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Epidemics
jo u r n al h om ep ag e :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
Estimation
of
age-specific
rates
of
reactivation
and
immune
boosting
of
the
varicella
zoster
virus
Isabella
Marinelli
a,b,
Alies
van
Lier
c,
Hester
de
Melker
c,
Andrea
Pugliese
a,
Michiel
van
Boven
c,∗aDipartimentodiMatematica,UniversitàdiTrento,Trento,Italy
bBasqueCenterforAppliedMathematics,Bilbao,Spain
cCentreforInfectiousDiseaseControl,NationalInstituteforPublicHealthandtheEnvironment,Bilthoven,TheNetherlands
a
r
t
i
c
l
e
i
n
f
o
Articlehistory:
Received24May2016
Receivedinrevisedform1November2016
Accepted7November2016
Availableonline22November2016
Keywords: Varicella Zoster Transmissionmodel Statisticalinference Vaccination
a
b
s
t
r
a
c
t
Studiesintotheimpactofvaccinationagainstthevaricellazostervirus(VZV)haveincreasinglyfocusedon herpeszoster(HZ),whichisbelievedtobeincreasinginvaccinatedpopulationswithdecreasinginfection pressure.ThisideacanbetracedbacktoHope-Simpson’shypothesis,inwhichaperson’simmunestatus determinesthelikelihoodthathe/shewilldevelopHZ.Immunitydecreasesovertime,andcanbeboosted bycontactwithapersonexperiencingvaricella(exogenousboosting)orbyareactivationattemptof thevirus(endogenousboosting).Hereweusetransmissionmodelstoestimateage-specificratesof reactivationandimmuneboosting,exogenousaswellasendogenous,usingzosterincidencedatafrom theNetherlands(2002–2011,n=7026).Theboostingandreactivationratesareestimatedwithsplines, enablingthesequantitiestobeoptimallyinformedbythedata.Theanalysesshowthatmodelswith highlevelsofexogenousboostingandestimatedorzeroendogenousboosting,constantrateoflossof immunity,andreactivationrateincreasingwithage(tomorethan5%peryearintheelderly)givethebest fittothedata.Estimatesoftheratesofimmuneboostingandreactivationarestronglycorrelated.This hasimportantimplicationsastheseparametersdeterminethefractionofthepopulationwithwaned immunity.Weconcludethatindependentevidenceonratesofimmuneboostingandreactivationin personswithwanedimmunityareneededtorobustlypredicttheimpactofvaricellavaccinationonthe incidenceofHZ.
©2016TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCC BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Varicellazostervirus(VZV)isaherpesviruscausingadisease knownaschickenpoxorvaricella.Afterresolvingofthesystemic primaryinfection,thevirusremainslatentlypresentinthehost. Theviruscanreactivatelaterinlife,resultinginadiseaseknownas herpeszoster(HZ)orshingles.Datingbacktotheworkof Hope-Simpson (1965), it hasbeen hypothesizedthat boosting ofthe immunesystemof alatentlyinfected personby contactwitha person experiencing chickenpoxcould lower theprobability of HZ developing at somepoint later in life (Brissonet al., 2010; Guzzettaetal.,2013;Ogunjimietal.,2013,2014;Thomasetal., 2002).Iftrue,suchexogenousimmuneboostingcouldhave pro-foundimplicationsfor vaccination programsaimedat reducing chickenpox, astheseare expectedtoreducevirustransmission
∗ Correspondingauthor.
E-mailaddress:michiel.van.boven@rivm.nl(M.vanBoven).
andexogenousimmuneboosting.Overtheyearsthishasbecome apopularhypothesis toexplainandpredictthedynamicsofHZ in unvaccinated and vaccinated populations, and an important reserveforcountriestointroduceVZVvaccination(Bonannietal., 2009;Brissonetal.,2010;Guzzettaetal.,2013;Polettietal.,2013). Whileearlierstudiesfocusedexclusivelyontheimplicationsof exogenousboosting(Bettaetal.,2016;Brissonetal.,2010;Guzzetta etal.,2013,2016;Ogunjimietal.,2015;Polettietal.,2013;vanLier etal.,2015),Hope-Simpson’soriginalhypothesisstatesthatboth exogenousandendogenousboosting,byreactivationattemptsof thevirusthataresuccessfullycounteredbytheimmunesystem ofthehost,contributetothemaintenanceofimmunity.Although dataarescarce,somestudiesindeedshowsubclinicalreactivation inspecificpopulations,mostlikelyinresponsetostress(Cohrsetal., 2008;Ljungmanetal.,1986;Mehtaetal.,2004;Papaevangelou etal.,2013;Wilsonetal.,1992).Itisthereforeofinterestto eval-uatetheabilityofendogenousandexogenousboostingtogether toexplaintheavailableHZincidencedata,andtoobtaininsight intherelativestrengthofendogenousversusexogenousimmune http://dx.doi.org/10.1016/j.epidem.2016.11.001
1755-4365/©2016TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/
Fig.1.Overviewofvaricellazostervirus(VZV)seroprevalence(i.e.,fractionofthepopulationinfected)(left-handpanel)andestimatedforceofinfection(right-handpanel)
(vanLieretal.,2015).Theshadedarearepresentsthe95%confidencebandoftheforceofinfection.
boosting. Here we build on earlier studies (Betta et al., 2016; Brissonetal.,2010;Guzzettaetal.,2013,2016;Polettietal.,2013; Ogunjimietal.,2015;vanLieretal.,2015)toobtainestimatesof keyparametersdeterminingthedynamicsofVZVinunvaccinated andvaccinatedpopulations.Specifically,weuselong-termHZ inci-dencedatafromtheNetherlands(2002–2011,n=7026cases(van Lieretal.,2015))toestimateage-specificratesofimmune boost-ing,lossofimmunity,andreactivationinapopulationwithoutVZV vaccination.Ourapproachextendsearlieranalysesby(i)including thepossibilityofendogenousnexttoexogenousimmuneboosting, and(ii)allowingtheage-specificratesofreactivationandimmune boostingtohaveflexibleshapes,usingnaturalcubicsplines.Inthis manner,theshapesandvaluesoftheratefunctionsareoptimally informedbythedata,incontrasttopreviousstudiesthathavemade restrictiveassumptionsfromtheonset(e.g.,Guzzettaetal.,2013; vanLieretal.,2015).
Theanalyses reveal that rates of loss of immunity arehigh (0.048–0.10peryear),thatreactivationratesarelowupto40years, increasestronglyto0.03–0.10peryearatoldage,andthatthe frac-tionofthepopulationwithwanedimmunitythatispronetoHZ decreasesfrom45–55%atage25to10–40%atage80,anddepends sensitivelyonwhetherornotendogenousboostingistakeninto account.Ouranalysesshowthatexogenousboostingisstronger thanendogenous immuneboosting in childrenand adultswith youngchildren,andthatitisconceivablethatendogenous boost-ingexceedsexogenousboosting inotheragestrata.We discuss theimplicationsforvaccinationprogramsaimingtoreduce chick-enpox,andpossiblefutureavenuestoobtainfurtherquantitative informationonthemagnitudeoftheHope-Simpsoneffect.
2. Methods
2.1. Varicellaprevalence
The prevalence of VZV by age is based on Dutch serologi-caldata,derivedfromthesecondcross-sectionalserosurveillance study(PIENTER2),conductedamongpeopleaged0–79yearsinthe Netherlandsin2006/2007(VanDerKlisetal.,2009).The seropreva-lenceofVZVinthePIENTER2studyisquantitativelyverysimilar totheseroprevalenceintheearlierPIENTER1study(conductedin 1995/1996),indicatinglittlechangeoftheageatinfectionwith VZVovertheperiod1996–2006(deMelkeretal.,2006;VanLier etal.,2013).Toavoidtheinterferenceofmaternalantibodieswe onlyincludethe6251samplesofparticipantsaged6monthsor older.TheVZVseroprevalencedataprovidethebasisfor estima-tionofthebasicreproductionnumberofVZVusingtheso-called socialcontacthypothesis(vanLieretal.,2015).Hereweusethe earlierestimatesoftheforceofinfectionasinputforestimationof theratesofreactivationandimmuneboosting.Thedataandearlier modelfitareshowninFig.1,andcanbedownloadedinthedata supplementofanearlierpublication(vanLieretal.,2015).
2.2. Herpeszosterincidence
Sentineldataontheincidenceofgeneralpractitioner(GP) con-sultationsduetoHZbyageintheperiod2002–2011(basedon 7026cases)areprovidedbyNIVEL,theNetherlandsInstitutefor HealthServicesResearch(vanLieretal.,2015).Themajorityof HZ patients consult their GP becauseit is a painful condition. BecauseHZ complaintsarehighly specificand accompaniedby typicallesions,witha positivepredictivevalueof clinical judg-mentof90.8%(95%CI:87.3–94.3%),weexpectthatmisclassification ofthediagnosisbytheGPisrare(Opsteltenetal.,2007).Hence, we usethesentinel GPdataas a proxy for thetotal incidence of HZin thecatchmentpopulation. The HZdata canbe down-loadedinthedatasupplementofanearlierpublication(vanLier etal.,2015),andshowsnoincreaseordecreaseofincidenceover time, and norelative increase or decrease in anyagegroup in thestudyperiod.TheHZdataarethemainsourceofinformation onwhichestimatesoftheratesofreactivationandboostingare based.
2.3. Modelstructureandanalysis
ThemathematicalmodeltodescribeprogressiontoHZisbased onearlierstudies(Guzzettaetal.,2013;Polettietal.,2013;vanLier etal.,2015).Weextendthesemodelsby(i)includingendogenous nexttoexogenousimmuneboosting,asinHope-Simpson’soriginal hypothesis(Hope-Simpson,1965),and(ii)allowingfortheratesof reactivationandimmuneboostingtoberelativelyunconstrained, usingnaturalcubicsplines.
Thepopulationisassumedtobeindemographicequilibrium, andinfectiouscontacts(i.e.,contactssufficientfortransmission) betweenpersonsofdifferentagesaremadeaccordingtothesocial contacthypothesis.Thisimpliesthattheprobabilityofan infec-tiouscontactisproportionaltoobservedcontactpatternsinthe Netherlands(Mossongetal.,2008;vanLieretal.,2015).Wealso assume,asinearlierstudiesandbackedbyempiricalevidence,that primaryvaricellaisinfectiousandthatinfectiousnessofHZis neg-ligible(Guzzettaetal.,2013;Seiler,1949;vanLieretal.,2015). Next,weassumeinourmainanalysesthatreactivationcanoccurat mostonceduringalifetime(Hope-Simpson,1965;Oxman,2009). Wehavealsoconsideredanextensionofthemainmodelinwhich multiplereactivationsarepossible,and obtainedsimilarresults (AppendixE).Forthisreason,wefocushereonthemodelwith atmostonereactivationinaperson’slifetime(Fig.2).
Further,theduration ofvaricella andHZ episodes(weeksto months)areshortcomparedwiththehumanlifespan(decades), and we may therefore use the short-disease approximation in which the varicella and HZ states are not explicitly modeled (Goeyvaertsetal.,2010).Finally,asinearlierstudies(Guzzettaetal., 2013;Polettietal.,2013)butcontrastingwithothers(Marziano etal.,2015),wemodelthedistributionofpersonsoverclassesby
Fig.2.Schematicofthemodelundertheshort-diseaseapproximation.S(a)denotes
theage-specificprevalenceofuninfectedpersons,V(a)theage-specificprevalence
oflatentlyinfectedpersonswithsufficientimmunitytopreventreactivation,W(a)
theprevalenceoflatentlyinfectedpersonswithwanedimmunity,andR(a)the
prevalenceofpersonswhohaveexperiencedHZ.Theforceofinfectionandrate
ofexogenousimmuneboostingaregivenby(a)andz(a),andtheratesoflossof
immunity,endogenousimmuneboosting,andreactivationaregivenby(a),(a),
and(a).
agewithafixedcontactstructure,withdemographiccomposition asintheNetherlandsduringcollectionofthecontactdata.
Fig.2shows a schematicof themodel,and Eqs.(1) and (2) giveamathematicaldescription.Individualsarebornsusceptible toVZVinfection(intheSclass).FromtheScompartment individ-ualsareinfectedatanage-specificrate(a)(theforceofinfection), afterwhichtheyentertheVcompartment(latentlyinfectedwith immunitytoreactivation).Immunitytoreactivationislostatarate (a),andgainedatratesz(a)and(a)byexogenousand endoge-nousboostingrespectively.Reactivationfromthelatentlyinfected compartmentwithwanedimmunityWoccursatarate(a),and individualswhohaveexperiencedHZafterreactivationenterthe Rcompartment.
Mathematically,themodeldynamicsarespecifiedbythe fol-lowingsetofordinarydifferentialequations
dS(a)
da =−(a)S(a), dV(a)
da =(a)S(a)−(a)V(a)+(z(a)+(a))W(a), dW(a)
da =(a)V(a)−(z(a)+(a)+(a))W(a), dR(a)
da =(a)W(a),
(1)
withforceofinfection(a)givenby (a)=q
M 0c(a,s)(s)S(s)ds, (2)
andinitialconditionsS(0)=1,V(0)=0,W(0)=0,andR(0)=0.Inthe above,M=100(yr)isthemaximumage,c(a,s)isthecontactrate betweenindividualsofagesandthoseofagea,andqisthe propor-tionalityparameterthathasbeenestimatedfromseroprevalence data.Specifically,withproperscalingofthecontactfunctionthe proportionalityparameterqcanbeequatedwiththebasic repro-ductionnumberR0.Throughoutwetake ˆR0=8.7(95%CI:8.1–9.3), asestimatedusingseroprevalencedata(vanLieretal.,2015).Notice thatthisimpliesthatintheNetherlandsinfectionofinfantsand children occursata relativelyyoungage(Nardone etal.,2007; VanLieretal.,2013).Noticefurthermorethattheabove implic-itlyimpliesthatwedonotusetheexplicitEq.(2)intheestimation procedure,butratherinserttheestimateofforceofinfection(a)
inEq.(1)(Fig.1)asobtainedearlier(vanLieretal.,2015).Inthis manner,allotherestimatesoftransitionratesarefullycompatible withtheserologicaldata(Fig.1),atthe(slight)costthattheforceof infectionmaynotbeoptimallyinformedbytheHZincidencedata (vanLieretal.,2015).2.4. Discretization
Statisticalinferenceoftransitionratesisbasedonadiscretized approximationoftheexpandedstatespace(Goeyvaertsetal.,2010; Guzzettaetal.,2013;vanLieretal.,2015).First,weextendthe statespacebykeepingtracknotonlyofwhetherindividualsare latentlyinfectedwithorwithoutimmunitytoreactivation(i.e.,in
compartmentsVandW),butalsobycountingthenumberoftimes thatindividualshavepassedthroughtheVandWcompartments. Hence,ViandWirefertoindividualswhoareintheVandW com-partmentsforthei-thtime.WecanthensolveEq.(1)recursively forallViandWi(i=1,2,...)intermsoftheforceofinfection(a). Nextweapproximatethesystembyassumingthatindividualscan onlycyclethroughtheViandWi statesforacertainnumberof times.Backedbypreliminaryanalyseswetakethisnumbertobe5, astheerrorsointroducedremainssmallwhencomparedtoresults withhighermaximalnumberofcycles(notshown).Finally,we dis-cretizethesolutionsforViandWitobeabletotiethemodeledto theobservedincidenceofHZ.SincetheHZincidencedataare avail-ableinagegroupsof1year,wealsotakeagegroupsof1yearin thediscretization.DetailsaregiveninAppendixA.
2.5. Naturalcubicsplines
Flexibleestimationofthetransitionrates(a),(a),and(a) isensuredbyusingnaturalcubicsplines(deBoor,2001).Acubic splineisasmoothfunctionthatisconstructedfromcubic poly-nomial pieces that are joined at certain knots, and that have continuousfirstandsecondderivatives(deBoor,2001).Forcubic splines the number of polynomial basis functions (pieces) is m=n+3,wherenisthenumberofageintervals.For simplicity, weassumethroughoutthattheintervalsareofthesamelength.
Next,eachratef(a)canbewrittenasalinearcombinationof elementsofaB-splinebasis{Bj}m
j=1 f(a)= m
j=1 ˇjBj(a), whereˇT=(ˇ1,...,ˇm)arethecoefficientsofthebasisfunctions thatneedtobeestimated.Basedontheabove,thediscretized age-specificratesfT=(f(a1),...,f(a100))aregivenby
f=Bˇ,
whereB=(bij)=Bj(ai).Weevaluatetheperformanceofmodelsfor arangeofvaluesfornusingAkaikeinformationcriterion(AIC)and Bayesianinformationcriterion(BIC)toselecttheoptimalnumber ofageintervals(BurnhamandAnderson,2003).AppendixBshows resultsforn=4,...,7.Theanalysessuggestthatn=6isoptimal (i.e.,knotsatxi= 100i
6 years,withi=0,1,...,6),andwewilluse thisvaluethroughout.Hence,m=9andBisa100×9matrix. 2.6. Estimationandmodelselection
Estimatesoftheparametersareobtainedbymaximizationof thelikelihoodoftheHZincidence.Asthenumberofcasesissmall comparedtothenumberofperson-years,bothwithregardtothe total number of cases and total number of person-years(7026 and 2,049,362)andthenumbersperagegroup,we usea Pois-sonlikelihoodwithparameteriWiNi,whereiWiistheexpected (modeled)incidencerateofHZinagegroupi,andNiisthenumber ofperson-years.Confidenceintervalsofparameters(,,,andz) arecalculatedbydirectresamplingofthedata.Here,200 resam-pleddatasetsaregeneratedassumingthatthenumberofHZcases ineachoftheage-classesfollowsaBinomialdistributionwith bino-mialtotalsNiandparameterIi/Ni),whereNiandIiarethereported numberofindividualsandreportednumberofcasesinage-classi, respectively.Throughoutweuseageclassesof1year,sothatwith theexceptionoftheyoungestageclasses(<4years)eachageclass containsmorethan50cases.Foreachoftheresampleddatasets theparametersarere-estimated,andpercentilebootstrap confi-denceintervalsarecalculated.ModelselectionisbasedonAICand
Table1
Overviewofmodelfits.Asuiteofsimplificationsofthebaselinemodel(Scenario1)areconsideredinwhichoneormoreoftheparametersareconstantinsteadofbeing
modeledwithsplines.Shownarethemaximizedlog-likelihood(log(Lmax)),theassociatedAICandBICdifferenceswiththebest-fittingmodelScenario6*inthemaintext,
andthedegreesoffreedom(df).Thebest-fittingmodeldoesnotincludeendogenousimmuneboosting(≡0),andhasaconstantrateoflossofimmunity( ˆ=0.076year−1).
SeeTable2foranoverviewofparameterestimates.
Scenario Constantrates Label log(Lmax) AIC BIC df
1 – −347.5 16.0 60.2 28
2 −348.0 0.8 24.2 20
3 −352.7 10.4 33.7 20
4 −351.9 8.7 32.1 20
5 , −360.2 9.2 11.8 12
6 , Modelwithendogenousboosting −354.8 −1.6 0.98 12
6* ≡0, Modelwithoutendogenousboosting −356.6 0 0 11
7 , −383.7 56.3 58.9 12
8 Allrates −1363.7 2000.3 1982.1 4
Thevalueshavebeenplottedinboldfacepurelytoemphasizethemodelwithhigheststatisticalsupport.
BICdifferences (Burnhamand Anderson,2003).Allanalysesare
performedusingR3.2(RDevelopmentCoreTeam,2008). 2.7. Scenarios
Weconsiderasuiteofscenariostoestimatetherateoflossof immunitytoreactivation((a)),therateofendogenousimmune boosting((a)),therelativesusceptibilityoflatentlyinfected indi-vidualstoexogenousimmuneboosting(z),andthereactivationrate ((a)).Thescenariosdifferbywhethertheratesareassumed con-stantorestimatedwithage-dependentsplines.Inthebaseline sce-nario(Scenario1)allratesareestimatedwithsplines.InScenarios 2–4onerateparameterisassumedconstantatatime,inScenarios 5–7tworatesareassumedconstant,andinScenario8allthreerates areconstant.Finally,basedonpreliminaryanalyseswealsoinclude aScenario(Scenario6*)inwhichtheexogenousreactivationrateis assumedtobezero(≡0).Allthesescenariosincludeexogenous boosting.Tovalidatethischoice,wealsoconsiderasimilarsuite ofscenariosthatdonotincludeexogenousboosting(seeAppendix C).Asexpected,thesemodelsgiveaworsefittothedata.
3. Results
Table1givesanoverviewoffitsofthedifferentScenariosbased onAICandBICdifferences.Scenarioswithallratesassumed con-stant(Scenario8)orwithconstantratesofreactivationandlossof immunity(Scenario7)donotperformwell(AICandBICwith thebestfittingmodelarewellover50inbothcases).Betterfitsare obtainedwithmodelsinwhichallparametersareestimatedwith splines(Scenario1),inwhichoneoftherateparametersisassumed constant(Scenarios2–4),orinwhichboththerateofendogenous boosting()andthereactivationrate()areconstant(Scenario 5).Overall,thebest-fittingmodelshaveaconstantorzerorateof endogenousimmuneboosting,constantrateoflossofimmunity, andage-specificrateofreactivation.Wehenceforthfocusonthese twomodels,whichwewilllabelthemodelwith(Scenarios6)and without(Scenarios6*)endogenousboosting.
Parameterestimatesofthebest-fittingmodelswithand with-outendogenous boosting are given in Table2 and Fig.3.Both scenariosproducegoodandsimilarfitstotheHZincidencedata (Fig.3).Waningofimmunityoccursonatimescalethatvariesfrom 20years( ˆ=0.048peryear(95%CI:0.033–0.068))inthemodel withendogenousboostingto13years( ˆ=0.076peryear(95%CI: 0.063–0.087))inthemodelwithoutendogenousboosting.Further, whiletheendogenousreactivationrateis0inthemodelwithout endogenousboosting,itisestimatedatˆ=0.093peryear(95%CI: 0.069–0.11)inthemodelwithendogenousboosting. Susceptibil-ityoflatentlyinfectedindividualsrelativetothesusceptibilityof uninfectedpersonsperinfectiouscontactisestimatedtobe approx-imately1(ˆz=1(95%CI:0.0.998–1)inthemodelwithendogenous
boostingand ˆz =0.987(95%CI:0.986–0.990)inthemodelwithout endogenousboosting),thisindicatesthataninfectiouscontactthat wouldleadtoinfectionofanuninfectedpersonisalsosufficientto leadtoimmuneboosting.Asaconsequence,exogenousimmune boostingalmostequalstheforceofinfection(Fig.1),andin chil-drenandintheagegroupofadultswithyoungchildren(≈30–40 years)exogenousimmuneboostingexceedsendogenousboosting (ifpresent).Inolderadults(>40years)andolderchildrenandyoung adults(<30years)whohave alowestimatedforceofinfection, exogenousimmuneboostingisexpectedtobelessprominent,and endogenousboosting exceeds exogenousboosting inthemodel withimmuneboosting, perhapswithexceptionofgrandparents withyounggrandchildren(≈65years).
Estimatedreactivationratesareage-dependentinboth scenar-ios withhighstatisticalsupport,increasingwithagefrom,say, 40yearsonwards.Specifically,estimatesareapproximately0.10 peryearinelderlyinthemodelwithendogenousimmune boost-ing and over 0.02 per year in the model without endogenous immuneboosting(Fig.3).Interestingly,thereactivationratesare substantiallyhigherinthemodelwithendogenousboostingthan in themodelwithout endogenous boosting fromtheageof 60 yearsonwards,whiletheestimatedfractionthatisavailablefor reactivation( W(a))is substantiallyhigherinthemodelwithout endogenousimmuneboosting(Fig.3).Hence,bothscenariosmake differentinterpretationsonthefractionofthepopulationthatis availableforreactivationinolderadultsandelderly.Hence,while themodelsprovide similarfits totheHZ incidencedata, there aresignificantunderlyingdifferenceswithregardtotheratesof endogenous immune boosting, reactivation, and fraction ofthe populationthatisavailableforreactivation.
Although it is expected that the majority of cases of HZ is reported to a GP (Opstelten et al., 2007), some degree of under-ascertainmentisconceivable.Additionally,eventhoughHZ symptomsarehighlyspecific,itispossiblethatsomereportedcases ofHZmayhavebeenmisdiagnosed.Toinvestigatetherobustness oftheaboveresultstounder-andover-ascertainment,wehave reanalyzedthedataassuming10%under-ascertainmentand10% over-ascertainment.Theresultsoftheanalysesarelaiddownin AppendixD(seeTable5andFig.5fordetails)andshowthatthe resultswitheitherunder-orover-ascertainmentarequantitatively closetotheresultsobtainedinTables1and2andFig.3.
4. Discussion
Our analyses provide support for the hypothesis of Hope-Simpson(1965)thatimmuneboostingaffectstheincidenceofHZ. Ourresultsaddtoearlierstudies(Bettaetal.,2016;Guzzettaetal., 2013,2016;Polettietal.,2013;vanLieretal.,2015)byshowing thatimmuneboostingisapotentforceinamodelthatincludes both endogenous and exogenous boosting, and that allowsthe
Table2
Parameterestimatesofthescenarioswithandwithoutendogenousimmuneboosting(cf.Table1).Shownaremaximumlikelihoodestimateswith95%confidencebounds
forparametersthatareassumedconstant.SeeFig.2forparametersestimatedwithsplines.
Scenario Parameter
(year−1)(95%CI) (year−1)(95%CI) z(95%CI)
6 0.048(0.033–0.068) 0.093(0.069–0.11) 1(0.998–1)
6* 0.076(0.063–0.087) – 0.987(0.986–0.990)
Fig.3.HZincidencedataandoverviewofthebest-fittingmodels(left-handpanels:withendogenousimmuneboosting,right-handpanels:withoutendogenousimmune boosting).ShownaretheobservedandpredictedincidenceofHZinbothscenarios(toppanels;dots:data;lines:prediction),thefractionofthepopulationwithwaned immunity(W(a))(middlepanels),andtheestimatedreactivationrates((a))(bottompanels).Linescorrespondtomaximumlikelihoodestimatesandshadedareasto bootstrapped95%confidenceranges.AlsoshownarethereactivationratesasestimatedforItaly,Finland,England(andWales),andGermanyinmorerestrictiveanalyses (Brissonetal.,2010;Polettietal.,2013).
reactivationratetotakeflexibleshape.Ourquantitative estima-tionsfurthershowthatinmostagegroupstherateofexogenous immuneboostingissubstantiallyhigherthantherateof endoge-nousimmuneboosting,butalsounderlinethatevenlowestimated rates of endogenous immune boosting significantly affect the distribution of latently infected individuals with and without waned immunity. More generally, the distribution of latently infected individualswithand withoutimmunitytoreactivation is shaped by an intricate interplay between the age-specific ratesatwhichimmunityislostandgained(seealsoBettaetal., 2016;Guzzettaetal.,2016),andevensmallratesofendogenous boostingaffectthisdistributionsignificantly.Themainreasonis that thereis nodirect information underpinning which partof theadultpopulationislatentlyinfectedwithandwithoutwaned immunity. The implication is that not all individuals who are latently infected may be available for developing HZ, but that theexactfractioncannotbeestimatedwithprecision.Hence,for
publichealthconsiderableuncertaintyremainsaboutthepotential effectivenessofHZvaccinationofelderly.
Waningofimmunityand,toalesserextent,immuneboosting arewelldocumented,andhavebeenincludedinanagent-based transmissionmodeltoanticipatetheimpactofvaricellavaccination of HZ incidence (Ogunjimi et al., 2015).In this model, immu-nity is a continuous quantity, and each individual is equipped withhis/herownlevelof(cellular)immunity.Thiscontrastswith our model in which immunity to reactivation is either com-plete(compartmentV) orcompletelyabsent(compartment W). Fromabiologicalperspective,theearliermodel(Ogunjimietal., 2015) is biologically more plausible, while our current model issimplerandmoreamenabletoformalstatisticalinference.In both models, much of theremaining variation and uncertainty derives from the fact that there is very little data topinpoint exactly which fraction of the adult population is available for reactivation.
Anumberofassumptionsneedscrutiny.First,wehaveassumed, asinearlierstudies(e.g.,Guzzettaetal.,2013;Polettietal.,2013), thattheforceofinfectionisknown.Herewehavetakentheforce ofinfectionasestimatedfromDutchseroprevalencedata(vanLier etal.,2015).Ideally,allparametersofinterestwouldbeestimated inonego,takingintoaccountboththeseroprevalenceandHZ inci-dencedata.Thisis,however,computationallydemanding.Further, ithasbeenarguedthatthebasicreproductionnumberandforceof infectionarestronglyidentifiedbyserologicaldata,thereby reduc-ingtheneedtoincludealldatainonesynthesizinganalysis(van Lieretal.,2015).Earlierstudieshavetakenasimilarapproachin whichtheforceofinfectionisestimatedfirst,andtheninasecond stepthetransitionratesareestimatedfromHZincidence(Guzzetta etal.,2013;Polettietal.,2013;vanLieretal.,2015).
Second,ouranalysesarebasedonthearbitraryassumptionthat thesplineshavefiveinteriorknots.Thisseemstoproduce reason-ablyflexibleshapesofthereactivationratewithoutgrossly overpa-rameterizingthemodel.Itshouldbenoted,however,thatwhether ornotmodelswithconstantorflexibleratesofboosting,lossof immunity,andreactivationarepreferredbyinformationcriteria suchasAICorBICdependsnotonlyonthedatabutalsoonthe num-berofadditionalparameters(andhencethenumberofknots).A computationallymoreinvolvedalternativecouldforinstancemake useofpenalizedsplinesinwhichapenaltyisusedtoavoid exces-sivefluctuationsoftheratefunctions(EilersandMarx,1996).This, however,wouldalsonecessitateestimationoftheeffectivenumber ofparameters,andcannotaddressthequestionwhetheritislikely ornotthatcertainparametersareconstant,asisinvariablyassumed inmodelingstudies.Inasensitivityanalysiswehavedecreasedand increasedthenumberofknotsandobtainedquantitativelysimilar results,showingthatourresultsdonotdependsensitivelyonthe degreeoffreedomoftheratefunctions(AppendixB).
Fig.3showsthereactivationratesasestimatedusingourmodel from Dutch HZ incidence data together with estimates of the reactivationratesasestimatedearlierusingHZincidencedatafrom Germany,Finland,Italy,andEngland(Brissonetal.,2010;Poletti etal.,2013).Interestingly,ourestimatesareclosetoearlier esti-matesbyBrissonandcolleaguesuptotheageof60(Brissonetal., 2010).Inolderagegroupsourestimatesofthereactivationrates aresubstantiallyhigherinthescenariowithendogenousimmune boosting.Thiscanbeattributedtothefactthatinthisscenariomore olderpersonshavefunctionalimmunityduetoendogenous boost-ing,andhigherreactivationratesarerequiredtoexplaintheHZ incidencedata.Alsointerestingisthefindinginourstudyandthe earlierstudiesbyBrissonandcolleaguesandPolettiandcolleagues thattheestimatedreactivationratedecreasesinchildhooduntilthe ageof23years.Intheearlierstudiesonecouldhavearguedthatthis isaconsequenceoftherelativeinflexibilityofthefunction describ-ingthereactivationrate.Thisexplanation,however,doesnothold inouranalyses,andsuggeststhatthehighestimatedreactivation ratesinchildrenmayreflectatruebiologicalphenomenon.A rea-sonableexplanationmightbethatforsomereasontheimmune systemofchildrenhasnot(yet)acquiredfullcontrolofthevirus.
Ourresultshavepotentialimplicationsforlarge-scalevaricella andcombinedvaricellaandzostervaccinationprograms.Ourtwo mainscenarios(Fig.3)makedifferentexplanationsonhowthe observedzostercasesareproduced.In themodelwith endoge-nousboosting thefractionof personswithwanedimmunity is smallwhilethereactivationrateishigh,whileinthemodelwithout endogenousboostingthefractionwithwanedimmunityishigher butthereactivationrateislower.Indiscriminatezoster vaccina-tionisexpectedtopreventsimilarnumberofcasesofHZinboth cases.However,thisexpectationmaynotholdanymoreif large-scalevaricellavaccinationis addedtothemix.Inthis case,the forceofinfectionisexpectedtobereducedtoasimilarextentin bothscenariosasestimatesoftheexogenousboostingparameter
zareveryclose(Table2).Asaresult, theimpactof exogenous boostingis expectedtobereducedtoasimilarextentafterthe introductionofvaricellavaccination.However,immuneboosting isstillafactorinthemodelwithendogenousboosting(cf.Table2). Asaconsequence,onewouldexpectthatpersonsthatare exoge-nouslyboostedintheabsenceofvaricellavaccinationcouldstillbe boostedendogenouslyinthepresenceofvaricellavaccination.This couldpotentiallyleadtoalessdramatictransientincreaseinthe incidenceofHZaftertheintroductionofvaricellavaccinationthan predictedearlier(Brissonetal.,2010;Guzzettaetal.,2013,2016; Karhunenetal.,2010;SchuetteandHethcote,1999;vanLieretal., 2015).Thedownsidewouldbethattheaddedvalueofzoster vac-cinationwouldalsobediminished.Ofcourse,inanidealscenarioit wouldbepossibletotargetzostervaccinationspecificallytothose personswithwanedimmunitywhoareatriskofdevelopingzoster.
Acknowledgements
We thank thepersons includedin the PIENTER2 serological studyfortheirparticipation,andPiero Manfrediforstimulating discussions.ThisresearchissupportedbytheBasqueGovernment throughtheBERC2014–2017programand bySpanishMinistry ofEconomyandCompetitiveness MINECO:BCAMSeveroOchoa excellenceaccreditationSEV-2013-0323.
AppendixA. Modelsolutionanddiscretization
WebaseourestimationproceduresontheODEsspecifiedby Eqs.(1)and(2).Intheanalysesweuseearlierestimatesoftheforce ofinfection(a)(vanLieretal.,2015),andobtainmaximum likeli-hoodestimatesofotherparametersusingthediscretizedsolution ofEq.(1).Specifically,thepopulationissplitintoJ=100ageclasses of1year.
Asafirststepweexpandthestatespacebykeepingtrackof thenumberoftimesthatanindividualhasbeenintheVandW compartments(Guzzettaetal.,2013).Hence,Vi(a)representsthe fractionofthepopulationofageawhoareintheVcompartment forthei-thtime(i=1,2,...).SeeFig.4foraschematicofthemodel withexpandedstatespace(seealsoGuzzettaetal.,2013).TheODEs ofEq.(1)forV(a)andW(a)arereplacedintheexpandedmodelby
dV1(a)
da =(a)S(a)−(a)V1(a) dW1(a)
da =(a)V1(a)−(z(a)+(a)+(a))W1(a) ···
dVi(a)
da =(z(a)+(a))Wi−1(a)−(a)Vi(a) dWi(a)
da =(a)Vi(a)−(z(a)+(a)+(a))Wi(a),
(3)
withi=2,3,...andappropriateinitialconditions.Intheabove, the(total)fractionsV(a) andW(a)inthelatentlyinfected com-partmentsaregivenbyinfinitesumsV(a)=
∞i=1Vi(a)andW(a)= ∞i=1Wi(a). Reasonable approximations for these sums can be obtainedbyassumingthatthereisafinitebutsufficientlylarge numberoftimesnthatanindividualcanpassthroughtheVand
Wcompartmentsduringitslifetime. Hence,Vi(a)and Wi(a)are approximatedby V(a)≈ n
i=1Vi(a) and W(a)≈ n
i=1 Wi(a),
withn∈Z+.Inpractice,theseapproximationsworkwellalready forsmalln(Guzzettaetal.,2013).Herewetaken=5,whichgives resultsthat quantitatively closetothose obtainedwhen taking n=10(resultsnotshown).
TheODEswithexpandedstatespacecanbesolvedrecursively, startingwithS(a),V1(a),andW1(a).Thesolutionisgivenby
S(a)=e −
a 0 (s)ds V1(a)= a 0 (s)S(s)e − a s (r)dr ds W1(a)= a 0 (s)V1(s)e − a s [z(r)+(r)+(r)]dr ds Vi(a)= a 0 (z(s)+(s))Wi−1(s)e − a s (r)dr ds Wi(a)= a 0 (s)Vi(s)e − a s [z(r)+(r)+(r)]dr ds (4)withi≥2andS(0)=1theonlynon-zeroinitialcondition.Itfollows thatV(a)andW(a)canbeapproximatedby
V(a)≈V1(a)+ n
i=2 1(Wi−1)(a) W(a)≈W1(a)+ n i=2 (1◦2)(Wi−1)(a), where 1(f)= a 0 (z(s)+(s))f(s)e − a s ()d ds 2(f)= a 0 (s)f(s)e − a s [z()+()+()]d ds.Next, we discretize Eq. (4) as follows. If the limits of the J=100ageclassesaredefinedbythevectora=(0,1,...,J),then thej-thclass contains allpersonswithageintheinterval [a[j], a[j+1]), wherea[j] denotesthej-thelementofa.Usingthis nota-tion,theforceofinfection(a)andtherates(a),(a),and(a) are replaced by their discretizedcounterpart j, j, j,and j, and Vi andWi (i,=1,...,n,j=1,...,J)inthej-thage-classare givenby (V1)j= j
k=1 k k−kexp − k−1 i=1 i(ai+1−ai) exp − j i=k+1 i(ai+1−ai)[exp(−k(ak+1−ak))−exp(−k(ak+1−ak))]
(W1)j= j
k=1 k zk+k+k(V1)kexp − j i=k+1 (zi+i+i)(ai+1−ai) [1−exp(−(zk+k+k)(ak+1−ak))] (Vi)j= j k=1 zk+k k (Wi−1)kexp − j i=k+1 i(ai+1−ai) [1−exp(−k(ak+1−ak))] (Wi)j= j k=1 k zk+k+k(Vi)kexp − j i=k+1 (zi+i+i)(ai+1−ai)AppendixB. Optimalnumberofageintervals
Todeterminetheoptimalnumberofageintervalswehavere-analyzedthedatausingthescenariosofTable1,whilevaryingthenumber ofageintervalsfromn=4ton=7(5–8knots).TheresultsareshowninTable3below.
Table3
Overviewofanalyseswithvaryingnumberofageintervals.Seemaintextfordetails.
Numberofageintervals Scenario Ratesassumedconstant log(Lmax) AIC BIC df
4 1 – −355.4 19.7 48.2 22 2 −357.2 11.3 24.3 16 3 −359.5 15.9 28.8 16 4 −356.8 10.6 23.5 16 5 , −370.3 25.5 22.9 10 6 , −362.2 13.3 15.9 10 6* ≡0, −382.3 51.5 51.5 9 7 , −413.9 112.7 110.0 10 5 1 – −354.9 24.7 61.1 25 2 −353.7 8.4 26.6 18 3 −357.9 16.7 35.0 18 4 −355.5 12.0 30.2 18 5 , −370.9 28.8 28.7 11 6 , −369.0 26.9 29.4 11 6* ≡0, −380.0 46.9 46.9 10 7 , −395.4 77.8 77.7 11 6 1 – −347.5 16.0 60.2 28 2 −348.0 0.8 24.2 20 3 −352.7 10.4 33.7 20 4 −351.9 8.7 32.1 20 5 , −360.2 9.2 11.8 12 6 , −354.8 −1.6 0.98 12 6* ≡0, −356.6 0 0 11 7 , −383.7 56.3 58.9 12 7 1 – −352.9 32.8 84.9 31 2 −346.9 2.7 31.3 22 3 −353.8 16.5 45.1 22 4 −353.3 15.6 44.2 22 5 , −356.0 2.8 8.0 13 6 , −358.7 8.3 13.5 13 6* ≡0, −403.5 95.8 98.4 12 7 , −373.4 37.7 42.8 13
Thevalueshavebeenplottedinboldfacepurelytoemphasizethemodelwithhigheststatisticalsupport.
AppendixC. Amodelwithoutexogenousboosting
Inthemaintextwefocusonmodelscenarioswithexogenousimmuneboosting,asputforwardbyHope-Simpson(1965).Herewe
provideasensitivityanalysisshowingthatmodelscenarioswithoutexogenousboostinghavelowempiricalsupportcomparedtoscenarios withexogenousboosting.InTable4wepresentlog(Lmax),andAICandBICdifferencesofthesemodelswiththebest-fittingmodelScenario 6*inthemaintext.
Table4
Overviewoffitsofmodelscenarioswithoutexogenousboosting(cf.Table1).Asuiteofsimplificationsofthebaselinemodel(Scenario1)areconsideredinwhichoneor
moreoftheparametersareconstantinsteadofbeingmodeledwithsplines.Shownarethemaximizedlog-likelihood(log(Lmax)),theassociatedAICandBICdifferenceswith
thebest-fittingmodelScenario6*inthemaintext,andthedegreesoffreedom(df).
Scenario Ratesassumedconstant log(Lmax) AIC BIC df
1 – −360.5 40.0 81.7 27 2 −351.9 6.76 27.6 19 3 −358.9 20.6 41.5 19 4 −358.8 20.4 41.2 19 5 , −373.8 34.5 34.5 11 6 , −366.7 20.2 20.2 11 6* ≡0, −379.5 44.0 41.4 10 7 , −387.5 61.8 61.8 11 8 allrates −2996.8 5266.5 5248.3 3
AppendixD. Under-andover-ascertainment
Tofurtherinvestigatetherobustnessofourestimates,weanalyzetwoadditionalscenariosassuming10%under-ascertainmentand10%
over-ascertainment.Specifically,weincreaseordecreasetheobservednumberofHZinfectionsby10%,andre-estimatetheparameters.
Table5andFig.5showresultsofthebest-fittingScenarios6and6*. Table5
Parameterestimatesofmodelscenarioswithunder-orover-ascertainmentandwithorwithoutendogenousimmuneboosting(cf.Table1).Shownaremaximumlikelihood
estimateswith95%confidenceboundsforparametersthatareassumedconstant.SeealsoFig.5.
Scenario Assumption Parameter
(year−1) (year−1) z
6 10%over-ascertainment 0.048 0.11 1
6 10%under-ascertainment 0.053 0.080 1
6* 10%over-ascertainment 0.076 – 0.981
6* 10%under-ascertainment 0.078 – 0.987
Fig.5.Modelfitsinthepresenceofunder-andover-ascertainment(left-handpanels:withendogenousimmuneboosting;right-handpanels:withoutendogenousimmune boosting).ShownaretheobservedandpredictedincidenceofHZinbothscenarios(toppanels),thefractionofthepopulationwithwanedimmunity(W(a))(middlepanels), andtheestimatedreactivationrates((a))(bottompanels).Dashedlinescorrespondtomaximumlikelihoodestimatespresentedinthemaintext(Fig.3)forthescenarios.
Continuouslinesrefertomaximumlikelihoodestimatesassuming10%under-ascertainment(blue)and10%over-ascertainment(green).(Forinterpretationofthereferences
tocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)
AppendixE. Amodelwithmultiplereactivations
Themodelinthemaintextassumesthatonlyonereactivationcantakeplaceinaperson’slife.Hereweinvestigatewhethertheresults soobtainedstillholdinamodelwithmultiplereactivations.ThemodelstructureisshowninFig.6.Inordernottoovercomplicatethe modelweassumethatreactivationleadstoimmuneboosting,andthatimmunitywanesatanage-specificrate(a)thatdoesnotdepend onthenumberofreactivationeventsthatapersonhasalreadygonethrough.TheresultsoftheanalysesaregiveninTables6and7and Fig.7(cf.Tables1and2,andFig.3).Overall,theseanalysesgivesimilarfitstothedataasanalyseswithmodelsthatdonotincludemultiple
reactivation(Table7),andmainlydifferwithrespecttotheestimated(unobserved)fractionofthepopulationintheWcompartment (Fig.7).
Fig.6. Schematicofthemodelwithmultiplereactivations.S(a)denotestheage-specificprevalenceofuninfectedpersons,V(a)theage-specificprevalenceoflatently
infectedpersonswithsufficientimmunitytopreventreactivation,andW(a)theprevalenceoflatentlyinfectedpersonswithwanedimmunity.Theforceofinfectionand
rateofexogenousimmuneboostingaregivenby(a)andz(a),andtheratesoflossofimmunity,endogenousimmuneboosting,andreactivationaregivenby(a),(a),
and(a).Theage-specificincidenceofreactivationisgivenby(a)W(a).
Table6
Parameterestimatesofthescenarioswithandwithoutendogenousimmuneboostingandmultiplereactions(cf.Table2).Shownaremaximumlikelihoodestimateswith
95%confidenceboundsforparametersthatareassumedconstant.
Scenario Parameter
(year−1)(95%CI) (year−1)(95%CI) z(95%CI)
6 0.084(0.073–0.100) 0.10(0.093–0.13) 0.978(0.977–0.984)
6* 0.042(0.025–0.044) – 0.990(0.989–0.994)
Table7
Overviewoffitsofmodelscenarioswithmultiplereactivations(cf.Table1).Asuiteofsimplificationsofthebaselinemodel(Scenario1)areconsideredinwhichoneormore
oftheparametersareconstantinsteadofbeingmodeledwithsplines.Shownarethemaximizedlog-likelihood(log(Lmax)),theassociatedAICandBICdifferenceswiththe
best-fittingmodelScenario6*inthemaintext,andthedegreesoffreedom(df).
Scenario Ratesassumedconstant log(Lmax) AIC BIC df
1 – −347.7 16.3 60.6 28 2 −353.1 11.1 34.5 20 3 −352.7 10.4 33.8 20 4 −351.2 7.4 30.8 20 5 , −360.6 10.1 12.7 12 6 , −354.8 −1.4 1.2 12 6* ≡0, −356.6 0.2 0.2 11 7 , −383.0 54.9 57.5 12 8 Allrates −1176.0 1626.0 1607.7 4
Thesetofordinarydifferentialequationsofthisdynamicalsystemisgivenby
dS(a)
da =−(a)S(a),
dV(a)
da =(a)S(a)−(a)V(a)+(z(a)+(a)+(a))W(a),
dW(a)
da =(a)V(a)−(z(a)+(a)+(a))W(a).
FollowingthesamereasoningasinAppendixA,wearriveatthefollowingsetofdiscretizedequations
(V1)j= j
k=1 k k−kexp − k−1 i=1 i(ai+1−ai) exp − j i=k+1 i(ai+1−ai)[exp(−k(ak+1−ak))−exp(−k(ak+1−ak))]
(W1)j= j
k=1 k zk+k+k(V1)kexp − j i=k+1 (zi+i+i)(ai+1−ai) [1−exp(−(zk+k+k)(ak+1−ak))] (Vi)j= j k=1 zk+k+k k (Wi−1)kexp − j i=k+1 i(ai+1−ai) [1−exp(−k(ak+1−ak))] (Wi)j= j k=1 k zk+k+k(Vi)kexp − j i=k+1 (zi+i+i)(ai+1−ai)Fig.7.HZincidencedataandoverviewofthebest-fittingmodels(left-handpanels:withendogenousimmuneboosting,right-handpanels:withoutendogenousimmune
boosting).ShownaretheobservedandpredictedincidenceofHZinbothscenarios(toppanels;dots:data;lines:prediction),thefractionofthepopulationwithwaned
immunity(W(a))(middlepanels),andtheestimatedreactivationrates((a))(bottompanels).Linescorrespondtomaximumlikelihoodestimatesandshadedareasto
bootstrapped95%confidenceranges.Alsoshownarethecorrespondingestimatesinthemodelwithoutmultiplereactivation(Fig.3).
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