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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
Optimally capturing latency dynamics in models of tuberculosis transmission
Romain Ragonnet
a,b,∗, James M. Trauer
a,c,d, Nick Scott
b,c, Michael T. Meehan
e, Justin T. Denholm
a,d,f, Emma S. McBryde
a,eaFacultyofMedicine,DentistryandHealthSciences,UniversityofMelbourne,Australia
bBurnetInstitute,Australia
cSchoolofPopulationHealthandPreventiveMedicine,MonashUniversity,Australia
dVictorianTuberculosisProgram,Melbourne,Australia
eAustralianInstituteofTropicalHealthandMedicine,JamesCookUniversity,Australia
fRoyalMelbourneHospital,Melbourne,Australia
a r t i c l e i n f o
Articlehistory:
Received13February2017
Receivedinrevisedform30May2017 Accepted14June2017
Availableonline16June2017
Keywords:
Tuberculosislatency Mathematicalmodelling Riskofdiseaseactivation Parameterestimation
a b s t r a c t
Althoughdifferentstructuresareusedinmoderntuberculosis(TB)modelstosimulateTBlatency,it remainsunclearwhethertheyareallcapableofreproducingtheparticularactivationdynamicsempiri- callyobserved.Weaimedtodeterminewhichofthesestructuresreplicatethedynamicsofprogression accurately.Wereviewed88TB-modellingarticlesandclassifiedthemaccordingtothelatencystruc- tureemployed.Wethenfittedthesedifferentmodelstotheactivationdynamicsobservedfrom1352 infectedcontactsdiagnosedinVictoria(Australia)andAmsterdam(Netherlands)toobtainparameter estimates.Sixdifferentmodelstructureswereidentified,ofwhichonlythoseincorporatingtwolatency compartmentswerecapableofreproducingtheactivationdynamicsempiricallyobserved.Wefound importantdifferencesinparameterestimatesbyage.Wealsoobservedmarkeddifferencesbetween ourestimatesandtheparametervaluesusedinmanypreviousmodels.Inparticular,whentwosuc- cessivelatencyphasesareconsidered,thefirstperiodshouldhaveadurationthatismuchshorterthan thatusedinpreviousstudies.Inconclusion,structuresincorporatingtwolatencycompartmentsand age-stratificationshouldbeemployedtoaccuratelyreplicatethedynamicsofTBlatency.Weprovidea catalogueofparametervaluesandanapproachtoparameterestimationfromempiricdataforcalibration offutureTB-models.
©2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Tuberculosis (TB) is a major health issue with10.4 million active cases and 1.8 million deaths worldwide in 2015 (WHO, 2016).Furthermore,aroundonequarteroftheworld’spopulation isestimatedtobeinfectedwithTB(HoubenandDodd,2016),rep- resentingahugereservoirofpotentialdisease.Accordingly,fully understandinglatentTBinfectioniscrucialforassessingthefuture epidemictrajectory and designing effectiveTB control policies.
Despitethis, muchreinfection occursinhighincidencecohorts, hampering accurate estimation of latency dynamics. Therefore insightsintotheactivationdynamicsfollowingasingleinfection episodeofMycobacteriumtuberculosisprovidedbyrecentstudies
∗Correspondingauthorat:85CommercialRoad,BurnetInstitute,Melbourne,VIC 3003,Australia.
E-mailaddress:[email protected](R.Ragonnet).
inverylowtransmissionsettingsareparticularlyvaluable(Trauer etal.,2016a;Slootetal.,2014).Theseworksprovidedetailedinfor- mationonpatternsofactivation,highlightingthatmostactivecases occurwithinthefirstfewmonthsofinfection.
MathematicalmodellinghasinformedTBcontrolprogramsby simulatinginterventions,orbyexplainingthemechanismsunder- lyingobservedepidemiologicaltrends(VynnyckyandFine,1997;
Gomesetal.,2004;Castillo-ChavezandFeng,1997;Abu-Raddad etal.,2009;Cohenetal.,2008;Dye,2012;Menziesetal.,2012;
Traueretal.,2016b),yetlittleisknownaboutwhethersuchmod- ellinghasbeen abletocapturelatencydynamics accurately.In thepast, TB modelshave beenconstructed tocapturethelife- longprobability ofdisease and,although somemodelsallowed for markeddifferencesbetweenthe earlyand late dynamicsof infection(Dowdyetal.,2013;Linetal.,2011;AparicioandCastillo- Chavez,2009),estimatesfortheassociatedparametershavenot beenfitcloselytolongitudinaldata.Despitethis,ithasbeenshown
http://dx.doi.org/10.1016/j.epidem.2017.06.002
1755-4365/©2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.
0/).
thatwhenmodellinginfectious diseases,itis criticaltoemploy appropriatedistributionsoflatentperiods(Wearingetal.,2005).
Focusingonemerging infectiousdiseases,WallingaandLipsitch furtherdemonstratedthatcapturingthemeanofthegeneration timesisnotsufficienttocharacterisetransmissionaccurately,as theshapeofthedistributionofthegenerationintervalsalsoplays acriticalroleininfectiondynamics(WallingaandLipsitch,2007).
AlthoughTBisanancientdisease,itsepidemiologyiscontinuously evolving.Inparticular,changesinTBepidemiologyinresponseto emergingphenomena,suchasintroductionofdrug-resistantforms ofTBorstrongercontrolprograms,arelikelytoaffecttheshape ofthegenerationtimedistribution.Therefore,therecentdetailed characterisationof TB activationdynamicsrepresent a valuable opportunitytoreview and improvemodelling practices forthe simulationofTBlatency.
Compartmentaldynamictransmissionmodels−themostcom- montypeofTBmathematicalmodel−simulateTBlatencywith various levels of complexity. While some modellers employ a singlelatencycompartmentthatprecedestheactivediseasecom- partment (Colijn et al., 2008; Blower and Chou, 2004), others incorporateasecondlatencycompartmentinordertocapturetwo differentratesofprogressionfromlatentinfectiontoactivedis- ease(Hilletal.,2012;Cohenetal.,2006;Traueretal.,2014).When twolatencycompartmentsareincorporated,theycaneitherbe positionedinseriesorinparallel,involvingdifferentunderlying assumptionsregardingtheprogressionpathwaystoactivedisease.
First,theserialstructureimpliesthatnewlyinfectedindividuals remainathighriskofdiseaseduringtheinitialphaseandthen, ifTBactivationhasnotoccurred,theytransitiontoanothercom- partmentwheretheirriskofdevelopingactiveTBisreduced.By contrast,withaparallelcompartmentalstructure,theunderlying assumptionisthataproportionofinfectedindividualsbelongto ahighriskcategory,whiletheremainderareatlowerriskofTB disease.WhileTBmodellinghasbeenusedextensivelyforover40 years,itremainsunclearwhichofthesestructuresarebestadapted tothenaturalhistoryofTB.
Inthisstudy,weaimtodeterminethemostappropriatemodel structurestosimulateTB latencyandprovideestimates forthe parametersassociatedwiththesestructuresacrossdifferentage categories.Weusethedistributionoftheestimatedtimesfrom infectiontoTBactivationin1352infectedcontactsofindividuals withactivepulmonaryTBfromVictoria(Australia)andAmster- dam(Netherlands)tocalibratethelatencystructuresofdifferent candidatemodelstothedynamicsobservedinthedata.
2. Methods
2.1. Literaturereview
Oursearchwasbasedontheliteraturereviewofmathematical and economic TB modelling articles provided by the TB Mod- ellingandAnalysisConsortium,availableonlineathttp://tb-mac.
org/Resources/Resource/4(seeAppendixinSupplementaryfilefor moredetails).Fromthisdatabaseweidentifiedall88publications reportingtheuseofadeterministiccompartmentaltransmission dynamicmodel.Allselectedpaperswerereviewedindependently bytwoauthors(RR,JMT)whoclassifiedthemanuscriptsaccording tothestructureusedtomodelTBlatency.Thesetwoindependent investigationsledtothesameclassificationwhichispresentedin theAppendixinSupplementaryfile(TableS1).
2.2. Analyticalsolution
Foreachlatencystructurefoundintheliterature,weassoci- atedabasicdynamicmodelcomprisedofthelatencystructurein
combination with compartments representing susceptibility to infectionandactivedisease.We thenfoundanalyticalsolutions for the TB activation dynamics corresponding to each model.
Namely,consideringthat individualswereinfected attimet=0, wedeterminedtheproportionI(t) ofinfectedindividualsthathad developedactiveTBaftereachtimet(t≥0).Analyticalexpressions arealsopresentedforthetotal proportionofinfectedindividu- alsprogressingtoactivedisease,obtainedbycalculatingthelimit ofI(t) astapproachesplusinfinity.Thedetailedmethodusedto obtaintheanalyticsolutionsisdescribedintheAppendixinSup- plementaryfile.
2.3. Datausedtocalibratethemodels
Themodelsdescribedabovewerecalibratedtoindividualdata onclosecontactsofindividualswithactivepulmonaryTBnotified intheAustralianstateofVictoriafromJanuary2005toDecember 2013.Thesedataarederivedfromaverylowendemicsettingand weredescribedindetailbyTraueretal.Theyconsistof613infected contactsofwhom67(10.9%)developedactiveTBduringthestudy period(Traueretal.,2016a).Toenhanceourdataset,wealsoused thepublisheddata onclose contactsof pulmonary TB patients fromAmsterdam(Netherlands)notifiedbetween2002and2011, asreportedbySlootetal.(Slootetal.,2014).Thesedatainclude 739infectedindividuals,ofwhom71(9.6%)developedactiveTB.
Thedetailedapproachesusedtodeterminebothdatesofinfection andactivationinindividualsinthetwostudiesarepresentedin therespectivemanuscripts.Theactivationtimesmeasuredinthese datawereusedtocalibratethedifferentmodels.Inordertovali- dateourapproachinvolvingmergingoftwodatasets,wepresent acomparisonoftheestimatesobtainedfromtheseparatefittings tothetwodatasets(AppendixinSupplementaryfileSection8.2).
TheapproachusedtoextractdatafromSlootandcolleagues’arti- cleisdescribedindetailintheAppendixinSupplementaryfile, alongwithavalidationanalysisoftheextractionmethodwhile thedistributionofthetimestoactivationmeasuredinthetwo datasets(VictoriaandAmsterdam)ispresentedinFig.S7(Appendix inSupplementaryfile).
Traueretal.alsoproposedanimputationmethodwhichtakes intoaccountthecensorshipformigration,death,andpreventive treatment(Traueretal.,2016a).Weusedthisapproach,whichis associatedwithhigherestimatesconcerningtheriskofTBactiva- tion,inasupplementaryanalysis.
2.4. Modelfitting
Modelfittingtodatawasmadebybuildingthesurvivallike- lihood defined asfollows. For a given model associated witha givensetofparameters,,weobtainananalyticalsurvivalfunc- tionS(t) whichrepresentstheprobabilitythatactivationhasnot occurredyetattimetgiventhatinfectionoccurredatt=0.This function is associated with a hazard function (t) definedby (t)=−S(t)/S(t),characterisingthechancethatprogression toactiveTBoccursatpreciselytimet,given survivaluptothat time.Then,foreachinfectedcaseiofourdataset,forwhomtidesig- natesthetimeofeitherTBactivationorendoffollow-up,wedefine anindividuallikelihoodcomponentby ifthecasewas notknowntodevelopactiveTB;and ifthecase effectivelyactivatedTBattimeti.Finally,weaimtomaximisethe multi-dimensionallikelihoodobtainedbymultiplyingalltheindi- viduallikelihoodcomponentstogether: .Thisproblem isequivalenttomaximisingthefollowinglog-likelihoodthatwe defineasthefittingscore: .
Anotherfittingmethodwasusedforvalidationandwhenthe datadidnotallowforthesurvivallikelihoodtobeutilised.Specif- ically,aleastsquaresoptimisationwasperformedtominimisethe
Fig.1.Representationofthedifferentmodelstructures.Solidlinesrepresentpro- gressivetransitionsbetweencompartmentsthatareparameterisedwithrates, whiledashedlinesrepresentinstantaneoustransitionsbetweencompartmentsthat areassociatedwithproportions.Theflowsrelatedtonaturaldeatharenotrepre- sentedandareassociatedwiththenaturalmortality.
distancebetweenthesurvivalcurvesgeneratedbythedataandby themodel.Thetime-pointsconsideredwhenperformingthisopti- misationwereequallyspacedbyonedayandrangedbetween0 andthemaximaltimetimeasuredinthedata.
Thetwomethodsintroducedaboveareconceptuallydifferent.
Ineffect,thesurvivallikelihoodapproachsearchesforthesetof modelparametervaluesthatmaximisestheprobabilityofobserv- ingthedata,givenaparticularmodelandaparticularparameter set.Incontrast,theleastsquaresoptimisationisbasedonameasure ofadistancebetweentwocurves:theactivationcurveobserved fromthedataandtheoneproducedbyamodel.Althoughthese twoapproachesarefundamentallydistinct,theyarebothformsof optimisationappliedtothemodel’sparametervalues.Thisoptimi- sationisbasedonaniterativemethodwhichallowstheparameters tovaryinamulti-dimensionalspace,fromwhichweretainonlythe parametersetthatyieldstheoptimalmeasure,i.e.thegreatestlike- lihood(fortheprobabilisticapproach)orthesmallestdistance(for theleastsquaresmethod).ThisprocesswasobtainedusingRv.3.3 anditsincorporatedfunction“constrOptim”.
Modelfittingwasperformedseparatelyusingthreeagecate- gories(“<5years old”,“5to14 years old” and“≥15years old”) inadditiontoapooledanalysisincludingthewholepopulation.
Inordertofullyexploretheparameterspacesand toreportall acceptableparametersets, weusedaMetropolis-Hastingsalgo- rithmpresentedindetailintheAppendixinSupplementaryfile.
3. Results
3.1. Literaturereview
Fromourreviewof 88publications related toTBmodelling, wefoundthat sixdifferentcompartmentalstructureshadbeen employedtomodelTBlatencyandtheseformthebasisofthefol- lowinganalysis.Thesixmodelsarerepresentedin Fig.1,along withthelabelsusedfor theassociated parameters.Thelevelof complexityrangesfroma modelincorporatingnolatencycom- partment(Model1)tomodelsbasedontwolatencycompartments positionedeitherinseries(Models4and5)orinparallel(Model 6). Some structures incorporate a bypass from the susceptible compartmenttotheactivediseasecompartment,allowingforcon- siderationofinstantaneousactivationofTBdiseaseafterexposure
(Models 1,3and 5).TableS1(AppendixinSupplementaryfile) reportsourclassificationofthe88reviewedstudiesaccordingto thelatencystructureemployed.
3.2. Modelcalibration
Fig.2presentsthedynamicsofTBactivationobtainedfromeach modelwhenoptimallycalibratedtoourdata(seeMethodspara- graph for fittingapproach). Onlythemodels incorporatingtwo latencycompartments(Models4,5and6)suitablyreplicatethe dynamicsofTBactivation.Model1,whichisnotshown,doesnot containanyfreeparametersandassumesthatallinfectedpatients progresstoactivediseaseimmediatelyafterexposure.Thisisnot compatiblewithourdatanorourknowledgeofthelifelongriskof TBdiseaseininfectedindividuals(Traueretal.,2016a;Slootetal., 2014).BothModels2and3produceunreasonablypoorfitstothe data.IntheAppendixinSupplementaryfileweshowthattheequa- tionsdescribingModels2and3onlyinvolveasingleexponential function,whichisnotsufficienttoreplicatethetwodistinctpat- ternsobservedinthedynamicsofactivation—ahighriskofdisease activationoverthefirstfewmonths,followedbyadramatically lowerrisk inasecondphase. Incontrast,in Models4, 5and6, whichincludetwolatencycompartments,theactivationdynam- icsaredrivenbytwoexponentialcomponentsthatareassociated withtwoindependentgrowthrates,leadingtoaccuratereplication ofthepatternsempiricallyobservedforeachagecategory.Thefit- tingscores(seeFig.2)forModels4,5and6indicatethatnone providesasignificantlybetterfitthantheothers;however,Model 5presentsahigherlevelofcomplexitythanModels4and6,as itinvolvesestimatingoneadditionalparameter.Sincetheinclu- sionofthisparameterdidnotimprovemodelfitting,weconsider thatitisunnecessarytoemploythisstructure.Thispropositionis stronglysupportedbyamoredetailedanalysisofModel5where differentapproachesallsupportthattheadditionalparameterdoes notimprovefitting(seeAppendixinSupplementaryfile).Accord- ingly,theremainderofouranalysisrelatesonlytoModels4and 6.
Table1presentstheparametervalues(withunitsofdays−1) obtainedfromthecalibrationofModels4and6tothedata.We observethattherateofre-activation()ismuchlowerintheage category“<5yearsold”thanintheotheragecategoriesforboth Model4andModel6.Incontrast,therateofrapidprogressionis higherinthe“<5yearsold”categorythanintheotheragecategories whenconsideringModel4;andwhen consideringModel 6,the proportionofinfected individualsexperiencingahighriskofTB disease(1−g)ishigherforchildrenthanfortheothercategories (35%for“<5y.o.”,19%for“5to14y.o.”,5%for“15y.o.andmore”, and9%forallagestogether).
ThelifelongriskofTBactivationininfectedindividualscanbe calculatedanalyticallyforthedifferentmodels(seeAppendixin Supplementaryfile,p.4).TheparametervaluesreportedinTable1 forModels4and6correspondtototalproportionsofactivation rangingbetween10%(“15y.o.andmore”)and32%(“<5y.o.”).
Given theverylow valuesobserved fortherateof reactiva- tion()inthe“<5yearsold”category,weundertookanadditional analysisinordertoexplorethepossibilityoftheabsenceoflate reactivationinthiscategory.Inthisanalysis,Models4and6were fittedtothedataundertheconstraintthat=0,withbothmod- elsfoundtoperformequallywellinabsenceofreactivation.Fig.3 presentsthecorrespondingsimplifiedmodelsaswellasthebestfits obtained.Afurtherexplorationofthecontributionofendogenous reactivationversusprimaryactivationispresentedintheAppendix inSupplementaryfile(Section3.)forModel4andforthedifferent agecategories.Thisanalysisdemonstratesthatendogenousreacti- vationcontributesverylittletotheburdenofactiveTB,especiallyin youngindividuals(“<5yearsold”and“5to14y.o.”).Althoughthis
Fig.2. CalibrationsobtainedwiththedifferentmodelsforthepercentageofactiveTBamonginfectedindividualsovertimesinceinfection.Theblacklinesandgreyshade representtheestimates(centraland95%CI)obtainedfromtheKaplan-Meieranalysisofourdata.ThebluelinerepresentsthepercentageofactiveTBovertimeobtained fromthedifferentmodels,whenoptimallycalibratedwiththesurvivallikelihoodmethod.Thex-scaleswerechoseninordertoallowforadecentvisualisationoftheearly stagesofinfectionanddonotcovertheentiretimewindowscorrespondingtothedataset.Fittingwasrealisedbymaximisingthefittingscore(FS).(Forinterpretationofthe referencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)
contributionismoresubstantialinthe“≥15yearsold”category,we estimatethatonly1%ofinfectedindividualsofthiscategorywould haveprogressedtoactivediseasetransitioningfromcompartment LBafterfiveorlessyearsfrominfection(usingEq.(23)inAppendix inSupplementaryfile).
OuranalysisoftheanalyticalsolutionsassociatedwithModel 4andModel 6demonstratedthatthetwo expressionscoincide iftheparametervalues ofthetwomodelssatisfythefollowing relationships:
ε6=ε4+46=4g6= 4 4+ε4−4
wherethesubscriptsindicatetowhichofModels4or6the parametersapply.Thisfindingindicatesthatthedynamicsofacti- vationsimulatedbythetwomodelsareidentical,differingonlyin thevalueoftheparametervaluesthatshouldbeapplied.
3.3. Probabilitydistributionofparameters
We useda Bayesianframework toinfertheposterior distri- butions for the parameters in themodel. Uniform priors were usedand toolsof Bayesianinference appliedincludingMarkov Chain Monte-Carlo exploration using the Metropolis-Hastings acceptancealgorithm.Theresultingposteriordistributionsforthe
Table1
Parameterestimates.Calibrationissuedfromthesurvivallikelihoodmaximisationappliedtothemergeddataset(VictoriaandAmsterdamdata).Pointestimatescorrespond totheparametersmaximisingthelikelihoodwhilevaluesintobracketsindicatethenarrowestintervalcontaining95%oftheacceptedvaluesduringtheMetropolis-Hastings simulation.Ratesarepresentedasdailyvalues.
ε g
Model4
Allages 1.1e−3 (8.4e−4–1.5e−3)
1.0e−2 (8.5e−31.4e−2)
5.5e−6 (2.5e−61.1e−5)
–
<5y.o. 6.6e−3 (4.4e−39.5e−3)
1.2e−2 (8.5e−31.8e−2)
1.9e−11 (5.0e−91.6e−5)
– 5–14y.o 2.7e−3
(1.7e−33.9e−3)
1.2e−2 (7.9e−31.6e−2)
6.4e−6 (6.7e−71.9e−5)
–
≥15y.o 2.7e−4 (1.6e−45.1e−4)
5.4e−3 (3.5e−31.1e−2)
3.3e−6 (1.9e−61.0e−5)
–
Model6
Allages 1.1e−2 (9.2e−31.5e−2)
– 5.5e−6
(3.4e−61.0e−5)
0.91 (0.890.93)
<5y.o. 1.9e−2 (1.2e−22.5e−2)
– 3.4e−11
(2.7e−92.0e−5)
0.65 (0.550.73) 5–14y.o 1.4e−2
(9.4e−31.8e−2)
– 6.4e−6
(8.0e−72.2e−5)
0.81 (0.750.86)
≥15y.o 5.6e−3 (3.8e−39.6e−3)
– 3.3e−6
(7.3e−79.3e−6)
0.95 (0.940.97)
Parameterestimates.Calibrationissuedfromthesurvivallikelihoodmaximisationappliedtothemergeddataset(VictoriaandAmsterdamdata).Pointestimatescorrespond totheparametersmaximisingthelikelihoodwhilevaluesintobracketsindicatethenarrowestintervalcontaining95%oftheacceptedvaluesduringtheMetropolis-Hastings simulation.Ratesarepresentedasdailyvalues.
Fig.3. SimplifiedmodelstructuresadaptedtosimulateTBlatencyinyoungchildren (<5yearsold)Here,itisassumedthatprogressionfromthepreviouscompartments LBofModel4andModel6toactivediseaseIcannotoccur.ThecompartmentLB
thereforebecomesaprotectedstatewhichislabelledRinthisillustration.Fitting wasrealisedbymaximisingthefittingscore(FS).
differentparametersofModels4and6arepresented(Table1).
Proposedstatisticaldistributionsthatcouldbeusedtogenerate similarsetsofparametersareavailableintheAppendixinSupple- mentaryfile,alongwiththeposteriordistributionsobtainedforall parameters.Theacceptablerangesofvaluesfortheparameterare wideforthecategory“<5y.o.”,duetothesmallnumberofreacti- vationcasesinourdataset.However,ouranalysisshowedthatthe rateofreactivationislimitedbyanupperboundofaround2.3e−5 inallcategories,whichrepresentsarelativelysmallannualriskof re-activationof0.8%.
Byanalysingthedistributionsofeachoftheretainedparameter setsinpairs,weobservedacollinearitybetweentheparameters andεforModel4.Thiscorrelation(representedinFig.4)sug- geststhatwhenindividualsareassumedtostabiliseinfectionmore rapidly(increases),therateofrapidprogressiontoTBdisease(ε)
tendstoincreasetocompensate.Thisaffinerelationshipbetween parametersandisalsodescribedthroughaformalmathematical analysis(AppendixinSupplementaryfile,Section10.1.2.).
Anotherresult provided bythe Metropolis-Hastings simula- tionwasthedistributionoftheaveragedurationspentinthefirst latencycompartmentforModel4.Thisquantitywasobtainedvia theformula+ε1+whererepresentsthenaturaldeathrate.Fig.5 presentsthedistributionofthesedurationsforthedifferentage categories.Theaveragedurationspentinthefirstlatencycompart- mentisestimatedat82daysforallagecategoriescombined,this durationbeingshorterforchildren(52daysfor“<5y.o.”category) thanforolderindividuals(70daysfor“5to14y.o.”and146days for“≥15y.o.”).
3.4. Validationandsensitivityanalyses
Ourfindingswereconsistentwhenweemployedanalternate fittingmethod(least squaresminimisation),performedseparate fitsforeachdataset(VictoriaandAmsterdam)insteadofthesingle mergeddataset,orusedalternateextractionmethodsforthedataof Slootetal.WhenfittingModel3usingleastsquaresminimisation, weobtainedamodelcalibrationthatdifferedfromthatobtained withthesurvivallikelihoodmethod(SeeAppendixinSupplemen- taryfile,Fig.S14).However,whilethissecondcalibrationallowed forabettersimulationofthelatestagesoflatencycomparedtothe baselinefittingmethod(Fig.2),itcompletelyfailedtocapturethe earlydynamicsofactivation.Moredetailsaboutthedifferentsen- sitivityanalysesareavailableintheAppendixinSupplementary file.
Finally, by using imputed dataas described in Traueret al.
accountingforthecensorshipformigration,death,andpreventive treatmentandthereforeassociatedwithhigherestimatesforthe riskofTBdisease(Traueretal.,2016a),ourconclusionsregarding optimalmodelselectionremainedunchanged,whileweobtained slightlydifferentoptimisedparametervalues(seeAppendixinSup- plementaryfile).Inparticular,inModel4,imputationledtoaslight increaseintherateofrapidprogression(ε)combinedwithaslight reductionintherateoftransitiontowardsthelatelatencycom- partment().ConcerningModel6,thebestcalibrationtoimputed datawasobtainedwithahigherproportionofinfectedindividuals transitioningtothehighriskcompartment(1−g)whiletherateof rapidprogression(ε)wasslightlyreduced.
Fig.4. RepresentationofthecollinearityobservedbetweentheparametersandεforModel4.Thereddotsrepresentthe10,000acceptedparametersetsobtainedfrom theMetropolis-Hastingssimulation.Theblacklinerepresentstheaffinemodelapproximatingthedatawithaleastsquareminimisation.(Forinterpretationofthereferences tocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)
Fig.5. DistributionofthetimesspentinthefirstlatencycompartmentLAformodel4.Distributionassociatedwith10,000acceptedparameterssetsfromtheMetropolis- Hastingssimulation.
3.5. Comparisonofourresultswithpreviousworks
Wereviewedtheparametervaluesthathadbeenemployedin thepreviousstudiesincorporatingthelatencystructuresofModel 4andModel6andcomparedthesevaluestoourestimates(see AppendixinSupplementaryfile,Section9).ConcerningModel4, wefoundthat therateofprogression fromcompartment LAto compartmentLB()wasconsiderablylowerthanourestimatein allpreviousstudies,whichtypicallyassumelongperiodsspentin earlylatency(2–5years).Similarly,therateoffastprogressionto activeTB(ε)usedintheexistingliteraturewasmuchlowerthanour estimate.ForModel6,theproportionofslowprogressors(g)found inpreviousworkswasclosetoourestimate,whereastherateoffast progression(ε)hadbeenmarkedlyunderestimated.Finally,while therateofslowprogressiontoactiveTB()wasgenerallyunder- estimatedinstudiesincorporatingeitherstructure(Models4or6), twostudiesusedpointestimatesthatfallinsideofthe95%CIwe report,andnineotherstudiesusedparameterrangesoverlapping our95%CI.
3.6. Rapidestimationoftheparametersforfutureworks
Atheoreticalanalysisoftheequationsgoverningthedynamic modelsallowedustodevelopamethodtoestimaterapidlythe differentparameters of Models4 and 6 from anydataset.This approachinvolvessimplegraphicalmeasurementsperformedon the curve representing the proportion of active cases among infectedindividuals over timefrominfection (denoted),such asthecurvesrepresentedinFig.2.Thedetaileddescriptionand demonstrationofthemethodisavailableintheAppendixinSup- plementaryfileandthemainresultsarepresentedhere.Theprofile ofcanbedecomposedintotwophases(seeFig.2anddetailed
descriptioninAppendixinSupplementaryfile).Themethodcon- sistsoffirstmeasuringtheinitialslopeof(denoteds0)aswell asthecharacteristicsofatangenttoonapointsituatedatthe beginningofthesecondrisephase(slopedenoteds,andy-intercept denotedy0).Then,estimatesforthedifferentparameters(ˆε,,ˆ ˆ and ˆg)canbeobtainedbyusingthethreemeasures(s0,sandy0) asfollows:
ForModel4: ˆε=s0ˆ= εˆ
y0 −εˆ−ˆ=s
ˆ
+εˆ+
ˆ
ForModel6: ˆε= s0
y0 −ˆv= s0s
s0−y0(s0+)gˆ= εˆ−s0 ˆ ε−ˆv wheredesignatesthenaturalmortalityrate.
4. Discussion
4.1. Maincontributionsofthestudy
Inthisstudy,wedeterminethemostappropriatemodelstruc- turesforaccuratelysimulatingTBlatency.Wedemonstratethatof thestructuresemployedinthepast,onlythosethatincorporatetwo compartmentsforlatentinfectionareabletoreproducethespecific dynamicsofTBactivation.Suchapproachesthereforeinvolvetwo differentlevelsfortherateofactivation,allowingformoreTBcases tooccurafterrecentinfectionthanthroughlatereactivation.This workalsoprovidesadetailedandflexible catalogueofparame- tervaluesassociatedwiththeretainedmodelstructures,thatwas validatedbytheuseoftwoindependentfittingmethodsleadingto similarestimatesandthathighlightedmarkedgapswithparameter valuesemployedinpreviousworks.Thisstudymaybecomearefer-
enceforcalibrationoffutureTBmodelsandourapproachinvolving estimationswithandwithoutagestratificationwillallowourfind- ingstobedirectlyappliedwhethermodelsareage-structuredor not.
4.2. Serialversusparallelstructure
Wedemonstratethatthetwocompartmentsrequiredtomodel TBlatencycouldbeplacedeitherinseriesorinparallel,andwould leadtoidenticalactivationdynamics.Thedifficultyinmakingthis distinctionbetweenthetwomodelsisunfortunatebecausethey reflecttwoalternatebiologicalmechanismsthatcouldexplainthe higherburdenofdiseaseobservedafterrecentinfection,eachlead- ing to differentrecommendations for TB prevention strategies.
First,theserialstructuremodelsadecreasingriskofactivationover time ineveryindividual,indicating thatpreventivecareshould betargeted at themostrecent infections as a prioritythrough interventionssuchascontacttracing.Ontheotherhand,thepar- allelstructuresuggeststhatindividualpredispositionsmaymake someinfectedindividualsmoreorlesslikelytodevelopdisease, regardlessofthetimefrominfection.Ifthissecondscenariowas verifiedwithepidemiological data, theseresultswould suggest that identifyingand findingpriority populationscould dramat- ically enhanceefficiency ofinterventions. Somefactorssuchas HIV-infection,diabetesorsmokingarealreadyknowntoenhance theriskofTBactivation(Bishwakarmaetal.,2015;Houbenetal., 2010;Horsburghetal.,2010;JeonandMurray,2008),althoughour parameterestimatesundertheparallelstructurescenariosuggest thataround9%ofinfectedindividualswouldbelongtoahigh-risk groupinwhichtheriskofTBactivationwouldbeashighas2000- foldthatofthelow-riskgroup.Alternatively,fortheseriesstructure ofthelatencycompartments,rapididentificationandearlytreat- mentofinfectedpeoplewouldbethepriority.
4.3. Agedependency
Ourestimationshighlightimportantdiscrepanciesbetweenage categories,withparameterestimatesforyoungchildren(<5years old)differingmarkedlyfromtheotheragecategories.Inparticular, ourstudysuggeststhatforyoungchildren,therateofdiseaseacti- vationinthehigh-riskpopulation(earlylatencyforserialstructure orhigh-riskgroupforparallelstructure)ismuchgreaterthanfor 15+yearolds.Incontrast,amongthelowerriskpopulation(late latencyforserialstructureorlowriskgroupforparallelstructure), youngchildrenseemtobeatlowerriskofactivationthantheother individuals.Thesefindingsindicatethattheyoungestinfectedpop- ulationpresentsamuch higher riskof TBactivationearlyafter infection,whiletheirriskofactivationreducesdramaticallyinthe laterstagesofinfection.Previousworkshavealreadyidentifiedage asanimportantfactorinfluencingthenaturalhistoryofTB(Trauer etal.,2016a;Slootetal.,2014;VynnyckyandFine,1997),andour analysisbringsadditionalinsightthatreinforcestheimportance ofprovidingyoungchildrenwithearlypreventivetreatmentfor infectionaslatetreatmentswouldbeuseless.Byrevealingsuch age-specificcharacteristics,ourstudyalsoimpliesthatfutureTB modellingworks shouldincorporateage-stratifiedstructures in ordertoreplicateTBactivationdynamicsaccurately.Ouranalysis alsodemonstratesthatsimplifiedmodelstructuresincorporating noreactivationareadaptedtosimulateTBlatencyinthe“<5years old”category,suggestingthatasignificantproportionofinfected childrenwillneverreactivate.Dependingonthemodelstructure employed,theseyoungindividualscouldeitherbecomeimmune aftersomedurationofinfection(serialstructure)orbeprotected immediatelyafterinfection(parallelstructure).
4.4. Comparisonwithpreviousworks
Whentwolatencycompartmentsarepositionedinseries,the rateof transitionfromearlylatencytolate latencyreflects the periodfor whichindividuals remainathighriskofdisease.We estimatethattheaverageearlylatencyperiodis52,70daysand 146days,forchild(<5),adolescents(5–14)andadults,respectively.
Mostpreviousworksemployingtheserialstructurehaveused5 yearstodefineearlylatency,basedonapreviousconventionto defineprimarydiseaseversusendogenousdisease(Vynnyckyand Fine,1997;Holm,1969),andthereforemayhavegreatlyoveres- timated theactualduration ofhigh risk.Our review of articles showedthatthemodelswiththeselongdurationsathighriskwere alsoassociatedwithlowerratesoffastprogressiontoactivedis- easethantheratesestimatedfromourstudywhichcompensates forthelongearlylatencyandleadstosimilaroverallrisksofacti- vationoveralifetime.However,wehaveshownthatourapproach, whichhasdramaticallyshorterdurationsforthetimespentinearly latencyandathighrisk,isrequiredinordertoreproducetheprofile ofactivationdynamicsaccurately.
Ourestimatefortherateoflatereactivationcorrespondstoa riskof0.20casesper100person-yearsforallages,butismuch loweramongthe‘<5years old’category.However,althoughour analysisclearlyhighlightsdifferentpatternsofreactivationbyage, accurateestimationofthereactivationparameterforthe‘<5years old’ category was limited by thesmall number of reactivation casesobservedinthissub-group,whichexplainsthewidevari- ationintheassociatedconfidenceintervals.Whilepreviousworks reportedsomewhatlowerestimatesforthelatereactivationrate forallages,rangingfrom0.04to0.16casesper100person-years (VynnyckyandFine,1997;Horsburghetal.,2010;Comstocketal., 1967;Ferebee,1970),ourfindingssuggestthattheseresultscan onlybeinterpretedinthecontextofagemixinthestudies,and wecanspeculatethatsomeofthisvariabilitymaybeaccounted forbydifferentproportionsof“<5yearolds”inthestudypopula- tions.Detaileddatabyagearenotavailablefrompreviousworks soformalanalysisbyagehasnotbeenpossible.Anotherpossible explanationforthedifferencebetweenourestimateandprevious valuesforthereactivationrateisthatthedefinitionofinfectionin ourdatasetsmaybemorespecificthanwhatwasusedinprevious studies.Inparticular,looserdefinitionsforinfectionmayinclude falsepositiveswhichwouldtendtoincreasethenumberofpersons
“atrisk”andthereforereducetheinferredreactivationrate.
WhileourstudyshowsthatpreviousTBmodelshavenotalways incorporatedtheoptimalstructureorparameterisationtoaccount forthespecificpatternsobservedinTBactivationdynamics,the potentialconsequencesofusingsuboptimalapproachesremains undetermined.However,ananalysisbyDowdyandcolleaguesof themostinfluentialparameterstoTBdynamicsdemonstratedthat theparametersdescribingearlylatencyareamongthosewiththe greatestimpactonmodelpredictionsofsteady-stateTBincidence (Dowdyetal.,2013).Thissuggeststhatitiscriticaltoreproduce earlydynamicsofTB infectioncloselyinordertoprovideaccu- rateinsightintotheepidemicstrajectory.Therefore,futureworks investigatingtheconsequencesofemployinginappropriatemodel structuresorparameterisationareneeded.
4.5. Potentialforotherfutureworks
OuranalysiswasbasedondatafromverylowTBendemicset- tingswherere-infectionisexpectednottoplayanimportantrole (Wangetal.,2007).Thisallowedourestimationtofocusonthe potentialprogressiontoactivediseasefollowingasingleinfection eventandtoavoidtheconfusionthatwouldemergefromrepeated infectionswhenestimatingthetimefrominfectiontoactivation.In theeventthatsimilardatasetsbecameavailableinhigherendemic
settings, a follow-up study could be conducted by integrating our“re-infectionfree”parameterisationintoamodelthatwould includeanadditionalpathwayforre-infectionduringlatency.By fittingsuchamodeltothenewdata,there-infectionparameter couldthenbeestimated,thusprovidingvaluableinsightsintothe relativecontributionofre-infectioncomparedtotheriskoffirst infection.
Itisimportanttorememberthatthecalibrationspresentedin thisstudyareassociatedwiththespecificmodelsthatweselected.
Accordingly,suchestimatescouldnotdirectlybeusedinmodels thatincorporateadifferentstructureorthatdonotbelongtothe category ofdeterministic compartmentaltransmission dynamic models.However,providedthatboththestructureandthenature ofthemodelcorrespondtothoseusedinthisstudy,ourworkcould alsobeusedtore-estimateparametersassociatedwithmanydiffer- entsettings.Tothisend,weprovideastep-by-stepmethodwhich allowsforrapidestimationofthedifferentmodelparametersby performingsimplemeasuresonthereactivationfailurecurve.Con- sequently,ourstudycouldalsobeusedtoinformmodelsinsettings thatpresentdifferentcharacteristics,suchashighHIV-endemic settingswheretheactivationdynamicsareknowntobedifferent (Houbenetal.,2010;Horsburghetal.,2010).
One natural limitation of this work is that it is linked to theepidemiological challengesoftiminginfectionandreactiva- tionaccurately.However,theestimateddurationsthatweusein this studyare derived fromtwo recently publishedworks that employedrigorousdefinitionsforbothdatesofinfectionandacti- vation(Traueretal.,2016a;Slootetal.,2014).
5. Conclusions
Onlymodelsemployingtwolatencycompartmentsareableto reproduceTBlatencydynamicsaccurately.We provideparame- tervaluestooptimallysimulateepidemiologicalobservationsin suchmodelsalongwithanapproachtoobtainingsuchvaluesfrom future epidemiological studies. Our analysis alsoreinforces the importanceofage-stratificationforcapturingthedramaticdiffer- encesbetweenagegroupsinpatternsofreactivation,whichimply fundamentalbiologicaldifferencesbetweenagegroups.However, wealsodemonstratethatdataofthetypethisanalysisisbased uponcannotbeusedtodeterminetheidealconfigurationforthe twolatencycompartments.
Competinginterests
Wehavenocompetinginterests.
Author’scontributions
RR,JMTandESMdesignedthestudy.RRperformedtheanalysis.
RR,JMT,NS,MTM,JTDandESMinterpretedtheresults.RR,JMT,NS, MTM,JTDandESMwrotethemanuscript.
Funding
RRwassupportedbyanAustralianGovernmentResearchTrain- ingProgramScholarship.
Acknowledgments
WethankthenursesandotherstaffoftheVictorianTBProgram forcollectingthedataInparticular,wewouldliketothankNompilo MoyoandEe-LaineTaywhocompiledthedataset.
AppendixA. Supplementarydata
Supplementarydataassociatedwiththisarticlecanbefound,in theonlineversion,athttp://dx.doi.org/10.1016/j.epidem.2017.06.
002.
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