COVID-19 Case Rates in the UK: Modelling Uncertainties as Lockdown Lifts

Abstract

Background: The UK was one of the countries worst affected by the COVID-19 pandemic in Europe. A strict lockdown from early 2021 combined with an aggressive vaccination programme enabled a gradual easing of lockdown measures to be introduced whilst both deaths and reported case numbers reduced to less than 3% of their peak. The emergence of the Delta variant in April 2021 has reversed this trend, and the UK is once again experiencing surging cases, albeit with reduced average severity due to the success of the vaccination rollout. This study presents the results of a modelling exercise which simulates the progression of the pandemic in the UK through projection of daily case numbers as lockdown lifts. Methods: A simulation model based on the Susceptible-Exposed-Infected-Recovered structure was built. A timeline of UK lockdown measures was used to simulate the changing restrictions. The model was tailored for the UK, with some values set based on research and others obtained through calibration against 16 months of historical data. Results: The model projects that if lockdown restrictions are lifted in July 2021, UK COVID-19 cases will peak at hundreds of thousands daily in most viable scenarios, reducing in late 2021 as immunity acquired through both vaccination and infection reduces the susceptible population percentage. Further lockdown measures can be used to reduce daily cases. Other than the ever-present threat of the emergence of new variants, the most significant unknown factors affecting the profile of the pandemic in the UK are the length and strength of immunity, with daily peak cases over 50% higher if immunity lasts 8 months compared to 12 months. Another significant factor is the percentage of unreported cases. The reduced case severity associated with vaccination may lead to a higher proportion of unreported mild or asymptomatic cases, meaning that unmanaged infections resulting from unknown cases will continue to be a major source of infection. Conclusions: Further research into the length and strength of both recovered and vaccinated COVID-19 immunity is critical to delivering more accurate projections from models, thus enabling more finely tuned policy decisions. The model presented in this article, whilst by no means perfect, aims to contribute to greater transparency of the modelling process, which can only increase trust between policy makers, journalists and the general public.

Full text

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Systems2021,9,60.https://doi.org/10.3390/systems9030060www.mdpi.com/journal/systems
Article
COVID‐19CaseRatesintheUK:ModellingUncertainties
asLockdownLifts
ClaireBrereton
1,
*andMatteoPedercini
2
1
ChildHealthResearchCentre,UniversityofQueensland,Brisbane,QLD4101,Australia
2
MillenniumInstitute,2200PennsylvaniaAveNW,Washington,DC20037,USA;
mp@millennium‐institute.org
*Correspondence:claire.brereton@uq.edu.au;Tel.:+61‐419‐901‐107
Abstract:Background:TheUKwasoneofthecountriesworstaffectedbytheCOVID‐19pandemic
inEurope.Astrictlockdownfromearly2021combinedwithanaggressivevaccinationprogramme
enabledagradualeasingoflockdownmeasurestobeintroducedwhilstbothdeathsandreported
casenumbersreducedtolessthan3%oftheirpeak.TheemergenceoftheDeltavariantinApril
2021hasreversedthistrend,andtheUKisonceagainexperiencingsurgingcases,albeitwithre‐
ducedaverageseverityduetothesuccessofthevaccinationrollout.Thisstudypresentstheresults
ofamodellingexercisewhichsimulatestheprogressionofthepandemicintheUKthroughprojec‐
tionofdailycasenumbersaslockdownlifts.Methods:AsimulationmodelbasedontheSusceptible‐
Exposed‐Infected‐Recoveredstructurewasbuilt.AtimelineofUKlockdownmeasureswasusedto
simulatethechangingrestrictions.ThemodelwastailoredfortheUK,withsomevaluessetbased
onresearchandothersobtainedthroughcalibrationagainst16monthsofhistoricaldata.Results:
ThemodelprojectsthatiflockdownrestrictionsareliftedinJuly2021,UKCOVID‐19caseswill
peakathundredsofthousandsdailyinmostviablescenarios,reducinginlate2021asimmunity
acquiredthroughbothvaccinationandinfectionreducesthesusceptiblepopulationpercentage.
Furtherlockdownmeasurescanbeusedtoreducedailycases.Otherthantheever‐presentthreat
oftheemergenceofnewvariants,themostsignificantunknownfactorsaffectingtheprofileofthe
pandemicintheUKarethelengthandstrengthofimmunity,withdailypeakcasesover50%higher
ifimmunitylasts8monthscomparedto12months.Anothersignificantfactoristhepercentageof
unreportedcases.Thereducedcaseseverityassociatedwithvaccinationmayleadtoahigherpro‐
portionofunreportedmildorasymptomaticcases,meaningthatunmanagedinfectionsresulting
fromunknowncaseswillcontinuetobeamajorsourceofinfection.Conclusions:Furtherresearch
intothelengthandstrengthofbothrecoveredandvaccinatedCOVID‐19immunityiscriticalto
deliveringmoreaccurateprojectionsfrommodels,thusenablingmorefinelytunedpolicydecisions.
Themodelpresentedinthisarticle,whilstbynomeansperfect,aimstocontributetogreatertrans‐
parencyofthemodellingprocess,whichcanonlyincreasetrustbetweenpolicymakers,journalists
andthegeneralpublic.
Keywords:COVID‐19;UK;vaccination;immunity;policy;systemdynamics;modelling;uncertainty

1.Introduction
TheCOVID‐19pandemicisanunprecedentedglobalcrisis.Theunusualnatureof
theSARS‐CoV‐2virus,whichcanbedeadlyforonepersonwhilsthavingnosymptoms
foranother,wasmisunderstoodbyscientistsandpolicymakersduringtheearlystages
ofthepandemic,leadingtounderestimationofcasenumbersandfocusoncontrolof
symptomaticinfections[1].Modellingstudies[2,3]andresearchontheprevalenceof
COVID‐19antibodiesintheUKpopulation[4]indicatedearlyonthatconfirmedcases
werelessthanhalfoftrueinfectionestimates,andthisrealityisreflectedinglobal
Citation:Brereton,C.;Pedercini,M.
COVID‐19CaseRatesintheUK:
ModellingUncertaintiesas
LockdownLifts.Systems2021,9,60.
https://doi.org/10.3390/
systems9030060
AcademicEditors:OzSahin
andRussellRichards
Received:10May2021
Accepted:21July2021
Published:6August2021
Publisher’sNote:MDPIstaysneu‐
tralwithregardtojurisdictional
claimsinpublishedmapsandinstitu‐
tionalaffiliations.

Copyright:©2021bytheauthors.Li‐
censeeMDPI,Basel,Switzerland.
Thisarticleisanopenaccessarticle
distributedunderthetermsandcon‐
ditionsoftheCreativeCommonsAt‐
tribution(CCBY)license(http://crea‐
tivecommons.org/licenses/by/4.0/).

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pandemicplanningguidance[5]andinthecontinuinguseofmeasuressuchaslock‐
downs,whichrestrictsocialcontactirrespectiveofknowninfectionstatusacrossanentire
population.
TheUnitedKingdom(UK)wasoneofthecountriesworstaffectedbyCOVID‐19in
thedevelopedworld,characterizedbyaslowinitialresponse,lackofbordercontrols,
changingregionalguidanceandeaseofmovementbetweenregions[6].TheUKismade
upoffourcountries—England,Scotland,WalesandNorthernIreland,eachwiththeau‐
tonomytoestablishtheirownCOVID‐19controls—butas84%ofthepopulationresides
inEngland,theprofileofthepandemicinEnglandandthemeasurestakentherearethe
mostsignificantdriveroftheUK’sCOVID‐19statistics.Theescalatingnumberofcases
anddeathsintheUKledtotheirbeingthefirstcountrytogiveauthorisationforemer‐
gencyuseofthePfizer/BioNTech(PB)vaccine.Thevaccinationprogrammestartedon8
December2020andcommittedfundsforaninitial30milliondoses[7].TheAstraZeneca
(AZ)vaccinewasauthorisedonthesamebasisforrolloutcommencing4January2021,
with100milliondosesordered.Thesevaccinesdeliveredthecapabilitytoimmunise50
millionpeople,effectivelycoveringtheentireeligiblepopulationoftheUKfortwodoses
each[8].ByendJune2021,78millionvaccinationshadbeenadministered,with33million
peoplefullyvaccinated.TheModernavaccinewasalsoapprovedbytheUKGovernment
[9],andinmid‐April2021,itstartedrollingouttounder30yearoldsasanalternativeto
AZ.
Massvaccinationhastwomainobjectives:toprotectindividualsfromdeathandse‐
vereillnessandtoincreasethenumberofimmuneindividualstothepointwhereenough
peopleareprotectedfromthevirustoprotectthepopulationasawhole(herdimmunity).
Forbothvaccinatedandrecoveredindividuals,thelongevityofprotectionfrominfection
andthedegreeofprotectionconferredarestilluncertain.Thelevelofpopulationprotec‐
tionrequiredforherdimmunityintheUK,oranyothercountry,hasbeenestimatedbut
isasyetunknown.
AstheCOVID‐19pandemichasevolved,newstrainshaveemerged,andintheUK,
theAlphavariantandDeltavarianthavesuccessivelybecomedominant.Eachofthese
strainshavebeenmoreinfectiousthantheirpredecessors,increasingthechallengesto
healthsystems.
Modellingstudieshavereachedanewlevelofpublichealthimportancein2020/2021
aspolicymakershaveseentheirvalueforpredictingandanalysingthefutureprogression
oftheCOVID‐19pandemicandallowingacomparisonofinterventionsandpolicydeci‐
sions.Therearebroadlytwomodellingapproachesbeingused.Mechanistic(dynamic)
modelssuchastheImperialCollegeLondon(ICL)model[10]reflecttheunderlyingtrans‐
missionprocessandcontainnon‐linearfeedbackloopsanddelays,enablinglongerterm
projectionandinferenceoftheresultsofchangingassumptionsorscenarios[11].Statisti‐
calmodels,forexampletheInstituteforHealthMetricsandEvaluationmodel[12],use
regressionbasedormachinelearningmethods.Thesemodelsdonotaccountforhow
transmissionoccursandarethereforenotsowellsuitedforlongtermprojectionsabout
epidemiologicaldynamics.TheScientificAdvisoryGroupforEmergencies(SAGE)inthe
UKusesanumberofmodelstoinformitsadvice[13].Inordertosupportabroadpublic
debateontheupcomingprecautionarymeasuresagainstCOVID‐19,wedevelopasimu‐
lationmodelwiththreepurposes:
1. toinvestigatethelikelyeffectsoflockdowneasingontheUKpandemic,exploring
theremaininguncertaintiesonvaccineefficacyandpost‐infectionimmunity;
2. toestimatetheunknownproportionofCOVID‐19casesintheUKandtheroleof
unknowncasesinthespreadofthedisease;
3. toincreasethetransparencyofthemodellingandanalysisprocess,byfocusingon
containingthemodeldetailcomplexityandclearlyestablishingtheimplicationsof
differentassumptions.


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2.Background
2.1.RecoveredandPost‐VaccinationImmunity
AstheCOVID‐19epidemiccontinuesintheUK,recoveredpopulationimmunityis
building.Thereisgrowingconsensusamongstresearchersthatrecoveredimmunitywill
notbelifelongandmaybeineffectiveagainstnewstrains.Seasonalcoronavirusessuchas
COVID‐19,whichinfectmucosalsurfacesanddonothaveaviremicphase,typicallyre‐
sultinantibodyresponsesthataredetectedformonthsorafewyears[14].Estimatesof
thelongevityofrecoveredimmunityrangefromatleast5monthstomorethan12months
[15–17].Thelongevityandlevelofprotectionofpost‐vaccinationimmunityisnotneces‐
sarilythesameasthatofrecoveredimmunityandwillalsobecomebetterunderstood
withelapsedtime,aswilltheprotectionwhichitgivesagainstemergingvariants.The
firststudiesspecifictoCOVID‐19reportedthatintheshortterm,recoveryfrominfection
gave83%protection(95%CI76–87%)fromreinfectionforatleast5months[18,19].Results
fromnewerUKpopulationresearchreleasedinApril2021showed70%(95%CI62–77%)
protectionfromreinfectionaftereitherinfectionorvaccination[20].Clinicaltrialscon‐
tinuetoinvestigatevaccineefficacy,theprotectiveeffectofpastinfectionandtheeffec‐
tivenessofbothvaccinesandpastinfectionagainstemergingCOVID‐19strains.
2.2.TransmissibilityafterVaccination
Vaccineefficacyhasthreecomponents:preventionofinfection,reductionofdisease
severityandpreventionoftransmission[21].Resultsfromclinicaltrialsfocusonpreven‐
tionandseverityofinfection,whichisdirectlymeasurable,ratherthanonpreventionof
transmission.Forthisstudy,therelevantcomponentofvaccineefficacyisitseffectiveness
inprotectingagainstonwardstransmissionofthevirus.ResearchshowsthattheUK’s
vaccinationprogrammehasresultednotonlyinprotectionfrominfectionbutalsoina
lowerviralburdenifinfected,leadingtoamuchhigherproportionofasymptomaticand
mildinfections.Comparisonofviralburdeninvaccinatedandunvaccinatedgroups
showsa65%decreasethreeweeksafteronedoseofeitherAZorPB,anda70%decrease
1weekafteraseconddose[20].Viralburdencanbeusedasaproxyforpost‐vaccination
transmissibilitydecrease,whichisnotdirectlymeasurable.
2.3.Known,UnknownandAsymptomaticCases
AsymptomatictransmissionisrecognisedasasignificantcontributortotheCOVID‐
19pandemic,bothfrompre‐symptomaticindividualsandfromthosewhoneverdevelop
symptoms[22,23].Theeffectofvaccinesinreducingtheseveritytoasymptomaticormild
diseasemayalsomeanthatmorecasesgoundetectedinthecommunity,contributingto
increasedtransmission[24].Atleast50%ofnewinfectionsareestimatedtohaveorigi‐
natedfromexposuretoindividualswithinfectionbutwithoutsymptoms[25].Evidence
suggestsa42%lowertransmissionrateforasymptomaticcases[26,27].Itisbroadly
acknowledgedthatthereismassiveglobalunder‐reportingofsymptomaticCOVID‐19
casesformanyreasonsrangingfromperceptionoflowpersonalriskfromCOVID‐19in‐
fectiontolackoftrustinhealthservices,lackoftestingcapacityandadesiretoavoidthe
negativeconsequencesofenforcedisolation[28].Theunknownproportionofcasesisthus
likelytobehigherthanthetrulyasymptomaticproportionandthemodellingexercise
usesoptimisationtechniquestoestimatethisunknownproportion.
3.MethodandDataSources
3.1.ModelDevelopment
WedevelopedadynamicmodeloftheCOVID‐19pandemicbasedontheestablished
Susceptible‐Exposed‐Infected‐Recovered(SEIR)compartmentalinfectiousdiseasemodel
structure[29].Themodel,showninFigure1,wasconstructedusingStellaArchitectsoft‐
waresuppliedbyiseesystems,Lebanon,NH,USA.

Systems2021,9,604of24



Figure1.SEIRmodelofCOVID‐19pandemicinUK.

Systems2021,9,605of24


TheSEIRsystemstructureisbasedonareinforcingfeedbackloopofexponentially
growinginfectionsovertime,balancedbyaneventualreductionofsusceptibleindividu‐
alsduetodeathorincreasingpopulationimmunity.Speedoftransmissionistrackedby
thecalculatedreproductionnumber,Rt,withdailycasenumbersreducingwhenRtfalls
below1(R0,initialreproductionnumber,isoftenusedincorrectlyinplaceofRt).
Themodelincludestheeffectsofthesocialdistancingandinfectionspreadmeasures
usedtocontrolthespreadofCOVID‐19.Infectionsareclassifiedasknownorunknown,
withtheparametersassociatedwithcontactratesgivendifferentvaluesdependingon
known/unknownstatus.Theeffectsofavaccinationprogramme,whichreducesthesus‐
ceptiblepopulation,andtheeffectsofrecoveredimmunitydrop‐off[30],whichincreases
thesusceptiblepopulation,arealsoincluded.
Themodelconsistsofstocks,flowsandauxiliaryvariablesincludingintermediate
calculationsforthedeterminationofflows.Stocksrepresentlevelsorstatevariables,in‐
cludingthenumbersofpeopleinthedifferentinfectiousstatesorthenumbersofvaccine
dosesavailable;thesearerepresentedbyrectangles.Flowsrepresenttheratesatwhich
peopleanddosestransitionbetweenstatesandarerepresentedbyvalvesymbols.These
ratesaredeterminedbytimeconstantsorprobabilityestimatesofmovingtoonestateor
another.ThemodelcapturesthefundamentaldriversoftheCOVID‐19pandemicand
doesnotprovidespatialorindividual‐leveldisaggregation.Itslackofdetailcomplexity
ismeanttoprovidetransparencyinthemodellingandanalysisprocess,whilstallowing
theexplorationofabroadrangeofalternativescenarios.
Themodelrunsfrom1February2020,whenthetotalpopulationissusceptible,to31
December2021,withatimestepof6h.Individualsacquiretheinfection,incubatethe
diseaseduringaninitiallatentperiodandthenbecomeinfectious.Eachstageintroduces
adelayintothesystem.Anindividual’sinfectiousstateisatfirstunknown,then,asthe
diseasebecomessymptomatic,itbecomesknowninaproportionoftheinfectedpopula‐
tion.Someindividuals’infectiousstateisneverknowntohealthauthorities,eitherbe‐
causetheyareasymptomaticorbecausetheydonotrecognizeorwishtodisclosetheir
symptomsforvariousreasons.Mostinfectedindividualsrecover,withaproportionof
knowninfectedindividualsdying.Recoveredindividualsacquirealevelofprotectiveim‐
munity,whichreducesthesusceptiblepopulation.Themodelalsoprojectstheeffectsof
potentialfutureUKGovernmentinterventionsbysimulatingincreasedlockdownswhen
knowndailycasesriseabovethresholdlevels.Allequations,auxiliaryvariablevaluesand
initialvaluesofstocksarelistedinSupplementaryTableS1.
3.2.ModelDataSources
Theinfectionrateinthemodeliscalculatedfromthesusceptiblepopulationandthe
dailyinfectingcontactrate,whichisaffectedbysocialdistancing,hygieneandlockdown
measuresandissignificantlylowerforknowninfectedindividuals.Infectivityinthe
modelincreasesfrom5December2020andagainfrom13April2021,reflectingtheemer‐
genceofthe‘UKvariant’B.1.1.7,nowknownastheAlphavariant,whichwasmeasured
as35%morecontagious(95%CI2–69%)[31,32]andthenthe‘Deltavariant’,assumedto
betwiceascontagiousastheoriginalvirus.ThemodelusesdataforthePBandAZvac‐
cinesonly,astheModernavaccinehasnotyetbeendeployedinquantityintheUK.
Thevaluesoftheparametersusedinthemodel,showninTable1,wereestablished
intwoways:
1. Forparameterswherereliabledatawasavailablefrompublishedresearch,e.g.,virus
incubationtime,themedianvaluesfromtheresearchwereused;
2. Forparameterswheredatawaseitherunavailableorconsideredunreliable,thePow‐
elloptimisationmethodwasusedtocalibratethemodelandconfirmanarrow
spreadof95%confidenceintervals.


Systems2021,9,606of24


Table1.Majorparametervaluesusedinmodel.
ParameterValueUnitSource
Incubationduration(non‐infectiouslatentperiod)3.5Days[33]
Diseasedurationstage1unknown2Days[33,34]
Diseasedurationstage2known 8Days[33,34]
Diseasedurationstage2unknown 5Days[33,34]
Timefromknowndiseasetilldeath11Days[34]
VaccinerolloutspeedPB/AZ130,000/380,000Doses/day[35,36]
Vaccineprotectionagainstonwardstransmission
21daysafterdose1PB/AZ65%$‐ [20]
Vaccineprotectionagainstonwardstransmission7
daysafterdose2PB/AZ70%$‐ [20]
Lengthofimmunityaftervaccinationorrecovery 8$Months[15]
Maximumpopulationimmunity 70%‐ [37]
Averageimmunityprotectionpostrecovery70%$‐ [20]
Unknowninfectiousnessratio*72%$‐ [5,26,27,38–40]and
modeloptimisation
Unconstrainedinfectingdailycontactrateun‐
known0.56$‐ modeloptimisation
Unconstrainedinfectingdailycontactrateknown0.14$‐ modeloptimisation
KnownproportionestimateFebruary202121%$‐[2]andmodeloptimisation
Relativeinfectivityafteralphavariantidentified1.32$‐[32]andmodeloptimisation
Relativeinfectivityafterdeltavariantidentified2.0$ [41]
*StartingpointwasthebestestimateusedbyCenterforDiseaseControlandPreventionbasedonmultipleassumptions
andconflictingresearchpapers.$Valueusedforbasecaseofmodel.
3.3.LockdownEffectivenessTimelineEstimation
Associaldistancingandlockdownshaveproventobeoneofthemosteffectiveways
ofcombatingthespreadofthevirus[42],acompositemeasureoflockdowneffectiveness
basedonthetimelineofthevariousrestrictionsandtheireasingmeasureswasakeypart
ofthemodel.Thismeasureisknownasthe‘lockdownpercentage’.Itvariesthroughout
thelifeofthemodelandmeasuresthetimelineofsocialdistancing,maskwearingand
movementrestrictionmeasuresandvariesbetween0%and100%,where0%represents
societywithnorestrictionsinplaceand100%ahypotheticaltotalrestrictionscenariowith
nocontactandthereforenotransmissionofthevirus.
FromJanuary2021,theUKGovernmentimplementedasetofcountrylockdown
planswhichspecifiedstagedstepdownsseparatedbyaminimumoffiveweeks,with7
day’snoticeofeachchange[43]toenabletheobservationofthedatabeforeproceeding.
Thedatesofthemostsignificantmeasurestakenandthefutureplans[43]areshownin
Table2.Thelockdownpercentagetimelinewasestimatedfromthistableandcompared
withdatafromaUKsocialdistancingmeasuresadherencestudy[44].
Table2.Datesofsignificantmeasures.
Event Date
FirsttwoUKCOVID‐19casesconfirmed1February2020
UKGovernmentCoronavirusactionplan3March2020
FirstCOVID‐19death3March2020
Contacttracingabandoned12March2020
UK‐widelockdowneffected26March2020
PrimeMinisteradmittedtohospitalwithCOVID‐19symptoms4April2020
COVID‐19alertlevelssystemannounced1May2020

Systems2021,9,607of24


Lockdowneased,workersreturn,outdoorexercisewithanother13May2020
Lockdowneased,non‐essentialshopsreopen15June2020
Restaurantsandpubsopen4July2020
Restaurant‘eatouttohelpout’campaign3August2020
Oneofeverythreecasesin20–29‐year‐olds,fastgrowthinyoungerpeople7September2020
England—‘RuleofSix’announcedtocurbsocialgatherings14September2020
England—three‐tieralertframeworkimplemented14October2020
NorthernIreland—4‐week‘circuitbreaker’lockdownstarts16October2020
Wales—3‐week‘firebreak’lockdownstarts23October2020
Scotland—5‐tieralertsystemstarts2November2020
England—4‐weeknationallockdownstartsatnewtier45November2020
NewCOVID‐19strain(Alphavariant)B.1.1.7detectedinUK13November2020
England—4‐weeklockdownends3December2020
PBimmunisationrolloutstarts8December2020
LondonandScotland,newtier4lockdown 20December2020
Christmasonedaylockdownrelaxation25December2020
AZimmunizationrolloutstarts4January2021
England,Scotland—tier5lockdownto22February6January2021
England—lockdownextendedto8March27January2021
Schoolsreturn8March2021
Non‐essentialretail,outdoorhospitalityandattractionsreopen12April2021
NewCOVID‐19strain(Deltavariant)B.1.617.2detectedinUK15April2021
Indoorhospitalityandsportingeventswithlimitedcapacityreopen 17May2021
PlannedEnglandandScotland‘Freedomday’21Junedeferredto19July14June2021
FUTURECHANGES: 
England—mandatorymaskruleslifted,nightclubsreopen,fullcapacityevents19July2021
Scotland—levelzero,upto10peoplemeetindoors,nightclubsremainclosed 19July2021
3.4.ModelCalibrationandOptimisation
ThemodelwascalibratedagainsthistoricalUKCOVID‐19case,deathandvaccina‐
tiondataupto12July2021sourcedfromJohnsHopkinsUniversity[36].Calibrationwas
doneusinganoptimisationprocesstofindthemodelvariableswhichproducedthebest
fittothehistoricaldata.Thevariableswhichwereusedforoptimisationwere:theknown
andunknowninfectingcontactrates,theinfectiousnessratioofunknowntoknowncases
andtheknownproportionofcases.Thisoptimisationproducedthemodel‘basecase’
whichwasusedasthestartingpointforvaryinguncertainties.Optimisationwasalsoper‐
formedfordifferingimmunitylengthscenarios.TherelativeinfectivityoftheAlphavar‐
iantandtheDeltavariantwerecalibratedbylateroptimisations.
Aftercalibration,thefollowingvalidationcheckswereperformed:
 The‘newsusceptible’and‘recoveredsusceptible’stocksinthemodelwerevalidated
againstUKCOVID‐19antibodyprevalencestudiestoensurethatthepopulation
fractionofpeoplewithantibodies,whocanbepresumedtohaverecoveredfrom
COVID‐19,alignswiththemodelledfraction[4];
 ModelledUKcasefatalityrateswerecomparedwithhistoricaldatatoensurebroad
alignment[36];
 ThereproductionnumberRt,calculatedbythemodelovertime,wascomparedwith
studiesoftheinitialR0andtheongoingCOVID‐19Rtvaluestocheckconsistency
[45];
 Theunknowninfectiousnessratiowascomparedwithpreviousresearchtoensure
thatitwasatleastashighastheestimatedasymptomaticinfectiousnessratio[26,27].
Themajorassumptionsmadeinthemodelinadditiontotheassumedparametervalues
were:

Systems2021,9,608of24
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 TherelativeinfectivityincreasesattwopointsintimeduetothenewAlphaandDelta
variants;
 Vaccinationproceedsatasteadydailyrateinallscenariosandisofferedtothetotal
eligiblepopulationirrespectiveofwhetheranindividualisknowntohaverecovered
fromCOVID‐19;
 Themaximumachievablepopulationimmunityfractionof70%iscappedbyineligi‐
blepopulationsectors(pregnantwomenandmostchildrenunder18),vaccinehesi‐
tancy[37]andlogisticaldifficulties;
 Theseconddoseofavaccineisgiven12weeksafterthefirstdose;
 Theprotectiveeffectofthefirstdoseofthevaccineisestablished21daysafterad‐
ministration,andincreasedprotectionisestablished7daysaftertheseconddose;
 Theaveragetimelagbetweensymptomonsetandthereportingofapositivecaseto
thedatasourceis4days.
3.5.UncertaintyModelling
Havingestablishedthemodel‘basecase’throughcalibrationandvalidation,uncer‐
tainparametersinthemodelwerethenvariedbetweenthe95%CIsreportedinclinical
trials,enablingtheexplorationoftheeffectonfuturedailycaserates.Asummaryofthe
areasofuncertaintyinvestigatedisshowninTable3.
Table3.Scenariossimulatedinmodel.
Scenario
Immunity
LengthPost
Vaccination
andPost
Recovery
ProtectionFrom
InfectionGiven
byRecovered
Immunity
VaccineProtec‐
tion3Weeksaf‐
ter1stDose
VaccineTransmis‐
sionProtection1
Weekafter2nd
Dose
FutureKnown
Proportionof
Cases
LockdownChar‐
acteristics
BaseCase8months[15]70%[20]PB/AZ65%[20]PB/AZ70%[20]50%‐
Recoveredimmunity
protectionvariations8months62%/70%/87%
[15–17,19,20]PB/AZ65%PB/AZ70%50%‐
Vaccineprotectionvari‐
ations8months70%
PB/AZ
60%/65%/70%
[20]
PB/AZ
62%/70%/77%[20]50%‐
Knownproportionof
casesvariations 8months70% PB/AZ65%PB/AZ70%50%/37.5%/25%/
12.5%
‐
Lockdownsensitivity
variations8/12months70%PB/AZ65%PB/AZ70%50%
Delaysfrom3.5
to21days,
Casethresholds
from5000to
25,000,
Lockdownin‐
creasefrom25%
to50%
Thereisnopublishedresearchdataavailableforpost‐vaccinationimmunitylength,sothiswasassumedtobethesame
aspost‐recoveryimmunity.TheproportionofknownCOVID‐19casesmayreduceduetolowereddiseaseseverity;the
modelwasrunusingvaluesof0%,25%,50%and75%reductionintheabsenceofpublishedresearch.

Systems2021,9,609of24
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4.Results
4.1.ModelFittoActuals
Figure2showsthereportedhistoricalandmodelled7‐dayaveragesfortheUK’snew
knowndailyCOVID‐19casesfrom1February2020to12July2021.Theerrorstatistics
calculations(R2:=0.97,RMSPE=3.6%andTheil’sinequalitycoefficient=0.07)confirma
goodfitofthesimulatedresultstohistoricalactuals.Thelockdownpercentageisrepre‐
sentedasablacklinewithitsscaleontherightaxis.Theleftaxisshowsthescalesforthe
actualandmodellednewknowndailycasesanddeaths,withcasesclimbingto60,000in
January2021.Thex‐axismarkingsshowthebeginningofeachmonth.

Figure2.DailyUKreportedCOVID‐19cases1February2020to12July2021.
TheeffectofthefirstUK‐widelockdown,whichwasestimatedas75%effective[44],
canbeseeninApril2020,withknowncasenumberspeaking16dayslater.Thegradual
easingofthelockdownfrom5July2020resultedinanincreaseinknowncasesfromAu‐
gust2020,withtheUKGovernment‘Eatouttohelpout’schemeestimatedtohaveraised
infectionratesby8to17%[46].Thelockdownpercentageincreasedfrommid‐October
2020inresponsetorisingratesastheEnglishtieredalertsystemstartedandNorthern
IrelandandWalesimposed‘firebreaklockdowns’,followedbyregionalrestrictionsin
Scotlandandafour‐weekEnglishlockdownstarting5Novemberinanattempttoreduce
casenumbersbeforetheChristmasperiod.Theeffectoftheseconsolidatedlockdowns
wastoreducetheknowncasenumbersfrommid‐November2020for16days,onlyfor
themtoclimbfrom5December2020onwardsastheUKmovedintoitsholidayperiod.
TheemergenceofthemorecontagiousAlphavariantinDecember2020acceleratedthe
newcaserateandmadeastrictlockdowninJanuary2021necessarytocontainthe‘second
wave’.Thelockdownwaseffectiveinreducingcases,whichpeakedat60,000perday12
daysaftertheChristmaslockdownrelaxationandthenfellbelow2000perdayinMay
2021.However,theDeltavariant,whichbecamedominantintheUKinApril2021,com‐
binedwitheasingoflockdownrestrictionsinAprilandMay,reversedthedownwards
trendandcasesclimbedtoover30,000perdayinJuly2021.
TheoptimisationprocessdescribedinSection3.2calculatedarelativeinfectiousness
valueof72%forunknowncases,whichisintherangesupportedbytheresearch[5].The

Systems2021,9,6010of24


knownproportionof21%ofcasesattheendofJanuary2021wasalsoobtainedthrough
optimisation,assumingalogarithmicgrowthratefromthebeginningofthemodel’s
timeframe.Thisisintherangesupportedbyothermodels[2]andhelpstoexplainwhy
non‐discriminatorylockdownswereadoptedastheonlyeffectivemeansofcontrolling
thespreadofCOVID‐19beforevaccinesweredeveloped.Theknownproportionwasas‐
sumedtoincreaseto50%byendMarch2021ascasesfell,testingcapabilityimprovedand
self‐testingbecamemandatoryforcertainprofessions,e.g.,teaching.Thisassumptionwas
validatedbyacomparisonofreportedcasesagainstrandompopulationsampling.
4.2.ExploringUncertainty
ThescenariosidentifiedinTable3weresimulatedbyvaryingtheselectedvariables
whilstkeepingothervariablesat‘basecase’levels.
4.2.1.UncertainImmunityLength
The‘basecase’definedinTable3assumes8monthsaverageimmunity,eitherafter
vaccinationorrecoveryfrominfection[15],a65%reductionintransmissionprotection
afteronedose,a70%reductionaftertwodosesofeitherthePBorAZvaccineand70%
protectionfromreinfectionafterrecoveryfromCOVID‐19[20].Researchtodatereports
thatimmunityislikelytovarybetween5and12months[15–17,19],andTable4shows
thesimulatedscenarios.Immunityagainstemergingvariantsmaybedifferentandisnot
accountedforinthismodel.
Table4.Varyingimmunityscenarios.
ScenarioImmunity
Length
RecoveredImmun‐
ityProtection
VaccineProtection3
Weeksafter1stDose
VaccineProtection
1Weekafter2nd
Dose
FutureKnown
Cases
Immunitylengthvaria‐
tions 5/8/12months70%PB/AZ65%PB/AZ70%50%
Themodelwasrunfrom1February2020to31December2021tosimulatethe‘base
case’of8monthsimmunityandshorterandlongeraverageimmunitylengthsof5and12
months.Figure3showstheprojecteddailyknowncasesforthethreescenarios,assuming
asteppedlockdownpercentagedecreasefromMarch2021onwards,whichreducesto
20%inmidJuly2021accordingtothecurrentUKGovernmenttimelines[43].Thefigure
of20%assumesthatsomedistancingrestrictionsarestillinplaceuntiltheendof2021,
thatpeoplewillcontinuetoexercisecautionandthatbusinesseswillcontinueriskreduc‐
tionpoliciessuchasdisinfectionandmanagementofcrowds.

Systems2021,9,6011of24



Figure3.DailyUKCOVID‐19casesprojectedtoend2021withvaryingimmunitylengths.
Forthe‘basecase’,thesolidredlineinFigure3showsthemodel’sprojectionofa
continuingrapidincreaseinknowndailycases,drivenbyincreasedtransmissionoppor‐
tunitiesandanincreasedsusceptiblepopulationpercentageasthoseinfectedinearly2021
losetheirimmunity.ThispeaksinSeptember2021at260,000dailyknowncaseswhen
populationimmunitycreatedbybothvaccinationandrecoveryfrominfectionreduces
thesusceptiblepercentageandnumbersstarttofall.Thisprojectionisstarklydifferent
fromthepre‐Deltavariantscenario,whichisrepresentedbythedottedredline.Inthis
scenario,immunityfrombothvaccinationandrecoverywouldhavecontaineddaily
knowncasesbelow3000fromMay2021.Increasingtheaverageimmunitylengthto12
monthsisprojectedtocontainthesurgeto160,000dailyknowncases,peakinginOctober
2021.Ifimmunityonlylastsfor5months,thesurgeishigherandapeakof430,000daily
knowncasesisreachedinAugust2021.A5‐monthimmunityscenarioassumingnoDelta
variantwouldalsoseecasesrisingmoreslowly,peakinginDecember2021.The5‐month
immunityscenarios,however,seemunlikelyasactualknowndailycasesarenotsurging
fastenoughinJuly2021toalignwiththemodel’sprojections.
TheresultsshowninFigure3arebasedontheassumptionthatfromMay2021on‐
wards,50%ofcasescontinuetobedetectedduetoincreasedtestingcapability.However,
thisdetectionratemaywellbeunachievableatthesehighcaselevels,inwhichcasere‐
portedresultswouldshowlowernumbersthanthoseprojectedinthesimulation.
4.2.2.UncertainImmunityEffectiveness
Researchhasproducedarangeofeffectivenessresultsandconfidenceintervalsfor
bothrecoveredandvaccinatedimmunity.Table5showsthevaryingimmunityeffective‐
nessscenariossimulated.Thescenariosreflectthe95%CIrangeofpost‐vaccinationand
post‐recoveryimmunityprotectionfromtheresultsofclinicalresearch[20],assumingthe
‘basecase’forothervalues[15–17,19,20].The95%CIrangesforrecoveredandvaccinated
immunityaredifferent,andthisisreflectedinthescenariosused.Figure4showsthe
modelledprojectionsforthesescenarios.


Systems2021,9,6012of24


Table5.Varyingimmunityeffectivenessscenarios.
ScenarioImmunity
Length
RecoveredIm‐
munityProtection
VaccineProtection3
Weeksafter1stDose
VaccineProtection1
Weekafter2ndDose
Future
Known
Cases
Vaccineprotectionvar‐
iations 8months70%PB/AZ60%/65%/70% PB/AZ62%/70%/77%50%
Recoveredimmunity
protectionvariations 8months62%/70%/87% PB/AZ65%PB/AZ70%50%

Figure4.Dailyknowncaseprojectionswithvaryingimmunityprotection.
Figure4aprojectsthatifpost‐vaccinationprotectionfrominfectionisatthelower
boundaryof62%aftertwodoses,knowninfectionswillbuildto320,000inSeptember.
Usingthehigherboundaryof77%protectionaftertwodoses,themodelprojectsthat
knowndailycaseswillpeakat210,000beforedroppingasherdimmunityfrombothvac‐
cinationandrecoveryreducesthesusceptiblepercentage.
Figure4bshowstheprojectedrangeofknowncasesforrecoveredimmunityvaria‐
tion.Themodelprojectsthatthelowervalueofrecoveredimmunityof62%willresultin
adailyknowncasesurgeto280,000inSeptember2021,reducingto215,000withthe
highervalueof87%.AsdescribedinSection4.2.1,50%detectionatthesehighdailycase
numbersmaybeunachievable,whichwouldreducethereportedcasepeaks.
4.2.3.UncertainKnownProportion
TheresultspresentedsofarshowonlytheknownproportionofCOVID‐19casesin
theUK.Asvaccinationreducesnotonlythecasenumbersbutalsotheaveragecasese‐
verity,theunknownproportionmayincreasefurtherastheproportionofmildorasymp‐
tomaticcasesgrows,evenwithincreasedeaseandavailabilityoftesting.Table6shows
thescenariosmodelled.


Systems2021,9,6013of24


Table6.Varyingknownproportionscenarios.
ScenarioImmunity
Length
RecoveredIm‐
munityProtection
VaccineProtection3
Weeksafter1stDose
VaccineProtection1
Weekafter2ndDose
FutureKnownCases
Knownpro‐
portionvaria‐
tions
8months70%PB/AZ65%PB/AZ70%50%/37.5%/25%/12.5
%
Figure5a,bprojectthedailyknownandtotalcasesfor2021forthe‘basecase’sce‐
nariowiththepercentageofknowncasestounknownrangingfrom50%to12.5%.The
basecaseassumesthat50%ofcasesareknown.

Figure5.DailyknownandunknownUKCOVID‐19casesin2021withvaryingknownproportionassumptions.
Asexpected,theprojectedknowncasenumbersdropastheunknownproportion
rises.Theprojectedtotalcaseswouldbeexpectedtoincreasewhenalowerpercentageof
thecasesareknownbecausetransmissionisnotbeingmanagedthroughisolationofin‐
fectedindividuals.However,becauseunknowncasesareassumedtobelessinfectious
andofashorterdurationthanknowncases[26,27],a75%reductionintheproportionof
knowncases(from50%to12.5%)generatesonlya40%increaseintotalcasenumbers.
4.2.4.ModellingtheEffectofInterventions
TheUKGovernment’splannedlandmarkdateof21June2021,‘Freedomday’,when
maskscouldberemovedandothersignificantrestrictionswouldbelifted,wasmovedto
19JulyasdailycasenumbersstartedtoriseinMay2021[47].Thisrise,drivenbythemore
transmissibleDeltavariantandtheeasedrestrictions,raisesthequestionofwhetherfur‐
therlockdownsshouldbeconsidereddespitetheincreasingvaccinationnumbers.From
theresultsshowninFigures3–5,itcanbeseenthatvaryingimmunitylengthhasalarger
impactoncasenumberprojectionsthanvaryingvaccinationandrecoveredimmunity
protectionwithintheirlikelyranges.Therefore,potentiallockdownscenarioswereex‐
ploredwithdifferingimmunitylengthassumptions,asshowninTable7.


Systems2021,9,6014of24


Table7.Varyinglockdowninitiationscenarios.
ScenarioImmunity
Length
Recovered
Immunity
Protection
VaccinePro‐
tection1st
Dose
VaccinePro‐
tection2nd
Dose
Future
Known
Cases
Lockdown
DailyCase
Threshold
Lockdown%
Lockdowneffects
forvaryingimmun‐
itylengths
5/8/12
months 70%PB/AZ65%PB/AZ70%7days50,00020%addition
Figure6a,bsimulatetheeffectsofaGovernmentpolicywhichreactstodailyknown
casesrisingabove50,000byincreasinglockdownlevelsby20%.The20%isatheoretical
numberwhichcouldbemadeupofanumberofdifferentmeasures,e.g.,self‐isolation
restrictions,masks,numberlimits.A7‐dayreactiontimeisbuiltintothesimulation,in
linewithcurrentGovernmentpolicy.

Figure6.Lockdowninterventionswhencasesriseabove50,000.
Figure6aprojectsthatforan8‐monthimmunitylength,a3‐month‐longreturntothe
40%lockdownlevelwouldberequiredfromlateJuly2021toreturncasestobelow50,000.
Fora12‐monthimmunitylength,a2‐monthreturntothe40%lockdownlevelwouldbe
required,startingatasimilartime.Figure6bprojectsthatfora5‐monthimmunitylength,
the50,000‐casethresholdwillbebreachedinJulyandcontinuinglockdownattheJuly
levelswouldreducethepeakdailynumbersto250,000beforetheydropdowninNovem‐
ber2021.
4.2.5.LockdownPolicySensitivities
ThescenariosshowninTable8wereusedtosimulatethesensitivityofthelockdown
policytothelengthoftimebeforeinitiatinglockdown.


Systems2021,9,6015of24


Table8.Varyinglockdowninitiationdelayscenarios.
ScenarioImmunity
Length
RecoveredIm‐
munityProtection
VaccineProtec‐
tion1stDose
VaccineProtec‐
tion2ndDose
DelayBefore
Lockdown
LockdownDaily
CaseThreshold
Lock‐
down%
Lockdownde‐
layvariations
8months70%PB/AZ65%PB/AZ70%
3.5,7,10.5,
14,17.5,21
days
500025%addi‐
tion
5months70%PB/AZ65%PB/AZ70%
3.5,7,10.5,
14,17.5,21
days
500025%addi‐
tion
Figure7projectstheresultsofvaryingthelockdownnoticeperiodbetween3.5and
21daysafterknowncasesreach50,000.Figure7ashowsthatthe8‐monthimmunity‘base
case’witha20%increaseinlockdownpercentageresultsinashorterdelayandalower
peakincases.Thehighestpeakisprojectedforthe21‐dayleadtime.Figure7bshowsthe
samepatternforthe12‐monthimmunityassumption,withmaximumdailyinfections
reaching96,000fora3.5‐dayleadtimeand136,000fora21‐dayleadtime.

Figure7.Effectofvaryingtimetoinitiatelockdown.
ThescenariosshowninTable9wereusedtosimulatethesensitivityofthelockdown
policytothecasethresholdbeforeinitiatinglockdown.
Table9.Varyinglockdowncasethresholdscenarios.
ScenarioImmunity
Length
RecoveredIm‐
munityProtec‐
tion
VaccineProtec‐
tion1stDose
VaccineProtec‐
tion2ndDose
DelayBeforeLock‐
down
Lockdown
DailyCase
Threshold
Lockdown%
Lockdown
casethresh‐
oldvariations
8months70%PB/AZ65%PB/AZ70%7days
25,000,50,000,
75,000,
100,000
20%addition
12months70%PB/AZ65%PB/AZ70%7days
25,000,50,000,
75,000,
100,000
20%addition

Systems2021,9,6016of24


Figure8projectstheresultsofvaryingthedailyknowncasethresholdforinitiating
lockdownbetween25,000and100,000,assuminga7‐dayleadtimeasperthecurrentUK
Governmentpolicy.Figure8ashowsthat,forthe8‐monthimmunity‘basecase’,thelower
thecasethreshold,thelowerthepeakofdailycases.Inallscenarios,casesfallrapidlyas
thesusceptiblepercentagereducesduetoincreasingpopulationimmunityfromthelarge
numbersofrecoveredinfectionsandvaccinations.Figure8bshowsthesamepatternfor
the12‐monthimmunityscenariowithlowerpeaksbecauseofthegreaterlevelofretained
recoveredpopulationimmunity.

Figure8.Effectofvaryingnumberofknowncasesrequiredtoinitiatelockdown.
ThemodelwasusedtosimulateextremelockdownscenariosasshowninTable10.
Table10.Testingextremelockdownscenarios.
FigureScenarioImmunity
Length
Recovered
Immunity
Protection
VaccinePro‐
tection1st
Dose
VaccinePro‐
tection2nd
Dose
DelayBefore
Lockdown
Lockdown
DailyCase
Threshold
Lockdown%
9a
Longdelay
&highcase
threshold
8/12months70%PB/AZ65%PB/AZ70%21days100,00020%addition
9bSeverelock‐
down8/12months70%PB/AZ65%PB/AZ70%7days50,00040%addition
Theextremeeffectsofahighthresholdof100,000casesanda21‐daydelaybefore
lockdowninitiationwereprojectedinFigure9a;forthe8‐monthimmunitybasecase,the
casethresholdisreachedinAugust2021andlockdownisinitiatedinearlySeptember
2021,continuingfor2monthswithdailyknowncasespeakingat250,000.Forthe12‐
monthimmunityscenario,ashorterlockdownstartinginSeptemberisrequired,anddaily
casespeakat160,000.Figure9bprojectstheeffectofa40%lockdownincreaseratherthan
the20%usedinotherscenariosandshowshow,forthe8‐monthimmunitybasecase,
reducedtransmissionopportunitylowersdailycasesfromapeakof107,000tobelowthe
50,000‐casethreshold,requiringanotherlockdownphaseinlate2021toreducecase

Systems2021,9,6017of24


numbersagain.The12‐monthimmunityscenarioonlyrequiresonelockdowntocontrol
casenumbersasongoingvaccinationscontinuetoreducethesusceptiblepercentage.

Figure9.Extremesimulationsforlockdowns.
4.2.6.ChangeinSusceptiblePercentage
InFebruary2021,100%oftheUKpopulationwassusceptibletoinfectionwith
COVID‐19.Thesusceptiblepercentagedroppedaspeoplebecameimmuneeitherthrough
infectionorvaccination.ThemovementofthesusceptiblepercentageisillustratedinFig‐
ure10forimmunitylengthvariationscenarios,withandwithoutnewlockdowninter‐
ventionsafterJune2021,asshowninTable11.
Table11.Susceptiblepercentageillustrations.
FigureScenarioImmunity
Length
Recovered
Immunity
Protection
VaccinePro‐
tection1st
Dose
VaccinePro‐
tection2nd
Dose
DelayBefore
Lockdown
Lockdown
DailyCase
Threshold
Lockdown%
10aImmunityvari‐
ations
5/8/12
months70%PB/AZ65%PB/AZ70%‐ ‐ ‐
10b
Immunityvari‐
ationswith
lockdowninter‐
vention
5/8/12
months70%PB/AZ65%PB/AZ70%7days50,00020%addi‐
tion


Systems2021,9,6018of24



Figure10.Susceptiblepopulationpercentagewithdifferingimmunityandinterventions.
Figure10showsthesusceptiblepercentagereducingasthepandemicprogresses.
Thesteeperdownwardslopescorrelatewithperiodsofhigherinfectionratesduring
whichmorepeopleacquirerecoveredimmunity.InFigure10a,forthe5‐monthimmunity
scenario,thesusceptiblepercentagedropsslowlythroughApriltoJuly2021asincreasing
numbersarevaccinated.Itthenfallssteeplyto13%becausetheinfectionsurge,whichis
seeninFigure3,generatesrecoveredimmunitybeforeincreasinginSeptember2021as
thisimmunityerodes.The8‐monthand12‐monthimmunityscenariosfollowasimilar
patternbutwithlesspronouncedslopechanges.
Figure10bshowsthesusceptiblepercentagesforthethreeimmunityscenarioswith
lockdowninterventionsimplemented.Forallscenarios,lockdownsasillustratedinFigure
6arerequiredtoreducedailyknowncasesbelow50,000.Thesehavetheeffectofslowing
thesusceptiblepercentagereductionbyreducingcasenumbersandhencegeneratingless
recoveredimmunity.
5.Discussion
5.1.ImplicationsofFindings
TheUKGovernment’sapproachtotheCOVID‐19pandemicintheUK,thoughini‐
tiallyhesitant,turnedaroundinearly2021whenstronglockdownmeasureswereputin
placeandanambitiousvaccinationprogrammewascommenced.TheUK’saggressive
pursuitofvaccinationispayingoff,withhalfthepopulationfullyvaccinatedatthebe‐
ginningofJuly2021.WereitnotfortheemergenceoftheDeltavariant,assumingthat
immunitygainedfromeitherinfectionorvaccinationlastsatleast8months,theUK
wouldbeassuredthatitcouldliftrestrictionsandkeepCOVID‐19casenumbersatalow
levelthroughouttheremainderof2021.However,sharplyrisingcasenumbersinJuly
2021arechangingthelandscape,withhealthworkersonceagainfearfulofbeingover‐
whelmedbyCOVID‐19cases[48].Thevaccinationprogrammehasreducedboththe
transmissionandseverityofthedisease,meaningthathospitalisationanddeathrateswill
begreatlyreduced,butwithhalfthepopulationstillunvaccinatedorincompletelyvac‐
cinated,andthescenariosprojectinghundredsofthousandsofdailycases,dailydeaths
arelikelytoreachintothehundreds[36]withoutcontainmentmeasures.

Systems2021,9,6019of24


Themostsignificantinfluencerofongoinginfectionrates,otherthantheemergence
ofanothermoreinfectiousvariant,islikelytobethelengthofprotectionconferredby
vaccinatedandrecoveredimmunity.Immunitylengthisasignificantunknown,which
willonlybecomeclearerasresultsfromlongitudinalstudiesonvaccinatedandrecovered
individualsemerge.ThemodellingusedbytheUKGovernment’sSAGEadvisorygroup
[13]specificallyexcludeswaningimmunityandthefutureemergenceofvariants,sothese
aresignificantgaps.Therearenotoolstopredicttheprofileoffuturevariantsbutfurther
researchtounderstandimmunitylength,particularlyvaccinatedimmunity,whichhasa
moresignificantinfluenceintheUKthanrecoveredimmunity,iscriticalforinforming
policyandforreducingtheuncertaintysurroundingthevariousscenarios.
Ascasessurge,thevaccinatedsectorofthepopulationwillbeprotectedfromserious
illnessanddeathbutvaccinationstatusintheUKisuneven,withloweruptakeamongst
disadvantagedgroupsandethnicminorities,leavingthesegroupsvulnerable.Theunvac‐
cinatedpopulationwillonlybeeffectivelyprotectedthroughherdimmunity,whichre‐
searchindicateswillbereachedwithasusceptiblepercentageof30%orless[49,50].The
limitsonthepercentageofthepopulationabletobevaccinatedwillbecomethemain
constrainttoachievingherdimmunity.About22%oftheUKpopulationarenotcurrently
eligibleforvaccination(21%under18,0.7%pregnant),whichmeansthat90%ofeligible
adultsneedtobevaccinatedtoachievea70%total.Withthehighestinfectionprevalence
inteenagersand20–24‐year‐olds[47],extendingvaccinationstochildrenisalogicalnext
steptoincreasingherdimmunity,andfurtherresearchandtrialsonthesafetyandefficacy
ofvaccinesforchildrenandpregnantwomenarerequiredtoinformpolicy.Continuing
educationandreassuranceforthevaccine‐hesitantsectorofthepopulationisalsore‐
quiredtoaddressresistance.Itseemslikelythatforherdimmunitytobemaintained,reg‐
ularboosterdosesofCOVID‐19vaccinationswillbeneeded;thepracticeofimmunizing
newlyeligiblepeoplewillbeinsufficienttocontrolthespreadofthevirus.
Casesarelikelytoshiftfromknowntounknownbecauseofthereductionininfection
severitypost‐vaccination.Asnothingotherthanlockdownappearstoworkwhenthere
aremanyunknowncases,acapabilitywhichmaintainsorimprovestheproportionof
knowncasesisimportant.Thepotentialformoreunknowncases,exploredinSection
4.2.3,isaconcernandstrengtheningpolicieswhichencourageroutinetestingmitigates
againstthegrowingunknownproportion,andthustheunseenburdenofdisease.The
projectionsforknowncasesinFigures3and4arebasedontheknownproportionremain‐
ingat50%,whichiswhytheyaresohighinsomescenarios.
ThecurrentGovernmentpolicyof7day’swarningofachangeinlockdownstatus
seemsareasonablebalancebetweenpeople’sneedfornoticeandtheinfectiongrowth
whichtakesplaceinthose7days,althoughthereisacaseforreducingnoticetocurb
growth.Anyargumentforalowlockdowncasethresholdtocurbgrowthhasbeenover‐
takenbyeventsinJuly2021,withover50,000dailycasesbeingreported.Theloadonthe
healthserviceswillbeacriticalconsiderationindecisionsaboutfurtherrestrictions;mod‐
ellingthatisoutsidethescopeofthisarticle.
5.2.ModellingDiscussion
TheUKGovernment’sSAGEadvisorygroupusesthreemodelsfromtheImperial
CollegeLondon,WarwickUniversityandtheLondonSchoolofMedicineandTropical
Hygienegroups[13].Theassumptionsusedbythemodelsaredocumented,butthepublic
cannoteasilyseeorunderstandthemodelsortheprocessbywhichtheresultsareob‐
tained.Thisgeneratesmistrustandskepticism,especiallyastheincorporationofnewfac‐
torssuchastheemergenceoftheDeltavariantcannotbedoneinstantaneously.
Thismodel,whilstithasmorelimitationsthanthelargermodels,hastheadvantage
ofbeingabletobedisplayedononepage,makingitpotentiallymoreaccessibleandtrans‐
parent.Itisanaggregatedmodel,withnosplitintoagebandswiththeirdifferingprofiles
andvulnerabilities.Itdoesnotaccountforurban/ruraldifferencesorforcountrydiffer‐
enceswithintheUK.Manyaspectsofthesimulation,forexample,vaccinerolloutramp

Systems2021,9,6020of24


upandtheemergenceoftheAlphaandDeltavariants,aresimplified.However,itisa
usefultoolforrepresentingCOVID‐19transmissionintheUKandcanbeusedtoproject
theeffectsofpoliciesandinterventionsacrossarangeofuncertainties.
5.2.1.Uncertainty
Themodelisbasedonasetofsignificantassumptionsbasedonevolvingclinical
researchwhichsuggestsarangeofscenarios.Ofparticularimportanceare:
 lengthofrecoveredimmunity;
 vaccineefficacyinreducingtransmission;
 durationandrelativeinfectiousnessofasymptomaticandmildlysymptomaticcases;
 ongoinguncertaintyontheproportionofunknowncaseswhichcontinuetodrive
infections.
ThestrategiesfordealingwithuncertaintyinCOVID‐19modellingproposedby
Wang/Flessa[51]havebeenfollowedforthismodellingexercise.Itisevidentbothfrom
theresultsandthediscussionthatchangesinkeyassumptions,includingfuturelock‐
downpercentages,canhavesignificantimpactsontheprojectionsinthemodel.Changes
inthevaccinemixmayalsochangethemodelprojections.Everymonththatthepandemic
progresses,newresearchwithadirectbearingonthemodelassumptionsisproduced,so
thereisanopportunityforongoingrefinement.
5.2.2.ConfidenceintheResultsforGivenAssumptions
Animportantdecisioninthemodellingprocessiswhichvaluestofixasconstants
andwhichtodeterminethrougha‘tryforfit’calibrationprocess.Ifoneattemptstovary
alloftheassumedvaluesinthemodel,therearetoomanydegreesoffreedomtobeable
toobtainmeaningfulresults.Itiscertainlypossibletoobtainsimilarresultswithdifferent
parametervalues,inlinewiththeconceptofequifiniality,whichdemonstratesthatdif‐
ferentsetsofparameterscanleadtothesameorsimilarresults[52].Thereisabalance
betweenfixingassumptionstoreducethenumberofvaluesinplay,enablingameaning‐
fuloptimizationprocesstoberun,andchoosingtofixassumptionswhicharenotcertain
enough,introducingerrorintothemodel.Themethodusedinthisexercise,whichrelies
onfixingvalueswhichhaveresearchbackingandcalibratingtheothervaluesagainsthis‐
toricaldatathroughacurve‐fittingexercise,hasintroducedalevelofrigourtotheprocess.
5.2.3.ComparisonwithOtherModels
Asignificantdifferencebetweenthismodelandmanyothermodelsproducedisthe
inclusionoflossofimmunity.MostoftheearlierCOVID‐19modelsexcludedlossofim‐
munity,althoughStrubenrecognisesitasafactorwhichwill