedo¯highse¯sitivityLeNgN [Q]L it remai¯s a largely ope¯ questio¯ how time series data could be a¯alyzed to arrive at accurate a¯d precise predicM tio¯sL with practical use to healthcare pla¯¯ersN we i¯vestigate if a particular simple class of timeM i¯varia¯t fi¯ite impulse respo¯se HfirI models [R]6those withadelayeddeltaimpulserespo¯se6issufficie¯ttomodel therelatio¯betwee¯timeseriesdataNparticularlyLouraimis . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 31, 2021. ; https://doi.org/10.1101/2021.05.27.21257900doi: medRxiv preprint NOTE: This preprint reports new research that has not been certified by peer review and should not be used to guide clinical practice. R a¯dreas wacker et alN to i¯vestigate whether the simple fir model is sufficie¯t for relati¯g covidMQY cases Hdetected i¯fectio¯sIL icu admisM sio¯sLa¯dregistereddeathsi¯swede¯Nwethe¯demo¯strate how such simple models ca¯ be used for reco¯structi¯g the epidemiological evolutio¯ duri¯g times of measureme¯t u¯M certai¯tycausedbylimitedtestcapacityLaswellaspredictio¯ of icu dema¯d based o¯ case dataN 2 Methods RNQ data used i¯ this paperL the evolutio¯ of the pa¯demic is based o¯ the followi¯g official a¯d ope¯ly accessible time series reported by the swedish public health age¯cyZ – cases daily pcrMco¯firmed sarsMcovMR cases i¯ swede¯N the date refers to the registratio¯N – icu admissions daily ¯umber of icu admissio¯s i¯ swede¯ for patie¯ts with covidMQY at the give¯ dayN – deaths daily¯umberofdeathsi¯swede¯forperso¯s with a sarsMcovMR i¯fectio¯ at the give¯ dayN the data was extracted from [S] o¯ QT may RPRQ a¯d covers dates u¯til QS mayN N duetodelaysi¯reporti¯gLthelastSweeksfordeathdata a¯dthelastSdaysforicua¯dcasedataaredisregardedfrom statisticala¯alysisa¯ddisplayedbydottedli¯esi¯thisworkN due to i¯sufficie¯t testi¯g we also omit case data before QX ju¯e RPRP i¯ the model fitti¯gN duri¯g the first wave i¯ march5may RPRPL pcrMtesti¯g was esse¯tially focused to perso¯s admitted to hospital a¯d elderly care i¯ swede¯ due to limited testi¯g capacityN i¯ the first half of ju¯eL the swedish gover¯me¯t stro¯gly advocated the testi¯g of all perso¯swithsymptomsofcovidMQYa¯dsuppliedfi¯a¯cial assista¯ce to the regio¯s as of QQ ju¯e RPRPN we assume thatthishadfulle;ecto¯testi¯gafterafurtherweekLwhich justifies the date give¯ aboveN i¯ additio¯ to theordinary testi¯g of perso¯s with susM pectedcovidMQYi¯fectio¯Lresultsforsix randomizedstudM ies i¯ RPRP a¯d RPRQ have bee¯ published by the swedish public health age¯cy [T]N they ca¯ be used to estimate the prevale¯ce of covidMQY i¯ the populatio¯ at the correM spo¯di¯g timesN the studies co¯ducted RT5RX august RPRP a¯dRQ5RUseptemberRPRPdid¯otprovidea¯ypositivesamM plesL while RSL YL RTL a¯d TS positive cases where detected for RQ5RT april RPRPL RUM5RX may RPRPL SP november5T december RPRPL a¯d QR5QV april RPRQL respectivelyN test

were available for slightly less tha¯ S PPP perso¯s i¯ the studies of RPRP a¯d TWUX perso¯s for the study i¯ RPRQN the limited sample size results i¯ statistical u¯certai¯tyL i¯M dicated by the VX E co¯fide¯ce limits for the average asM sumi¯g a poisso¯ distributio¯ for the ¯umber of positively testedN sampli¯g bias might provide a reduced prevale¯ce for the two latest studies accordi¯g to the statistical a¯alysis performed i¯ the studies [T]N hereL we use the bare results based o¯ the ¯umber of positive casesN for compariso¯L we alsoprovidea¯estimatefortheicrbasedo¯sampli¯gMbias corrected dataN while deaths a¯d icu admissio¯s related to covidMQY ¯aturallyalsoappeari¯thecasedataLtheirtotal¯umberu¯til midMmay RPRQ sums up to o¯ly QNT E a¯d PNW E of the total casesL respectivelyN regardi¯g icu a¯d death dataL o¯e has totakei¯toaccou¯tthatapproximatelyWUEoficupatie¯ts survive [U]L a¯d former icu patie¯ts co¯tribute to the death tollwitho¯lyQSELasthereareapproximatelytwiceasma¯y deathsasicuadmissio¯sNfurthermoreLicuadmissio¯data a¯d death data relate to di;ere¯t age groupsZ while VY E of the icu patie¯ts are you¯ger tha¯ WP years oldL VX E of the deceased have reached at least the age of XPN Hall data from [S] extracted o¯ QT may RPRQNI the small overlap betwee¯ the groups ge¯erati¯g the casesL icu a¯d deaths time series suggests that statistical correlatio¯s betwee¯ the time series ca¯ be expected to reflect the li¯ks to their commo¯ causeZ a¯tecede¯t sarsMcovMR i¯fectio¯ i¯ the swedish societyN this motivates the fir model discussed i¯ secN RNS with three i¯depe¯de¯t filter fu¯ctio¯s for casesL icu admissio¯L a¯d deathsN furthermoreL data o¯ a¯tibody prevale¯ce from blood do¯ors a¯d health ce¯ter samples Hu¯related to covidMQYM specific testi¯gI have bee¯ aggregated a¯d published [V]N here we provide YU E co¯fide¯ce i¯tervals for covidM prevale¯ce based o¯ these data sourcesN RNR parameters used i¯additio¯tothedatao¯thecovidMQYevolutio¯a¯dprevaM le¯ces provided by the swedish public health age¯cy adM dressedi¯secNRNQLweapplythreefurtherparameterswithi¯ this studyZ – thetimei¯terval )i¯terval = QP± QdaysLduri¯gwhicha¯ i¯fectedperso¯showsapositivepcrresultNseesecNRNU for detailsN – the probability ?a¯tibody = P.YU± P.PUL that a previous sarsMcovMR i¯fectio¯ is detected by a¯ a¯tibody testN see secN SNR for detailsN – the average duratio¯g0 = QWbetwee¯ i¯fectio¯ a¯d the admissio¯ to icuN see secN SNS for detailsN note that )i¯terval a¯d?a¯tibody are used i¯depe¯de¯tly of each other i¯ two di;ere¯t determi¯atio¯s of the icrN both waysprovideesse¯tiallythesameresultLwhichstabilizesour resultsagai¯stsystematicerrorsi¯theseparametersN g0 o¯ly e¯ters HRRI a¯d the time axis i¯ figN SN . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 31, 2021. ; https://doi.org/10.1101/2021.05.27.21257900doi: medRxiv preprint estimati¯g the sarsMcovMR i¯fected populatio¯ fractio¯ a¯d the i¯fectio¯MtoMfatality ratio S RNS fi¯ite impulse respo¯se models fi¯ite impulse respo¯se HfirI models are a class of li¯ear filtersNHthei¯terestedreaderisreferredto[R]forathorough mathematical i¯troductio¯ to fir a¯d related li¯ear model structuresNI they describe the outcome of a timeMdepe¯de¯t observable Hsuch as a death rateI by a sum of precedi¯g data Hhere ¯umber of i¯fectio¯s at earlier datesIL which are weighted by a filter fu¯ctio¯N they are commo¯ly used for a¯alysi¯g epidemiological problemsL where the filter fu¯cM tio¯ could represe¯t for example the serial i¯terval distribuM tio¯N for practical applicatio¯sL the filter fu¯ctio¯ is ofte¯ ¯ot easy to obtai¯N here we show that the assumptio¯ of a dirac delta respo¯seL where the filter fu¯ctio¯ has o¯ly two freeparametersHdelaya¯damplitudeILallowsforaco¯siste¯t a¯alysis of the covidMQY evolutio¯ i¯ swede¯N a ce¯tral e¯tity for the evolutio¯ of a¯ epidemic is the ¯umber of ¯ew i¯fectio¯s˜G(C)L occurri¯g o¯ dayCN let ˜?2(C,g) be the probability for a¯ i¯fectio¯ starti¯g o¯ day C to ge¯erate a reported positive pcr testg days laterN this

i¯ the observatio¯ model ˜H2(C) = ∞Õ g=P ˜?2(C−g,g) ˜G(C−g)+ ˜42(C), HQI where ˜H2(C) de¯otes ¯ew cases o¯ dayCL a¯d˜42 is a zeroM mea¯u¯correlated¯oiseprocessreprese¯ti¯gstatisticalflucM tuatio¯s associated with the probability˜?2N the model has fi¯iteimpulsesi¯ce?2iside¯ticallyzeroforsufficie¯tlylarge g HeNgNa huma¯ lifetimeIL a¯d ca¯ be regarded practically as zero forg≫ QweekN the time i¯dexC has the u¯it of daysN i¯HQIitreprese¯tsthatthedetectio¯probabilitydistributio¯L defi¯ed through the depe¯de¯ce of the seco¯d time i¯dexgL may itself vary over timeN the use of the tilde∼ i¯ HQI is to disti¯guish u¯filtered measureme¯tsN historic observatio¯s˜H2(C) exhibit a clear weekday patter¯N for retrospective a¯alysis it is therefore customary to apply a ce¯tered WMday movi¯g average filterL H2(C) = Q W SÕ B=−S ˜H2(C−B), HRI tocompe¯sateforsuche;ectsNthroughoutthepaperwewill work with time series that have bee¯ subjected to filteri¯g accordi¯g to HRIN we will drop the∼ ¯otatio¯ but still write eNgN0cases1i¯steadof0filteredcases1i¯favorofreadabilityN withi¯ li¯ear system theoryL a model with the structure of HQI is referred to as a HstochasticI fi¯ite impulse respo¯se HfirI modelL implyi¯g Hcombi¯ed with HRII that aH2(C) ca¯ be described by a fi¯ite record ofG(C)N summatio¯ of ?2(C,g) over gL yields the expected i¯fectio¯MtoMcase ratio HicrIZ 12(C) = ∞Õ g=P ?2(C,g), HSI defi¯edastheprobabilitythataperso¯i¯fectedo¯day C will eve¯tually become detected a¯d registered as a caseN thece¯tralpoi¯tofthema¯uscriptisthatwei¯vestigate thehypothesisthat ?2(C,g)ca¯beadequatelymodeledusi¯g the delta fir model ?2(C,g) =12(C)X(g−g2), HTI where the discrete delta filter fu¯ctio¯ give¯ by X(C) =     Q, C= P, P, otherwise, HUI whereg2 istheaveragedelaybetwee¯i¯fectio¯a¯dcaseregM istratio¯N note that the model HTI is defi¯ed for the averaged qua¯tities HRIL where?2(C,g) does ¯ot display the weekday fluctuatio¯s i¯CL that are likely i¯˜?2(C,g)N assumi¯g that thegMdepe¯de¯ce of?2(C,g) is reaso¯ably well reproduced by its averageg2 a¯d sta¯dard deviatio¯fL the simplified model HTI is justified i¯ appN aN this relies o¯ the assumpM tio¯that?2(C,g) does¯otcha¯geo¯thescale fi¯C a¯dthat the seco¯d derivative ofG(C) is much smaller tha¯G(C)/fRN forthespecialcaseofa¯expo¯e¯tialevolutio¯for G(C)Lthis providesa¯accuracyofbettertha¯UEif fislesstha¯TVE of the doubli¯g timeL as already stated i¯ [W]N employi¯g HTIL the observatio¯ model HQI becomes H2(C) =12(C−g2)G(C−g2)+ 42(C). HVI themodelHVIassertsthatthe¯umberofcases H2(C)Ldetected through pcr testi¯g o¯ dayCL o¯ly depe¯ds o¯ the ¯umber of ¯ew i¯fectio¯sG(C−g2) that occurred g2 days earlierN furthermoreL the expected depe¯de¯ce is through a li¯ear scali¯g factorL the icrN RNT relati¯g the time series wei¯troducea¯alogousobservatio¯modelsforicuadmisM sio¯sH0 a¯d deathsH3Z H0(C) =10(C−g0)G(C−g0)+ 40(C), HWI H3(C) =13(C−g3)G(C−g3)+ 43(C), HXI where 10(C) is the i¯fectio¯ icu admissio¯ ratio HiiarI a¯d13(C) the i¯fectio¯ fatality ratio HifrIL where the time depe¯de¯ce de¯otes the i¯fectio¯ dateN the u¯derlyi¯g i¯fectio¯sG(C) are u¯k¯ow¯L a¯d ca¯¯ot beestimatedsolelyfromthemeasureme¯ts H2,H 0,H 3Lsi¯ce a¯ absolute refere¯ce frame agai¯st which to estimate the i¯dividualtimeMshiftsa¯dgai¯factorsis¯otavailableNhowM everL if we disregard the ¯oise termsL we ca¯ relate the cases a¯d icu admissio¯s time series through H0(C) = 10(C−g0) 12(C−g0)H2(C−g02), HYI . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 31, 2021. ; https://doi.org/10.1101/2021.05.27.21257900doi: medRxiv preprint T a¯dreas wacker et alN whereg02 =g0−g2 is the average delay betwee¯ the regisM tratio¯ as a case a¯d the admissio¯ to icu H¯ot ¯ecessarily for the same perso¯IN a¯alogouslyL we have H3(C) = 13(C−g3) 12(C−g3)H2(C−g32), HQPI withg32 =g3−g2 a¯d H3(C) = 13(C−g3) 10(C−g3)H0(C−g30), HQQI withg30 =g3−g0N eqsNHY5QQIca¯beco¯ve¯ie¯tlyfittedtodataNifthetimeM depe¯de¯ce of the1Mcoefficie¯ts is ¯egligibleL we ca¯ fit the ratio_ =10/12 a¯dg02 from HYI by mi¯imisi¯g the sum of squares ls 02(_,g 02) = Õ C [H0(C)− _H 2(C−g02)]R, HQRI whichresultsi¯thetwofitti¯gparameters _,g 02Ni¯orderto obtai¯ robust estimates we alter¯atively mi¯imise the modiM fied sum of squares lsmod 02 (_,g 02) = Õ C [H0(C)√ _ − √ _H 2(C−g02) ] R . HQSI as a third optio¯L we maximize the correlatio¯ coefficie¯t A(g02) = Í C[H0(C)− 6H0][ H2(C−g02)− 6H2]√Í C[H0(C)− 6H0]R √ [H2(C−g02)− 6H2]R HQTI toobtai¯g02 a¯duse10/12 = 6H0/ 6H2Ni¯allthreeapproachesL the timesC are chose¯ such that reliable data for bothH2(C− g02) a¯dH0(C) are availableN eqsN HQPLQQI are treated i¯ the same wayL where the i¯dices0,2 are replaced by3,2 a¯d 3,0 L respectivelyL i¯ the formulae aboveN noteL that each combi¯atio¯ of i¯dices applies a di;ere¯t time i¯terval due to the reliability co¯ditio¯N RNU calibrati¯g agai¯st ra¯domized pcr test data while HY5QQI establish relative relatio¯s betwee¯ the time series H2,H 0,H 3L a 0grou¯di¯g poi¯t1 is ¯eeded to obtai¯ absolutevaluesofthetimeshifta¯dscali¯gparametersofthe observatio¯modelsHV5XINra¯domizedpcrstudiesprovide such a grou¯di¯g poi¯tL where we use the data discussed i¯ secN RNQN from the ¯umber of positive pcr test results i¯ each studyL we ca¯ estimate the prevale¯ce#positive(C) by multiplyi¯g with the populatio¯ of swede¯ a¯d dividi¯g by the ¯umber of tested perso¯sN the prevale¯ce #positive(C) depe¯ds o¯ the probability ?positive(g)tohaveapositivetestresult gdaysafterbecomi¯g i¯fectedZ ˜#positive(C) = Õ g ?positive(g) ˜G(C−g). HQUI aftertimeMaveragi¯ga¯dusi¯gagai¯theimpulsefirmodel this provides #positive(C) =)i¯tervalG(C−gpositive), HQVI where )i¯terval = Õ g ?positive(g) HQWI is the average timeMi¯terval over which a positive test result is expected a¯d gpositive = Q )i¯terval Õ g g?positive(g) HQXI istheaveragedelayafterthetimeofi¯fectio¯Nfromthedata offigNRof[X]Lweextract )i¯terval = QP.Xdaysa¯dgpositive = QRdaysN a¯other study [Y] fou¯d)i¯terval = Y.UdaysN motiM vated by these ¯umbers we use)i¯terval = QP± Qdays a¯d makethesimplifyi¯gassumptio¯ gpositive =g2Ntheresulti¯g value forg2 from HRRI agrees well withgpositive = QRdaysL extracted from [X]IN a¯alogously to HYI we fit the relatio¯ #positive(C) = )i¯terval 1 9(C−gpositive)H 9(C−gpositive+g9) HQYI where 9 = 0,3 refers to the data sets for icu a¯d deathsN weapplyallthreefitti¯grouti¯esLagai¯¯eglecti¯gthetimeM depe¯de¯ce of1 9(C)6howeverLgpositive = g2 is kept fixedN Has we o¯ly have T ¯o¯Mva¯ishi¯g data poi¯ts for#positiveL the use of a seco¯d fitMparameter ¯ext to1 9 could provide spurious resultsI 3 Results SNQ scali¯g of data the symbols o¯ the upper pa¯el of figN Q show the daily swedish¯umbersofpositivelytestedperso¯sH¯amedcasesIL perso¯s admitted to i¯te¯sive care u¯its H¯amed icuIL a¯d deathsL see secN RNQ for detailsN these data were averaged over a seve¯ day period Hli¯esI i¯ order to avoid weekday fluctuatio¯sL resulti¯g i¯ the fu¯ctio¯sH2(C) HcasesILH0(C) Hicu admissio¯IL a¯dH3(C) HdeathsIL which are the mai¯ dataMsets used throughout this articleN here the timeC is chose¯ as the ce¯tral day of the averagi¯g periodN asdescribedi¯secNRNTLwedetermi¯edtheratios10/12L 13/12L a¯d13/10 as well as the correspo¯di¯g delays by the followi¯g procedureN for the fitti¯gL we disregarded the death data after Q february RPRQL as they are a;ected by the vacci¯atio¯ program a¯d thus a timeMdepe¯de¯ce for13(C) is expected hereN we also disregard the case data before QX ju¯eRPRPLwhe¯testi¯gbecameavailabletoallperso¯swith symptoms i¯ swede¯L see secN RNQN fi¯allyL we co¯sider the dataforcasesa¯dicuadmissio¯fromthelastthreedaysa¯d . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 31, 2021. ; https://doi.org/10.1101/2021.05.27.21257900doi: medRxiv preprint estimati¯g the sarsMcovMR i¯fected populatio¯ fractio¯ a¯d the i¯fectio¯MtoMfatality ratio U 03-01 05-01 07-01 09-01 11-01 01-01 03-01 05-01 date 0 2000 4000 6000 8000 10000 12000daily number cases 100*ICU 100*death 03-01 05-01 07-01 09-01 11-01 01-01 03-01 05-01 date 0 2000 4000 6000 8000daily number cases 193*ICU(5 days later) 68*deaths(13 days later) prevalence/16 Fig. 1 Upper panel: raw data HsymbolsI used i¯ this studyN Hthe ¯umbersoficuadmissio¯sa¯ddeathshavebee¯multipliedbyQPPto provide comparable ¯umbersNI the li¯es show the respective averages over a W day periodL where the ce¯tral day of the i¯terval was used i¯ the abscissaNLower panel:W day averaged data from upper pa¯el scaled by di;ere¯t factors as give¯ i¯ the lege¯dN the icuMdata are shiftedbyUdaystothelefta¯dthedeathMdataareshiftedbyQSdaysto the leftL so that they esse¯tially fall o¯ o¯e curveN dotted li¯es i¯dicate sectio¯s of data that are co¯sidered as i¯completeN the ora¯ge error bars provide scaled VX E Ho¯e ¯ormal sta¯dard deviatio¯I co¯fide¯ce levelsofra¯domizedpcrstudydataNallscali¯gfactorsa¯ddelaysare take¯ from the ls values i¯ tabN QN thedeathdatafromthelastthreeweeksasu¯reliableLaslate reportsarecommo¯overtheseperiodsNtheresultsaregive¯ i¯tabNQNwefi¯dthatallthreeapproachesprovideide¯tical timedelaysLwhichweregardasparticularlyreliableNalsothe fractio¯sbetwee¯theratiosagreefairlywellwithdeviatio¯s farbelowQPENi¯thefollowi¯gweapplythevaluesfromthe leastsquaresmethodlsLseesecNRNTLbutwe¯oteLthat¯o¯e of our results depe¯ds o¯ this choiceN the relatio¯s10/12· 13/10 =13/12 a¯dg02+g30 =g32 holdo¯lyapproximately as di;ere¯t time i¯tervals are used i¯ the fitti¯g due to the exclusio¯ of death data after Q february RPRQ a¯d case data before QX ju¯e RPRP addressed aboveN usi¯gthesescali¯gfactors 10/12 = Q/QYSa¯d13/12 = Q/VXas well as the respective time delaysL the lower pa¯el of figN Q shows that all three curves agree very well over theseco¯dwaveofnovemberRPRP5ja¯uaryRPRQNtheicu a¯d death curve also show a similar behavior at the first parameter ls lsmod corrN coe;N 10/12 PNPPUR PNPPUQ PNPPUQ g02 U U U 13/12 PNPQTW PNPQTV PNPQTQ g32 QS QS QS 13/10 RNWP RNVU RNUY g30 X X X )i¯terval 10 10 12 QUNY QVNP QUNR )i¯terval 13 13 12 QUNX QUNY QTNW Table 1 results for di;ere¯t fitti¯g procedures detailed i¯ secN RNT for ratios a¯d time delaysN i¯ the last two rows the fitted values for )i¯terval/10 a¯d)i¯terval/13 were multiplied with the factors from row Q a¯d S respectively to obtai¯ estimates for)i¯terval/12N wavemarch5mayRPRPLalbeittheratiobetwee¯icuadmisM sio¯ a¯d deaths appears to be slightly higher hereN the case ¯umbers are much lower due to the limited testi¯g before midMjulyN for the third wave march5may RPRQL the icu a¯d case curve agree very well Hthe dip i¯ cases arou¯d Q april may be attributed to decreased testi¯g arou¯d easterIN we also see that the death curve shows much lower values from arou¯d midMja¯uary RPRQL which coi¯cides with the start of the vacci¯atio¯ program i¯ swede¯ at the e¯d of RPRPN SNR compari¯g with ra¯domized pcr a¯d a¯tibody studies we also fitted the data from the V ra¯domized pcr studies to the icu a¯d death data Hhere we omitted the sixth studyL as the fatality was sig¯ifica¯tly reduced i¯ RPRQL most likely due to the vacci¯atio¯ programI a¯d fou¯d almost ide¯tical resultsi¯bothcasesLseetabNQNthusLtheprevale¯ceofpcrM detectable sarsMcovR i¯fectio¯s is about QV times higher tha¯ the ¯umber of cases Has reco¯structed by shifti¯g a¯d scali¯gthedeatha¯dicudataILseethelowerpa¯eloffigNQN as a¯ i¯fected perso¯ ca¯ be detected over a¯ average time i¯terval)i¯tervalL but is o¯ly registered o¯ce as a case with probability 12L this implies)i¯terval/12 ≈ QU.V± P.YNsee secN RNU for detailsN here we used the average of all values provided i¯ the two lowest rows of tabN Q a¯d used the maximal deviatio¯ as a¯ estimate for the errorN the time i¯tervalLa¯i¯fectedperso¯istestedpositiveseemstobeless well k¯ow¯N usi¯g)i¯terval = QP± QdaysL see secN RNUL this provides the icr based o¯ the prevale¯ce studies 1prevale¯ce 2 ≈ P.VT± P.QP HRPI we ¯ote that the sampli¯gMbias corrected data for the prevaM le¯ce results i¯ a slightly larger value1prevale¯ce_corrected 2 ≈ P.W± P.QQwith overlappi¯g error margi¯alN . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 31, 2021. ; https://doi.org/10.1101/2021.05.27.21257900doi: medRxiv preprint V a¯dreas wacker et alN 03-01 05-01 07-01 09-01 11-01 01-01 03-01 05-01 date 0 500 1000 1500 2000 2500total number in thousand with antibodies from vaccinations cases 193*ICU(5 days later) 68*deaths(13 days later) cases+364299 antibody (health centers) antibody (blood donors) Fig. 2 accumulated ¯umber of cases at a give¯ dateL where we added theestimatesbasedo¯scali¯gthedeatha¯dicudataNthedashedblue li¯e provides the case data where we added a¯ estimate for ¯umber of cases missed due to limited testi¯g before midMju¯eN the symbols with error bars show the results of a¯tibody tests performed for blood do¯ors a¯d blood samples from health ce¯ters [V]L where the fractio¯ of positive tests was multiplied by swede¯Gs populatio¯N note that the ¯umber of perso¯s with a¯tibodies i¯ march RPRQ i¯clude detected

from vacci¯atio¯sN thelowerpa¯eloffigNQshowsLthatthecasedataagree with the prevale¯ces divided by QV after testi¯g become widely accessible arou¯d mid of ju¯e RPRPN as both the scaledicua¯ddeathdataagreewellwiththeprevale¯cesat alltimesLweca¯takethesecurvesforestimati¯gthe¯umber ofcasesbeforemidMjulyLwherethecasedataare¯otreliable due to limited testi¯gN we fi¯d a clear plateau of total cases i¯ july5november RPRPLastherewerefew¯ewcasesi¯thisperiodNtheaverage plateau value Hat QN septI was TUP PPP Hreco¯structedI cases L with a¯ u¯certai¯ty Hbased o¯ the icu a¯d death dataI of about UU PPPN a¯tibody tests for di;ere¯t groups provide estimates of WPP PPP positive perso¯s is swede¯ both at the begi¯¯i¯ga¯dthee¯d oftheplateauLseefigNRNthisresults i¯12/?a¯tibody = P.VT± P.PXN the probability?a¯tibody to develop detectable a¯tibodies has bee¯ fou¯d to be above YP E [QP]L [QQ]N as it should ¯ot exceed QPP EL we assume ?a¯tibody = P.YU± P.PUL a¯d fi¯d 1a¯tibody 2 ≈ P.VQ± P.QQ HRQI this agrees very well with the di;ere¯t estimate HRPIL albeitdi;ere¯tstatisticala¯dsystematicerrorsHi¯particular the values of)i¯tervala¯d?a¯tibodyI e¯ter both ways to calcuM late12N thus we co¯sider12 = P.VSas a good estimate for the icr with the aware¯ess that a QP E error is ¯otu¯likelyN we ¯oteL that a relatively large ¯umber of perso¯s with a¯tibodies was fou¯d i¯ the study at the begi¯¯i¯g of march RPRQN hereL o¯e has to take i¯to accou¯t that a part of the a¯tibodies detected results from vacci¯atio¯sN at this time about WPP PPP perso¯s had bee¯ vacci¯ated i¯ swede¯ a¯d 03-01 05-01 07-01 09-01 11-01 01-01 03-01 infection date 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016estimated IIR and IFR 21 days average bc*ICU(17 days later)/case(12 days later) bc*death(25 days later)/case(12 days later) bc/193*death(25 days later)/ICU(17 days later) Fig.3 estimatedvaluesfortheiiar 10(C) Hgree¯Ia¯dtheifr 13(C) HredLmage¯taIbasedo¯HYLQPLQQINweassumeaco¯sta¯ticr12 = P.VS L which appears reliable for cases after QX ju¯e Hdotted curves i¯dicate thatearliercasesdataareappliedINthehorizo¯taldashedli¯esprovide the average valuesN a majority of them should have developed a¯tibodiesL whe¯ the data was collectedN SNS reco¯structi¯g the ifr a¯d the iiar assumi¯g the icr 12 = P.VSfor the time after ju¯e QXthL RPRPL we ca¯ estimate the variables10(C) a¯d13(C) from HYLQPIN at first we ¯eed the absolute delaysN here we rely o¯ thedataforicuadmissio¯Lwhichi¯averageoccursaboutQQ days after the o¯set of symptoms accordi¯g to the swedish i¯te¯sive care registry [QR]N furthermoreL it is k¯ow¯L that it takes about V days from the times of i¯fectio¯ to develop symptoms [QS]5[QU]N thus we useg0 = QWdays i¯ the folM lowi¯gN based o¯ the values o¯ tabN Q we get g2 = QRdaysLg0 = QWdaysL a¯dg3 = RUdaysN HRRI i¯figNSweplotthetimeMdepe¯de¯ceofthei¯fectio¯MtoM icuadmissio¯ratioHiiarI 10(C)byagree¯li¯eo¯thebasis of HYIN here we used RQ day averages to restrict fluctuatio¯sN we fi¯d that10(C)≈ P.STE is close to its averageL which co¯firms the quality of the scali¯gN the larger bump arou¯d late july occurs withi¯ a ra¯ge with small ¯umbers of icu admissio¯sHaverageofQNVi¯augustRPRPILsothatstatistical fluctuatio¯s ca¯¯ot be excluded hereN similarlyLweobtai¯theifr 13(C)fromHQPIasshow¯by theredli¯ei¯figNSNo¯averageLwehavePNWTEu¯tile¯dof RPRP Hbefore vacci¯atio¯s started to show e;ectsIL but there are pro¯ou¯ced cha¯ges over timeN for i¯fectio¯s i¯ august a¯dseptemberLtheifrismuchlowerNfori¯fectio¯safterQ ja¯uary RPRQL we see a disti¯ct decli¯e i¯ the ifr reachi¯g values of PNR E for i¯fectio¯s arou¯d Q april RPRQN the samebehaviorfortheifrca¯beobtai¯edfromfromHQQIas show¯bythemage¯tali¯ei¯figNSLwhich¯owextrapolates . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 31, 2021. ; https://doi.org/10.1101/2021.05.27.21257900doi: medRxiv preprint estimati¯g the sarsMcovMR i¯fected populatio¯ fractio¯ a¯d the i¯fectio¯MtoMfatality ratio W totimesbeforemidMju¯eu¯dertheassumptio¯Lthattheiiar remai¯ed esse¯tially co¯sta¯t i¯ this period as wellN 4 Discussion usi¯gope¯lyaccessibledatafromtheswedishpublichealth age¯cy a¯d the swedish i¯te¯sive care registryL our apM proach provides estimates for the i¯fectio¯MtoMfatality ratio HifrIL i¯fectio¯MtoMcase ratio HicrIL a¯d i¯fectio¯MtoMicu admissio¯ ratio HiiarIN we fi¯d that data for daily casesL daily icu admissio¯L a¯d daily deaths of i¯dividuals with co¯firmed i¯fectio¯ fall o¯ esse¯tially a si¯gle curve based o¯afirmodelwithadeltafilterfu¯ctio¯a¯dtimeMi¯varia¯t fit parametersL see the lower pa¯el of figN QN there are o¯ly two major wellMu¯derstood exceptio¯sZ HiI cases exhibit a poor match before midMju¯e RPRPL whe¯ free pcr testi¯g became broadly available i¯ swede¯ for all perso¯s with symptomsN HiiI there is a sharp relative decrease i¯ deaths coi¯cidi¯gwiththestartofthe¯atio¯alvacci¯atio¯program arou¯d the tur¯ of the year RPRP5RPRQN basedo¯thesefi¯di¯gswedemo¯stratethattheapproach ca¯ be used to retrospectively estimate the cases time seM ries prior to july RPRPL that would have bee¯ observable withabroadpcrtesti¯gprogrami¯placeNfurthermoreLby i¯corporati¯g data from six ra¯domized pcr studies co¯M ducted by the swedish public health age¯cy a¯d data for theprevale¯ceofa¯tibodiesLweobtai¯a¯absolutevaluefor the i¯fectio¯MtoMcase ratio icr ofP.VS± P.PWfor the time whe¯testi¯giseasilyavailabletoperso¯swithsymptomsi¯ swede¯NthisvalueislargerHbutcompatiblewithi¯itserrorI tha¯ the value PNUV obtai¯ed i¯ a study for icela¯d [QP]N it mea¯s that approximately SW E of all i¯fected remai¯ed u¯M detectedascasesNregardi¯gsystematicerrorsLtheicrvalue obtai¯ed from our study is reduced if both the time i¯terval for positive testi¯gL)i¯tervalL a¯d probability to develop meaM surable a¯tibodies after a¯ i¯fectio¯L?a¯tibodyL tur¯ out to be much less tha¯ QP days a¯d PNYUL respectivelyN o¯ the other ha¯dLlargervaluesarelesslikelyas ?a¯tibody = P.YUwaschoM se¯ close to its absolute maximum value Q i¯ the a¯alysisN howeverL a sampli¯g bias or falseMpositive tests may allow for deviatio¯sL which we ca¯¯ot judge hereN figNRi¯dicatesthatthetotal¯umberofcaseswouldhave bee¯arou¯dXRPPPPatthee¯dofRPRPLiftesti¯gi¯thefirst half of RPRP had bee¯ as available as i¯ the seco¯d halfN witha¯icrofPNVSLthisprovidesaboutQNSmillio¯i¯fected perso¯s i¯ swede¯ i¯ RPRPN this correspo¯ds to QSE of the populatio¯ a¯d is far below values required to reach herd immu¯ityN basedo¯theicrofPNVSLweobtai¯thei¯fectio¯toicu admissio¯ ratio iiar of PNSTE a¯d a¯ average i¯fectio¯ to fatality ratio ifr of PNWT E Hbefore the start of the vacciM ¯atio¯sIL see figN SN our value for the ifr is comparable to earlierstudies[QV]5[QX]Nnotethatpossiblesystematicerrors i¯ icrL as addressed aboveL a;ect the iiar a¯d ifr proporM tio¯allyN our a¯alysis shows that the iiar is rather co¯sta¯t i¯timeLatleastforthetimeafteraugustRPRPNi¯co¯trastLthe ifr varies much stro¯ger over timeN we attribute its decli¯e starti¯g for i¯fectio¯s arou¯d Q ja¯uary RPRQ to successful vacci¯atio¯ of elderly perso¯sN they domi¯ate the mortality i¯covidMQYLbutarelessreleva¯tforicuadmissio¯NhowM everL we lack expla¯atio¯ for the reduced ifr for i¯fectio¯s from midMjuly to midMseptember of RPRPN the close correlatio¯ betwee¯ casesL icu admissio¯sL a¯d deaths6dow¯ to a li¯ear scali¯g a¯d timeMshift o¯ce selfMtest had bee¯ made broadly available6warra¯ts furM ther i¯vestigatio¯N it could be explai¯ed through HiI timeM i¯varia¯ce of the icrL the iiarL the ifrL Hthe dip i¯ ifr i¯ summer is ¯ot visible i¯ figN QHbI due to the small ¯umM bersIa¯dthetemporaldistributio¯sdescribi¯gtheassociated flows[ HiiI variatio¯s i¯ the me¯tio¯ed e¯tities that have esM se¯tiallyca¯celledeachotherthroughoutRPRP[HiiiIexter¯al co¯fou¯ders providi¯g this ca¯celi¯g e;ectN case HiI would implythatthehealthcaresystemGsabilitytosavecovidMQY patie¯ts a¯d the impact from di;ere¯t virus mutatio¯s has ¯ot cha¯ged markedlyN case HiiI would be surprisi¯g for the time from august RPRPL where the ratio betwee¯ cases a¯d icu admissio¯ is largely co¯sta¯t i¯ timeL see figN SN thus it is most likely that12(C) a¯d10(C) are both co¯sta¯t i¯ thisra¯geNhoweverLforthefirsthalfofRPRPLthi¯gsareless obviousN the variatio¯s observed i¯ the mage¯ta curve may

from reductio¯s i¯10(C) a¯d13(C) HiNeNimproveme¯t i¯ healthcareI occurri¯g at di;ere¯t timesN case HiiiI would alsobe¯oteworthysi¯cethegroupsofdeceaseda¯dperso¯s admittedtoicucarehavemargi¯aloverlapwitheachotherN thestudiedtimeseriesalo¯eare¯otsufficie¯ttomapout the causatio¯ of the observed correlatio¯sL thus disti¯guishM i¯g betwee¯ HiI a¯d combi¯atio¯s of HiiI a¯d HiiiIN howeverL a¯ u¯dersta¯di¯g of the u¯derlyi¯g mecha¯ism could be obM tai¯ed by retrospectively tracki¯g i¯dividual traces co¯¯ectM i¯g the co¯sidered time series HeNgNperso¯s who have tested positiveL bee¯ admitted to a¯ icu or died with covidMQYIN importa¯tlyL if the correlatio¯ ca¯ be u¯derstood a¯d show¯ ¯ot to be coi¯cide¯talL it could co¯stitute the basis for a¯ accurate Q5R week predictor of the icu dema¯dN 5 Conclusion the summarizi¯g co¯clusio¯ from our observatio¯s is that importa¯t i¯sight i¯to ¯umerous aspects of a¯ o¯goi¯g epiM demic ca¯ be obtai¯ed by co¯sideri¯g the scali¯g betwee¯ di;ere¯t time seriesL where time shifts are crucialN this is based o¯ a¯ fir model with a delta filter fu¯ctio¯L which is show¯toworkwellNwedemo¯stratedthisbyreco¯structi¯g the daily ¯umber of cases for the first half year of RPRP peM riodsL where testi¯g was limited i¯ swede¯L a¯d extracti¯g time variatio¯s i¯ the i¯fectio¯ fatality ratioN . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 31, 2021. ; https://doi.org/10.1101/2021.05.27.21257900doi: medRxiv preprint X a¯dreas wacker et alN

this work was partially fu¯ded by the elliit strategic research areaN Conflict of interest the authors declare ¯o co¯flict of i¯terestN A Quality of the delta-impulse model after averagi¯g to remove weekly fluctuatio¯s a¯d ¯eglecti¯g fluctuaM tio¯s42(C)L HQI provides H2(C) = ∞Õ g=P ?2(C− g, g) G(C− g). HRSI while we do ¯ot k¯ow much about?2(C , g)L we make the reaso¯able assumptio¯ that it has a si¯gle peak a¯d does decay rather quick for largedelays gNi¯thiscaseitisreaso¯ablyreprese¯tedbyitsaverage 6g a¯d sta¯dard deviatio¯f defi¯ed as 6g(C) = Q 12(C) =Õ g=P g ?2(C , g) , HRTI fR(C) = Q 12(C) =Õ g=P ( g− 6g(C)) R?2(C , g) , HRUI where we used the ¯ormalisatio¯ from HSIN assumi¯gL thatG(C) is a smooth fu¯ctio¯L which appears reaso¯able after WMdays averagi¯gL we may approximate by a seco¯d order taylor expa¯sio¯ G(C− g)≈ G(C− g2)− G′(C− g2)( g− g2)+ Q RG′′(C− g2)( g− g2)R, HRVI whereLforgive¯CLtheaveragedelay g2 satisfies g2 = 6g(C− g2)Nthe¯ we obtai¯ from HRSI that H2(C)≈ ∞Õ g=P ?2(C− g, g) [ G(C− g2)− G′(C− g2)( g− g2) + Q RG′′(C− g2)( g− g2)R ] . assumi¯g ?2(C− g, g)≈ ?2(C− g2 , g)L as the detectio¯ probability should ¯ot cha¯ge withi¯ a timeMscale off for co¯sta¯t delaygL we fi¯d H2(C)≈ 12(C− g2) G(C− g2)+ f(C− g2)R R 12(C− g2) G′′(C− g2). HRWI thefirsttermisjustourdeltaMimpulsemodelLseeHTILwhiletheseco¯d term provides a correctio¯ less tha¯ U E if f(C− g2)R|G′′(C− g2)| G(C− g2) < P.Q. HRXI if the i¯fectio¯s show a¯ expo¯e¯tial behaviorG(C) = GP4A CL this provides fA < P.SQVor f < P.TVCdoubli¯g with the doubli¯g time Cdoubli¯g = l¯ R/AN the last relatio¯ was stated i¯ refN [W] for the apM plicability of the deltaMrespo¯se for expo¯e¯tial evolutio¯sN here we ge¯eralised this to arbitrary evolutio¯sG(C)L whereCdoubli¯g/l¯ Ris reM placed by the square root of the i¯verse relative seco¯d derivativeN Nomenclature acronyms fir fi¯ite impulse respo¯se [model] icr i¯fectio¯MtoMcase ratio icu i¯te¯sive care u¯it ifr i¯fectio¯MtoMfatality ratio iiar i¯fectio¯MtoMicu admissio¯ ratio ls leastMsquares [method] pcr polymerace chai¯ reactio¯ subscripts and modifiers 0 i¯fectio¯MtoMicu admissio¯ 2 case Hdetected i¯fectio¯I 3 i¯fectio¯MtoMfatality HdeathI ˜ weekly ce¯tral movi¯g average variables and symbols X diracGs delta distributio¯ _ iiarOicr f sta¯dard deviatio¯ g time delay Hu¯itZdaysI 1 gai¯ parameter 4 error or residual time series # number of i¯dividuals ? probability C time i¯dex Hu¯itZdaysI G new i¯fectio¯ time series H¯ot directly observableI H daily time series Hobservatio¯I

[Q] kN solteszL fN gustafsso¯L tN timpkaLet alNL 0the e;ect of i¯terve¯tio¯so¯covidMQYL1natureLvolNUXXLeRV5eRXLRPRPN OdNscOoNscOiNscZ10.1038/s41586-020-2405-7N [R] pN a¯tsaklis a¯d aN michelLlinear systemsN bosto¯L maZ birkhauserL RPPUN [S] swedish public health age¯cyL bekräftade fall i sverige – daglig uppdateringL RPRQN [o¯li¯e]N availableZhttps: //www.folkhalsomyndigheten.se/smittskydd- beredskap/utbrott/aktuella-utbrott/covid- 19/statistik-och-analyser/bekraftade-fall-i- sverige/ Hvisited o¯ may QTL RPRQIN [T] swedish public health age¯cyL undersökningar avseende förekomst av sjukdomL RPRQN [o¯li¯e]N availableZhttps: //www.folkhalsomyndigheten.se/smittskydd- beredskap/utbrott/aktuella-utbrott/covid- 19/statistik-och-analyser/undersokningar-och- datainsamlingar/forekomst-av-sjukdom/ Hvisited o¯ may QTL RPRQIN [U] kN stråli¯L eN wahlströmL sN waltherLet alNL 0mortality tre¯ds amo¯g hospitalised covidMQY patie¯ts i¯ swede¯Z a ¯atio¯M wide observatio¯al cohort studyL1the lancet regional health M europeL volN TL pN QPPPUTL RPRQNOdNscOoNscOiNscZ10.1016/j.lanepe. 2021.100054N [V] swedish public health age¯cyL undersökningar avseende genomgången infektionL RPRQN [o¯li¯e]N availableZhttps: //www.folkhalsomyndigheten.se/smittskydd- beredskap/utbrott/aktuella-utbrott/covid- 19/statistik-och-analyser/undersokningar-och- datainsamlingar/genomgangen-infektion/ Hvisited o¯ may QTL RPRQIN [W] jN walli¯ga a¯d mN lipsitchL 0how ge¯eratio¯ i¯tervals shape the relatio¯ship betwee¯ growth rates a¯d reproductive ¯umM bersL1proceedingsoftheroyalsocietyb:biologicalsciencesL volN RWTL ¯oN QVPYL ppN UYY5VPTL RPPWNOdNscOoNscOiNscZ10.1098/rspb. 2006.3754N . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 31, 2021. ; https://doi.org/10.1101/2021.05.27.21257900doi: medRxiv preprint estimati¯g the sarsMcovMR i¯fected populatio¯ fractio¯ a¯d the i¯fectio¯MtoMfatality ratio Y [X] lN kucirkaL sN lauerL oN laeye¯deckerLet alNL 0variatio¯ i¯ falseM¯egative rate of reverse tra¯scriptase polymerase chai¯ reactio¯5basedsarsMcovMRtestsbytimesi¯ceexposureL1 anM nals of internal medicineL volN QWSL ¯oN TL ppN RVR5RVWL RPRPN OdNscOoNscOiNscZ10.7326/M20-1495N [Y] zN huL cN so¯gL cN xuLet alNL 0cli¯ical characteristics of RT asymptomatic i¯fectio¯s with covidMQY scree¯ed amo¯g close co¯tacts i¯ na¯ji¯gL chi¯aL1science china life sciM encesL volN VSL ppN WPV5WQQL RPRPNOdNscOoNscOiNscZ10.1007/s11427- 020-1661-4N [QP] dN gudbjartsso¯L gN norddahlL pN melstedLet alNL 0humoral immu¯e respo¯se to sarsMcovMR i¯ icela¯dL1new england journal of medicineL volN SXSL ¯oN QXL ppN QWRT5QWSTL RPRPN OdNscOoNscOiNscZ10.1056/NEJMoa2026116N [QQ] jN da¯L jN mateusL yN katoLet alNL 0immu¯ological memory to sarsMcovMR assessed for up to X mo¯ths after i¯fectio¯L1 scienceL volN SWQL ¯oN VURYL RPRQNOdNscOoNscOiNscZ10.1126/science. abf4063N [QR] the swedish i¯te¯sive care registryL RPRQN [o¯li¯e]N availableZ https://www.icuregswe.org/data-- resultat/COVID-19-i-svensk-intensivvard HvisM ited o¯ may QSL RPRQIN [QS] cN mcaloo¯L aN colli¯sL kN hu¯tLet alNL 0i¯cubatio¯ period of covidMQYZ a rapid systematic review a¯d metaMa¯alysis of observatio¯al researchL1bmj openL volN QPL ¯oN XL RPRPNOdNscOoNscOiNscZ 10.1136/bmjopen-2020-039652N [QT] sN lauerL kN gra¯tzL qN biLet alNL 0the i¯cubatio¯ period of coro¯avirus disease RPQY HcovidMQYI from publicly reported co¯firmed casesZ estimatio¯ a¯d applicatio¯L1annals of interM nal medicineL volN QWRL ¯oN YL ppN UWW5UXRL RPRPNOdNscOoNscOiNscZ10. 7326/M20-0504N [QU] jN qi¯L cN youL qN li¯Let alNL 0estimatio¯ of i¯cubatio¯ period distributio¯ of covidMQY usi¯g disease o¯set forward timeZ a ¯ovelcrossMsectio¯ala¯dforwardfollowMupstudyL1scienceadM vancesLvolNVL¯oNSSLRPRPNOdNscOoNscOiNscZ10.1126/sciadv.abc1202N [QV] rNverityLlNokellLiNdorigattiLetalNL0estimatesoftheseverity of coro¯avirus disease RPQYZ a modelMbased a¯alysisL1the lancet infectious diseasesL volN RPL ppN VVY5VWWL RPRPNOdNscOoNscOiNscZ 10.1016/S1473-3099(20)30243-7N [QW] aN levi¯L wN ha¯agL nN owusuMboaiteyLet alNL 0assessi¯g the age specificity of i¯fectio¯ fatality rates for covidMQYZ sysM tematic reviewL metaMa¯alysisL a¯d public policy implicatio¯sL1 european journal of epidemiologyL volN SUL ppN QQRS5QQSXL RPRPNOdNscOoNscOiNscZ10.1007/s10654-020-00698-1N [QX] swedish public health age¯cyL the infection fatality rate of covidM19 in stockholmL RPRPN [o¯li¯e]N availableZhttps: //www.folkhalsomyndigheten.se/contentassets/ 53c0dc391be54f5d959ead9131edb771/infection- fatality-rate-covid-19-stockholm-technical- report.pdf Hvisited o¯ may QTL RPRQIN . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 31, 2021. ; https://doi.org/10.1101/2021.05.27.21257900doi: medRxiv preprint