The social contact hypothesis under the assumption of endemic equilibrium: elucidating the transmission potential of VZV in Europe

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Epidemics

jo u rn al h om ep age : w w w . e l s e v i e r . c o m / l o c a t e / e p i d e m i c s

The

social

contact

hypothesis

under

the

assumption

of

endemic

equilibrium:

Elucidating

the

transmission

potential

of

VZV

in

Europe

E.

Santermans

_,

_N.

_Goeyvaerts

_,

_A.

_Melegaro

_,

_W.J.

_Edmunds

_,

_C.

_Faes

_,

_M.

_Aerts

_,

P.

Beutels

_,

_N.

_Hens

a_{InteruniversityInstituteforBiostatisticsandStatisticalBioinformatics,HasseltUniversity,Diepenbeek,Belgium}

b_{CentreforHealthEconomicResearchandModellingInfectiousDiseases,Vaccine&InfectiousDiseaseInstitute,UniversityofAntwerp,Antwerp,Belgium} c_{DepartmentofPolicyAnalysisandPublicManagementandDondenaCentreforResearchonSocialDynamics,UniversitCommercialeL.Bocconi,Milan,}

Italy

d_{LondonSchoolofHygiene&TropicalMedicine,London,UnitedKingdom}

e_{SchoolofPublicHealthandCommunityMedicine,TheUniversityofNewSouthWales,Sydney,Australia}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received14June2014 Receivedinrevisedform 23December2014 Accepted30December2014 Availableonline10January2015 Keywords: Mathematicalmodel Mixing Contactdata Varicella Riskfactors

a

b

s

t

r

a

c

t

ThebasicreproductionnumberR0andtheeffectivereproductionnumberRarepivotalparametersin

infectiousdiseaseepidemiology,quantifyingthetransmissionpotentialofaninfectioninapopulation. Weestimatebothparametersfrom13pre-vaccinationserologicaldatasetsonvaricellazostervirus(VZV) in12Europeancountriesandfrompopulation-basedsocialcontactsurveysunderthecommonlymade assumptionsofendemicanddemographicequilibrium.Thefittotheserologyisevaluatedusingthe inferredeffectivereproductionnumberRasamodeleligibilitycriterioncombinedwithAICasamodel selectioncriterion.Foronly2outof12countries,thecommonchoiceofaconstantproportionality fac-torissufficienttoprovideagoodfittotheseroprevalencedata.Fortheothercountries,anage-specific proportionalityfactorprovidesabetterfit,assumingphysicalcontactslastinglongerthan15minare agoodproxyforpotentialvaricellatransmissionevents.Inallcountries,primaryinfectionwithVZV mostoftenoccursinearlychildhood,butthereissubstantialvariationintransmissionpotentialwith R0rangingfrom2.8inEnglandandWalesto7.6inTheNetherlands.Twonon-parametricmethods,the

maximalinformationcoefficient(MIC)andarandomforestapproach,areusedtoexplainthese differ-encesinR0intermsofrelevantcountry-specificcharacteristics.Ourresultssuggestanassociationwith

threegeneralfactors:inequalityinwealth,infantvaccinationcoverageandchildcareattendance.This illustratestheneedtoconsiderfundamentaldifferencesbetweenEuropeancountrieswhenformulating andparameterizinginfectiousdiseasemodels.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Introduction

Oneofthekeymeasuresofinfectiousdiseasetransmissionisthe basicreproductionnumberR0:theexpectednumberofsecondary casesperprimarycaseina“virgin”population(Diekmannetal., 1990).IfR0 is largerthan1 theinfectionmaybecomeendemic and thelarger R0, themore effort is required toeliminate the infectionfromthepopulation.AlthoughR0 isausefultheoretical measure,itisrarelyobservedinpractice.Theeffectivereproduction

∗ Correspondingauthor.Tel.:+32474927849.

E-mailaddress:eva.santermans@uhasselt.be(E.Santermans).

number R takes pre-existing immunity into account and thus reflectstheaveragenumberofsecondarycasesthatcanbeobserved inapartiallyimmune population.Thereareseveralmethodsto estimateR0andR(VynnyckyandWhite,2010).Inthisarticle,we focusonderivingR0fromtransmissionratesthatcanbeestimated fromserologicaldataundertheassumptionofendemic equilib-rium(AndersonandMay,1991).Adiseaseinendemicequilibrium, or steady state,mayundergo cyclical epidemics, but fluctuates aroundastationaryaverageovertime.WhitakerandFarrington (WhitakerandFarrington,2004a)haveshownthattheimpactof regularepidemiccycles,displayedbymanychildhoodinfections, canbeignoredwhenestimatingR0.Inthisequilibriumsetting,each infectiousindividualinfectsoneotherindividualonaverage,hence http://dx.doi.org/10.1016/j.epidem.2014.12.005

Risexpectedtobeequalto1(Diekmannetal.,1990).Again,ifR>1 theinfectionwillcontinuetospreadinthepopulationwhereasif R<1theinfectionwilldieout.

Weconsiderpre-vaccinationserologicaldataforthevaricella zostervirus(VZV)from12differentEuropeancountries(Nardone etal.,2007).VZVisoneoftheeightknownherpesvirusesthataffect humans.Primaryinfection withVZVresultsin varicella (chick-enpox)andmainlyoccursinchildhood.Ingeneral,thediseaseis benign,however,symptomsmaybemoresevereinadultsand com-plicationsmayoccurwhenvaricellaisacquiredduringpregnancy. VZV is highly contagious and transmitted through direct close contact withlesionsor indirectly throughair droplets contain-ingvirusparticles.TheincubationperiodfollowingVZVinfection rangesfrom 13 to18 days and each infected persontransmits thevirusforabout7days.Theantibodyresponsefollowing pri-maryinfectionwithVZVisbelievedtoinducelifelongprotection againstchickenpox.However,thevirusremainsdormantwithin thebodyandmayreactivateandgiverisetoherpeszoster(or shin-gles)afteryearstodecades(Milleretal.,1993).Inthisarticle,we willfocusonprimaryinfectionandignorereactivationleadingto zoster.

Estimatingtransmissionratesforanairborneinfectionsuchas VZVrequiresassumptionsontheunderlyingage-specificmixing patternsandR0 hasbeenshowntobehighlysensitivetothese assumptions(GreenhalghandDietz,1994).Indeed,serological sur-veys do not provide complete information about these mixing patterns,sincetheyreflecttherateatwhichsusceptibleindividuals becomeinfected,butnotwhoisinfectingwhom.This indetermi-nacypreventsassessment ofthevalidityof themixingpattern. Recently,attemptshavebeenmadetodealwiththis unidentifi-abilitybyexploitingknowledgeabouttherouteoftransmission (Farringtonetal.,2001;Unkeletal.,2014).However,thisrelieson thestrongassumptionthatinfectionsaretransmittedviathesame route.Theextenttowhichdifferentroutesoftransmissioncompete mayonlybeverifiedbyadditionaldatacollection.Inthisarticle,we informthemixingpatternwithdatafrompopulation-basedsocial contactsurveysandassumethattransmissionratesare propor-tionaltocontactrates.Socialcontactdatahavealreadyprovento beavaluableadditionalsourceofinformationwhenestimatingthe ‘WhoAcquiresInfectionFromWhom’(WAIFW)matrixandR0(see e.g.Wallingaetal.,2006;Ogunjimietal.,2009;Goeyvaertsetal., 2010).

Weusetheinferredeffectivereproductionnumberasamodel eligibilitycriterioncombinedwithAICasa modelselection cri-terion.To ourknowledge,Wallinga et al. (2001)were thefirst tousetheeffectivereproductionnumbertoassesthe plausibil-ity of differentmixing patterns.However, this is thefirst time thatRisexplicitlyusedasadeterminantinthemodelselection procedure. We evaluatehow constant and age-specific propor-tionalityfactorsaffectthefit totheserologyandtheestimated R0 values. Moreover, we assess the effect of age-specific het-erogeneityrelated toinfectiousness onmodeleligibilityand fit. Further,fromaselectedsetofdemographic,socio-economicand spatio-temporalfactors,weexplorewhichfactorsbestexplainthe between-countryheterogeneityinR0 usingtwonon-parametric methods:themaximalinformationcoefficient(MIC)andrandom forest.

The article is organized as follows. In Section “Materials and methods”, a descriptionof the serologicaland social con-tact surveys is provided, after which we elaborate on the dynamic model structure, estimation procedure and methods usedtodeterminepotentialrisk factorsforvaricella. InSection “Results”,we present theestimates of R0 and R under various model assumptions, and the results of the risk factor analy-sis. Finally in Section “Discussion”,the models and results are discussed.

Materialsandmethods Data

Serologicaldata.Inthisarticle,wereanalyzetheESEN2 (Euro-pean Sero-Epidemiology Network) data on VZV published by Nardone et al. (2007) together with newly available serology for Poland and Italy, totaling 13 serosurveys from12 different countriesincludingtwosamplesfromItaly(seeTable1).Atthetime ofseracollection,whichvariedbetween1995and2004,noneof theparticipatingcountrieshadintroducedauniversalVZV vaccina-tionprogram.Bloodsamplesweretestedusinganenzyme-linked immunosorbentassay(ELISA),therebyclassifyingthesamplesas seropositiveorseronegative(equivocalresultswereincludedas seropositive).Classificationisbasedontheobservedantibodylevel as compared tothecut-off level specifiedby themanufacturer ofthetest.Samplesizesrangefrom1268forPolandto4398for Germany,withsubstantialvariabilitybetweenthesurveyedage ranges.

Socialcontactsurvey.The spreadofairborneorclose-contact infectionsinapopulationisdrivenbysocialcontactsbetween indi-viduals.Recently,severalstudieswereconductedtomeasuresocial mixingbehavior,andReadetal.(2012)presentareviewofthe dif-ferentmethodologiesemployed.Thecross-sectionaldiary-based surveysthatwereconductedbetweenMay2005andSeptember 2006aspartofthePOLYMODproject,constitutedthefirst large-scaleprospectivestudytoinvestigatesocial contactbehaviorin eightEuropeancountries(Mossongetal.,2008).Participantswere recruited through random-digit dialing, face-to-face interviews orpopulationregisters,andcompletedadiaryabouttheirsocial contacts during one randomly assigned day. Participants were asked to record theage and genderof each contacted person, plus location,duration and frequency ofthecontact. Further,a distinctionbetweentwotypesofcontactwasmade:non-close con-tacts,definedastwo-wayconversationsofatleastthreewordsin eachothersproximity,andclosecontactsthatinvolveanysortof physicalskin-to-skintouching.Foranextensivedescriptionofthe survey,werefertoMossongetal.(2008).

Estimatingthebasicandeffectivereproductionnumber

Force of infection and mass action principle. To describe VZV transmission dynamics, a compartmental MSIR (Maternal protection-Susceptible-Infected-Recovered) model for a closed populationofsizeNwithfixeddurationof maternalprotection Aisconsidered,followingGoeyvaertsetal.(2010)andOgunjimi etal.(2009).Doingso,weexplicitlytakeintoaccountthefactthat newbornsareprotectedbymaternalantibodiesanddonottake partinthetransmissionprocess.Weassumethatmortalitydueto infectioncanbeignored,whichisplausibleforVZVindeveloped countries,andthatinfectedindividualsmaintainlifelong immu-nitytovaricellaafterrecovery.Further,demographicandendemic equilibriaareassumed,whichmeansthattheage-specific popula-tionsizesremainconstantovertimeandthatthediseaseisinan endemicsteadystateatthepopulationlevel.Underthese assump-tionstheage-specificprevalence(a)isgivenby:

(a)=1_−e−



A(u)du_,

where(a)istheage-specificforceofinfection,i.e.therateatwhich asusceptiblepersonofageaacquiresinfection.Thereisawide rangeofmethodsavailabletoestimate(a)fromseroprevalence data,seeHensetal.(2010)foranhistoricaloverview.

Sinceweaimtoestimatethebasicandeffectivereproduction numberforVZV,wedisentangletheforceofinfectionfurtherto thelevelofage-specifictransmissionrates.Letˇ(a,a)denotethe

Table1

Overviewoftheserologicaldataanddemographicparameters.

Country Datacollection Agerange(years) Samplesize Lifeexpectancy(years) Populationsize

Belgium(BE) 2001–2003 0–71.5 3251 77.6 10,309,722

Germany(DE) 1995/1998 0–79 4398 77.1 82,050,377

Spain(ES) 1996 2–39 3590 77.5 39,427,919

EnglandandWales(EW) 1996 1–20.9 2032 76.0 51,125,400

Finland(FI) 1997–1998 1–79.8 2471 76.7 5,146,965 Ireland(IE) 2003 1–60 2430 77.6 3,963,814 Israel(IL) 2000–2001 0–79 1543 76.2 6,223,842 Italy(IT’97) 1996–1997 0.1–50 3110 78.2 56,872,349 Italy(IT’04) 2003–2004 1–79 2446 80.3 5,788,0478 Luxembourg(LU) 2000–2001 4–82 2640 77.2 438,723 TheNetherlands(NL) 1995–1996 0–79 1967 77.0 15,493,889 Poland(PL) 1995–2004 1–19 1268 73.2 38,637,184 Slovakia(SK) 2002 0–70 3515 73.2 5,378,702

averagepercapitarateatwhichaninfectiousindividualofagea makeseffectivecontactwithasusceptiblepersonofagea,perunit time.Thekeyprinciplebehindtheestimationofˇ(a,a)isthe so-calledmassactionprinciple.IfthemeaninfectiousperiodDisshort comparedtothetimescaleonwhichtransmissionandmortality ratesvary,theforceofinfectioncanbeapproximatedby:

(a)≈ND_L



ˇ(a,a_)(a_)s(a_)m(a_)da_{, (1)} whereNdenotesthetotalpopulationsize,Lthelifeexpectancy, s(a)theproportionofpeopleinthepopulationofageathatare susceptible,andm(a)=exp{−



Giventhetransmissionratesˇ(a,a),followingDiekmannetal. (1990),thebasicreproductionnumberR0canbecalculatedasthe dominanteigenvalueofthenextgenerationoperatorgivenby G(a,a₎₌ND

L m(a)ˇ(a,a).

TheeffectivereproductionnumberRtakesintoaccountthe pro-portionofsusceptibleindividualsandisthedominanteigenvalue ofG(a,a)×s(a).

Mixingassumptions.Since(a)isaone-dimensionalfunctionof ageandˇ(a,a)makesupatwo-dimensionalfunction,additional assumptionsarenecessarytoestimatethetransmissionratesfrom seroprevalencedatausingthemassactionprinciple.Thetraditional approachofAndersonandMay(AndersonandMay,1991) strati-fiesthepopulationintoasmallnumberofageclassesandimposes differentmixingpatternsuponˇ(a,a).Thisistheapproachtaken intheexploratoryanalysisofNardoneetal.(2007).However,the choiceofthestructureimposedontheWAIFWmatrixaswellas thechoiceoftheageclassesareadhocandimpacttheestimation ofR0(GreenhalghandDietz,1994;VanEffelterreetal.,2009).We willconsideramorerecentapproachasproposedbyWallingaetal. (2006),byinformingˇ(a,a)withdataonsocialcontacts.Thisisalso theapproachtakenbyGoeyvaertsetal.(2010)whoexpressˇ(a,a) as

ˇ(a,a_)=q(a,a_)·c(a,a_),

wherec(a,a)isthepercapitarateatwhichanindividualofage amakescontactwithapersonofagea,perunitoftime,andq(a, a)aproportionalityfactorthatmaycapture,amongothereffects, age-specificsusceptibilityandinfectivity.

Inthisarticle,wecontrasttheconstantproportionality assump-tion,commonlyusedintheliteratureandreferredtoas“thesocial contacthypothesis”(Wallingaetal.,2006;Ogunjimietal.,2009; Melegaroetal.,2011),againstalog-linearfunctionoftheageofthe

susceptibleindividual,whichentailedanimprovementofmodelfit forVZVinBelgium(Goeyvaertsetal.,2010),thatisrespectively: log{q(a,a_)}=

0 and log{q(a,a)}=0+1a. (2) The contact rates c(a, a) are estimated fromthe POLYMOD contactsurveyusingabivariatesmoothingapproach,considering thosecontactswithskin-to-skintouchinglastingatleast15min sincethesecontactshavebeenshowntobemostpredictiveforVZV (Goeyvaertsetal.,2010;Melegaroetal.,2011).Forthecountries whoparticipatedinthePOLYMODproject,thecorresponding con-tactrateswereused,whereasfortheothercountriescontactdataof aneighboringcountryoracountrywithsimilarschoolenrollment ageswereused(cf.Table3inSupplementaryMaterial).Wepresent asensitivityanalysisintheSupplementaryMaterialtocompare thesead-hocchoiceswithamoreobjective selectionofcontact databymeansofAIC. Inthisanalysis,werepeattheestimation procedureforeachcountryseventimes,eachtimeforadifferent contactmatrix,andselect,percountry,fromthesesevenanalyses, theonethatresultsinthebestfittotheserologicaldata.Weobserve thattheeffectonR0remainswithinreasonablebounds,which indi-catesthatthechoiceofcontactdatahaslimitedinfluenceonour estimates.

Estimationprocedure.Inthisarticlewewillestimatetheforceof infectionusingmaximumlikelihoodestimationwiththeBernouilli log-likelihoodgivenby:

(;y,a)= n







Here,ndenotesthesizeoftheserologicaldatasetandyidenotes a binary variableindicating whether subject ihad experienced infectionbeforeageai.Thetransmissionratescannotbeestimated analyticallysincetheintegralEq.(1)hasnoclosedformsolution. However,itispossibletosolvethisnumericallybyturningtoa dis-creteageframework,assumingaconstantforceofinfectionineach 1-yearageinterval.Now,estimationproceedsasfollows:starting valuesfortheparametersareprovidedafterwhichthediscretized massactionprincipleisiterateduntilconvergenceandfinally,the resultingestimate ofthe forceof infectionis contrastedtothe serologyusingthelog-likelihood(3).Tocalculate95%confidence intervals,non-parametricbootstrapsareperformedonboththe contactdataandtheserologicaldatatoaccountforallsourcesof variability(Goeyvaertsetal.,2010).Thenumberofbootstrap sam-plespercountryisfixedat2000withconvergenceratesvarying between62%and100%.

SincesomecountrieslackserologicaldataonVZVintheolder agegroups,theoriginalserologyisaugmentedwithsimulateddata toavoidexcessvariabilityofthebootstrapestimates(Goeyvaerts

etal.,2010).ThesesimulationsaredrawnfromaBernouilli distri-butionwithmeanequaltotheseroprevalencefromthelast5age categorieswithatleast20observationsavailable.Thesizeofthe simulatedsamplesisdeterminedbythedemographyofthe popula-tion.Thismethodisplausiblefromanepidemiologicalpointofview sincetheVZVseroprofileisnotexpectedtodeclineafter20yearsof age.Basedontheaugmenteddata,post-stratificationweightsare calculatedusingcensusdataandincludedinthelikelihood.The lifeexpectancyLandtheage-specificmortalityrates_forevery countryareestimatedbasedondemographicdatafromtheyearof serologicaldatacollection(Eurostat,theOfficeforNational Statis-ticsforEnglandandWales,IsraeliBureauofStatisticsforIsrael) usingaPoissonmodelwithloglinkandoffsetterm(Hensetal., 2012).Toensureflexibility,aradialbasissplineisused.

ThedurationofmaternalimmunityisfixedatA=0.5years,while themeandurationofinfectiousnessforVZVistakenasD=7/365 years.Lastly,toreduceboundaryirregularitiesinducedby sparse-nessinthecontactdatafortheelderly,thecontactsurface,and hencetheserologicaldata,arerestrictedtothe0–69yearagerange. Asensitivityanalysisshowedlittleimpactonthepointestimates (resultsnotshown).

Model eligibility and indeterminacy. The estimated effective reproduction number ˆR and corresponding confidence interval allowus tocheck whethertheabove mixing patterns(2) con-formwiththeassumptionofendemicequilibrium.Inthissetting, eachinfectiousindividualinfectsoneotherindividualonaverage, henceRisexpectedtobeequalto1(Farrington,2003).Weusethis propertytoexcludethosemodelsforwhichRisestimatedtobe sig-nificantlydifferentfrom1.Furthermore,theeffectivereproduction numberallowsustomakeindirectinferenceabouttheage-specific heterogeneityrelatedtoinfectiousness,assuming

log{q(a,a)}=0+1a+2a, (4) whereaistheageoftheinfectiveindividual.Werefertothismodel astheextendedlog-linearmodel,inwhich2isreferredtoasan infectiousnesscomponent.Directinferencecanbetroublesome,as shownbyGoeyvaertsetal.(2010),sinceserologicalsurveysdonot provideinformationrelatedtoinfectiousness.Thisindeterminacy canbeillustratedasfollows:assumeforsimplicityˇ(a,a)=q(a, a)c(a,a)=q0q1(a)q2(a)c(a,a).Rewriting(1),thisimplies q0q1(a)= L(a)

ND



where(a),s(a)andc(a,a)canbeestimatedfromserologicaldata andsocialcontactdata,respectively.Thisimpliesthatwhenq0q1(a) isflexiblymodeled,theeffectofq2(a)ontheserologicalmodelis completelyabsorbedandthefitofthismodeldoesnotchangefor varyinginfectivitycurves.However,it doesaffecttheestimated valueofR0 andR.Wedealwiththisindeterminacybyletting2 varyoverafixedintervalandassessingtheeffecton ˆR.Thisway, thevalueof2canbedeterminedsuchthatRisnotsignificantly differentfrom1.ThisisillustratedinSection“Results”.

Elucidatingpotentialriskfactors

Toaddressthedifferencesintransmissibilitybetweencountries, a selection of 39 relevant country-specificvariables wasmade, comprisingdataondemography,childcare,populationdensityand weather(seeTable1inSupplementaryMaterial).Toinvestigate associationsbetweenR0 andthesevariables,twodifferent non-parametricapproachesareconsidered,whicharebrieflydescribed belowandmoreelaboratelyintheSupplementaryMaterial.

Maximal information coefficient. The maximal information coefficient (MIC) (Reshef et al., 2011) is a measure of two-variabledependence,designedspecificallyforrapidexplorationof

Fig.1. Estimatedbasicandeffectivereproductionnumberswith95%bootstrap percentileconfidenceintervalsforconstant(black),log-linear(gray)andextended log-linear(lightgray)proportionalityfactor.Foreachcountry,sizesofthedotsare proportionaltoAkaikeweights,hencelargerdotscorrespondtosmallerAICvalues. ThedottedhorizontallineindicatesthesingleeligiblevalueforRunderendemic equilibrium,whichisone.

high-dimensionaldatasets.TheMICispartofalargerfamilyof maximalinformation-basednon-parametricexplorationstatistics, whichcanbeusednotonlytoidentifyimportantrelationshipsin datasetsbutalsotocharacterizethem.

Randomforestapproach.Secondly,arandomforestapproachfor regressionisused(Breiman,2001),whichisaclassofensemble

methods–methodsthatgeneratemanyclassifiersandaggregate theirresults–specificallydesignedforclassificationandregression trees.Eachtreeisconstructedusingadifferentbootstrapsample ofthedataandeachnodeissplitusingthebestamongasubsetof predictorsrandomlychosenateachnode.Comparedtomanyother classifiers,thisturnsouttoperformverywellandisrobustagainst overfitting(Breiman,2001).Inaddition,ithasonlytwoparameters –thenumberofvariablesintherandomsubsetateachnodeand thenumberoftreesintheforest–andisusuallynotvery sensi-tivetotheirvalues.Weusetherandomforestalgorithmfromthe randomForestpackageinRwiththedefaultnumberoftrees(500). Thenumberofsplitvariablesisselectedsuchthatthehighest per-centageexplainedvarianceisobtained.Thepackageproducestwo measuresofimportanceofthepredictorvariables:“meandecrease inaccuracy”and“meandecreaseinnodepurity”.

Sensitivityanalysis.Totestthesensitivityofthisriskfactor anal-ysis,weappliedtheMICandrandomforestapproachtoestimates ofR0whenusingthebestfittingcontactmatrix.Thissensitivity analysisisincludedintheSupplementaryMaterial.Wecan con-cludethattheriskfactoranalysisisquiterobusttochangesinthe contactmatrix,asthemostimportantinfluentialfactorsdo not change.

Results

Basicand effective reproduction number.We apply the social contactdataapproachwithaconstantandage-specificlog-linear proportionality factor, as in (2), tothe 13 serologicaldata sets availableforVZV.Theestimatedbasicandeffectivereproduction numbersforboth modelsare presentedin Fig.1togetherwith 95%bootstrappercentileconfidenceintervals(alsoinTable3in SupplementaryMaterial). Thesize of thedots are proportional totheAkaikeweights(seeSupplementaryMaterial),hencelarger dotscorrespondtosmallerAICvalues.Theseestimatesare supple-mentedwithestimatesofthemeanageatinfectioninTable3in theSupplementaryMaterial.

Modelsareclassifiedaseligiblebasedonthe95%confidence intervalfortheeffectivereproductionnumber,andeligible mod-elsarecomparedbymeansofAIC.Whenthemodelwithlowest AICvalue is eligible,this model is selected.This resultsin the age-specificlog-linearproportionality factor beingpreferredfor Belgium,Denmark,Englandand Wales,Ireland,Israel,Italy,The NetherlandsandPoland.ForSpainandSlovakia,theconstant pro-portionalityfactorissufficienttoprovideagoodfit.ForFinland,the log-linearmodelispreferredintermsofAIC,butthismodelisnot eligible,whereasforLuxembourg,bothmodelsarenoteligible.In bothcases,theconstantandbasiclog-linearmodelarenotcapable ofprovidingagoodfittothedata.

Therefore,we considertheextended log-linearmodel in (4) forFinlandandLuxembourg.Fig.2presentstheprofilelikelihood estimatesof R0 and R asa function of 2. We observethat by includinganinfectiousnesscomponentintheproportionality fac-tor,theeffectivereproductionnumberRcanbeestimatedcloserto 1.NotethattheestimateofR0decreasesquitesubstantiallywith decreasing2,incontrasttoanincreaseinR.Thisreverserelation seemscounter-intuitive,butiscausedbyaninterplaybetweenq(a, a)ands(a).Now,byperforminganon-parametricbootstrapfor everyvalueof2 ona specificgrid,it ispossibletodetermine themaximalvalueof2 suchthat1iswithinthe95%bootstrap confidenceintervalofR.ThisisillustratedinFig.3.

The parameter estimates and confidence intervals for the extendedlog-linearmodelbasedonthesemaximalvaluesof2 arealsodisplayedinFig.1.Weobservethefollowing:forFinland, theextendedmodelhasanimprovedfitcomparedtotheconstant modelandisconformwiththeendemicequilibriumassumption.

Table2

TenfactorswiththelargestMICvalueofassociationwithR0,estimatedfromthe

finalmodelselectedforeachcountry,andcorrespondingSpearmancorrelation coefficientsS.

MIC S

1. Inequalityofincomedistribution 1.0 −0.64

2. Povertyrate 1.0 −0.73

3. %infantsvaccinatedagainstmumps 0.65 0.64

4. averagesquaremeterlivingareapp 0.59 0.42

5. %breastfeedingat3months 0.47 −0.21

6. %employedwomen25–49(min.1child0–5) 0.46 0.38

7. %infantsvaccinatedagainstpertussis 0.38 0.46

8. %infantsvaccinatedagainstrubella 0.36 0.51

9. %populationaged0–14 0.32 −0.22

10. Totalhealthexpenditure 0.32 0.51

ForLuxembourg,onlytheextendedmodeliseligible,andin addi-tion,ithasthelowestAICvalue.NotethattheestimateofR0for Luxembourgdecreasesconsiderably.

The estimated seroprevalence curves based on the selected modelforeachcountryarepresentedinFigs.4and5.Thefitted seroprofiles showa similarpattern acrosscountries,withmost infections occurring during early childhood and the estimated prevalenceapproachingoneasageincreases.However,the preva-lencedoesnotreachoneinallcountriesand,forexample,Italyhas amoreparticularprofile.LookingattheFOIcurves,thelargest esti-mateisobservedintheNetherlands(0.57year−1)attheageof5, followedbyLuxembourg(0.49year−1).ThelargestestimateofR0 isobtainedforTheNetherlands(7.60)andthelowestforEngland andWales(2.75).11outof13countrieshaveR0estimatedbelow 6.

Riskfactors.Thereisconsiderablevariationinestimatedbasic reproductionnumbers,andhenceintransmissibility,amongthe countriesunderconsideration.Therefore,weaimtoexplainthese differencesbyapplyingtheMICandrandomforestapproachona selectedsetof39relevantcountry-specificfactors(Tables1and2 inSupplementaryMaterial).Table2intheSupplementary Mate-rialcontainsthepairsofpotentialriskfactorswiththestrongest correlationgivenbytheSpearmancorrelationcoefficient.These correlationscanbeusedtointerprettherelationbetweenR0and certainfactors.

ThetenfactorswiththelargestMICofassociationwithR0,are presentedinTable2togetherwiththecorrespondingSpearman correlationcoefficients.Thisimplies,forexample,thatthehigher theinequalityofincome,thelowerR0.Resultsoftherandom for-estanalysisofR0aresummarizedinTable3wherethetenhighest scoringfactorsforbothimportancemeasuresaregiven.Comparing theresultsofbothanalyses,weobservethatfactorsrelatedtothe distributionofwealth(inequalityofincomeandpovertyrate), vac-cinationcoverageininfants(e.g.mumpsvaccinationcoverage)and childcareattendance(e.g.thepercentageofinfantsthatreceiveno formalcare)seemtobeassociatedwiththetransmissibilityofVZV. Discussion

Inthisarticle,weinvestigatedthetransmissibilityofVZVin12 Europeancountriesusingserologicalsurveydataandsocialcontact data.We contrastedthesocialcontacthypothesis,which is cur-rentlythemostusedapproachintheliterature,againstanapproach reflecting differences in characteristics related tosusceptibility andinfectivity.Furthermore,weintroducedtheeffective repro-ductionnumberasamodeleligibilitycriterionandweidentified whichcountry-specificsocio-demographicfactorsareimportant inexplainingdifferencesintransmissionpotentialbetween Euro-peancountriesusingtwonon-parametricapproaches:themaximal informationcoefficientandrandomforest.

Fig.2. ProfilelikelihoodestimatesofR0(leftaxis)andR(rightaxis)asafunctionof2,theparameterrelatedtoinfectiousness,forFinlandandLuxembourg.

Fig.3.ProfilelikelihoodestimatesofR(dots)withinterpolated95%bootstrappercentileconfidenceintervals(dashedlines)asafunctionof2,theparameterrelatedto

infectiousness,forFinlandandLuxembourg.Theverticaldottedlineindicatesthevalueof2forwhichtheupperconfidencelimitofRequals1(horizontaldottedline).

Table3

TenbestscoringfactorsobtainedbyarandomforestanalysisofR0,estimatedfromthefinalselectedmodelforeachcountry,andcorrespondingSpearmancorrelation

coefficientsS.

%increaseinMSE S Increaseinnodepurity S

1. Inequalityofincomedistribution −0.64 Inequalityofincomedistribution −0.64

2. Povertyrate −0.73 Povertyrate −0.73

3. Totalhealthexpenditure 0.51 Averagepopulationdensity 0.33

4. %0–2thatreceivenoformalcare −0.29 %0–2thatreceivenoformalcare −0.29

5. %infantsvaccinatedagainstmumps 0.64 Unmetmedicalneeds −0.31

6. %populationaged0–14 −0.22 Totalhealthexpenditure 0.51

7. %employedwomen(min.1child0–5) 0.38 Enrollmentrateschildren0–2 0.15

8. Averagesquaremeterlivingareapp 0.42 Averagesquaremeterlivingareapp 0.42

9. Averagepopulationdensity 0.33 %65+vaccinatedagainstinfluenza −0.19

Figs.4and5.Observedage-specificVZVseroprevalence(dots)andtheprofileestimatedfromthefinalmodelselectedforeachcountry(solidline).Thecorrespondingforce ofinfectionestimatesaredisplayedbythelowersolidline.

The social contact hypothesis provided a good fit to the VZV seroprevalence for only 2 out of 12 countries. The other countriesbenefitedfromanextended approachbyassumingan age-dependentproportionalityfactor,whichsupportsandextends earlierfindings of Goeyvaertset al. (2010)for VZVin Belgium. Thismayreflect theadditionalimportanceofage-specific char-acteristicsrelatedtosusceptibilityandinfectiousness,suchasthe meaninfectiousperiod.Furthermore,thesocialcontactdataare usedasproxiesforeventsbywhichaninfectionistransmitted. Hence,theproportionalityfactorcanalsobeconsideredasan age-specificadjustmentfactorrelatingthetruecontactratesunderlying infectiontothesocialcontactproxies.Alternatively,socialdataare difficulttocollectfromyoungchildren,withparentsfillingoutthe diaryontheirbehalf.Itmaywellbethattheyconsistently under-estimatethetruenumberofcontactsthatyoungchildrenmake.

Ouranalysisdirectlyimprovesupontheoriginalanalysisofthe ESEN2dataonVZVbyNardoneetal.(2007)whousedthe tra-ditionalAndersonandMayapproachbyimposinga3-parameter structureontheWAIFWmatrix(AndersonandMay,1991).Our methodof using R as a model eligibility criterion extends the approachofGoeyvaertset al.(2010)byaddressingthe indeter-minacyoftheinfectivityparameter.Ourresultscomplementthose ofMelegaroetal.(2011)whoanalyzedpartoftheVZVserology usingthesocialcontacthypothesisonly.Comparingtheestimated R0 values,wenoticethatourresultsingeneralsomewhatdiffer fromtheestimatesobtainedbyNardoneetal.(2007)andMelegaro etal.(2011).Thisisnot unexpected,since therearedifferences inmethodologyand it isknownthattransmissionassumptions havealargeimpactontheestimation ofR0.SeeTable4inthe SupplementaryMaterialforacomparativeoverviewoftheresults. TheresultsinFig.1indicatethattherearesubstantial epidemi-ologicaldifferencesbetweenEuropeancountries.Thisisimportant toconsiderwhenparametrizingmathematicalmodels.Childhood vaccinationcoverage(fordifferentvaccines),childcareattendance, populationdensityandaveragelivingareaperpersonwere pos-itivelyassociated with R0, whereas income inequality, poverty, breastfeeding,andtheproportionofchildrenunder14yearsof ageshowednegativeassociations.Whileitseemsintuitively logi-calthatgreaterchildcareattendanceandpopulationdensitylead tomorerapidspreadofvaricella,otherassociationsaremore dif-ficulttointerpret.Lesspovertyandincomeinequality,andhigher vaccinationcoveragesmaybeassociatedwithmoreaffluent soci-etiesinwhichwomenaremorelikelytobeemployedandchildren havemoreuniversalaccesstochildcareandkindergartenfroman earlyageon,facilitatingthespreadofVZV.

Inouranalyses,werelied ona fewassumptions. Firstofall, weassumedthattheserologicalstatusofanindividualisadirect measureofhis/hercurrentimmunityagainstVZV(Plotkin,2010). Further,weconsideredphysicalcontactslastinglongerthan15min tobeagoodproxyforpotentialvaricellatransmissioneventsas shownbyGoeyvaerts etal.(2010)for Belgium.Finally, ouruse ofRasa modeleligibility criterionreliedontheassumptionof endemicequilibrium.Thisassumptionissupportedbythe similar-ityintheresultsobtainedforthetwosamplesofItaly.Inaddition mostsurveysspantwo seasons,whichpartlycapturesany sea-sonalfluctuation.However,therearemanyfactorsthatcancause changesintheagedistributionofVZVcasesovertime,e.g.changes indemography,medicalpractice,socio-culturalfactorsetc. Look-ingatthismorerigorouslyrequiresanadditionalin-depthanalysis whichisthetopicoffutureresearch.However,togetasenseof theway ˆR changeswhendemographicorendemicequilibriumare perturbed,wepresentasensitivityanalysisintheSupplementary Material.We observethat ˆRincreaseswhenapercentageofthe newbornswouldhavebeenvaccinatedandwhenthenumberof birthswouldbeincreasing.Itdecreaseswhentheannualnumber ofbirthswoulddecrease.

Sincedirectinferencefortheinfectivityparameterishindered bythelackofinformationregardinginfectiousnessinthe serolog-icaldata,weestimatedthisparameterviaindirectinferenceusing theeffectivereproductionnumber.Thisindeterminacyillustrates that theuseofsocial contact datadoesnot completelyresolve theidentifiabilityissuesencounteredwhenestimatingmixing pat-ternsfromserologicaldata.Hence,furtherresearchisnecessaryto obtainadditionalknowledgeabouttheage-specificsusceptibility andinfectivityprofilesinordertoinformtheproportionalityfactor inthissocialcontactapproach.

Acknowledgments

ESacknowledgessupportfromaMethusalemresearchgrant fromtheFlemishGovernment.NGisbeneficiaryofapostdoctoral grantfromtheAXAResearchFund.NHacknowledgessupportfrom theAntwerpUniversityscientificchairinEvidence-Based Vacci-nology,financedin2009–2014byanunrestrictedgiftfromPfizer. AMis currently receivingfunding fromthe EuropeanResearch CouncilundertheEuropeanUnion’sSeventhFrameworkProgram (FP7/2007-2013)/ERCStartingGrant[AgreementNo.283955]. Sup-portfromtheIAP ResearchNetworkP7/06 oftheBelgianState (BelgianSciencePolicy)isgratefullyacknowledged.The computa-tionalresourcesandservicesusedinthisworkwereprovidedbythe HerculesFoundationandtheFlemishGovernment–department EWI.

ThisstudywasinitiatedaspartofPOLYMOD,aEuropean Com-missionprojectfundedwithintheSixthFrameworkProgramme, Contractnumber:SSP22-CT-2004-502084.

AppendixA. Supplementarydata

Supplementary data associated with this article can be found,intheonlineversion,athttp://dx.doi.org/10.1016/j.epidem. 2014.12.005.

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