Modelling and Bayesian analysis of the Abakaliki smallpox data

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ContentslistsavailableatScienceDirect

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

Modelling

and

Bayesian

analysis

of

the

Abakaliki

smallpox

data

Jessica

E. Stockdale,

Theodore

Kypraios,

Philip

D. O’Neill

∗

SchoolofMathematicalSciences,UniversityofNottingham,UnitedKingdom

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received27July2016

Receivedinrevisedform28October2016 Accepted7November2016

Availableonlinexxx

Keywords:

Smallpox Bayesianinference MarkovchainMonteCarlo Stochasticepidemicmodel Abakaliki

a

b

s

t

r

a

c

t

ThecelebratedAbakalikismallpoxdatahaveappearednumeroustimesintheepidemicmodelling liter-ature,butinalmostallcasesonlyaspecificsubsetofthedataisconsidered.Theonlypreviousanalysis ofthefulldatasetreliedonapproximationmethodstoderivealikelihoodanddidnotassessmodel ade-quacy.Thedatathemselvescontinuetobeofinterestduetoconcernsaboutthepossiblere-emergence ofsmallpoxasabioterrorismweapon.WepresentthefirstfullBayesianstatisticalanalysisusing data-augmentationMarkovchainMonteCarlomethodswhichavoidtheneedforlikelihoodapproximations andwhichyieldawiderrangeofresultsthanpreviousanalyses.Wealsocarryoutmodelassessmentusing simulation-basedmethods.Ourfindingssuggestthattheoutbreakwaslargelydrivenbytheinteraction structureofthepopulation,andthattheintroductionofcontrolmeasureswasnotthesolereasonfor theendoftheepidemic.Wealsoobtainquantitativeestimatesofkeyquantitiesincludingreproduction numbers.

1. Introduction

In1967,anoutbreakofsmallpoxoccurredintheNigeriantown ofAbakaliki.ThevastmajorityofcasesweremembersoftheFaith TabernacleChurch(FTC),areligiousorganisationwhosemembers refusedvaccination.AWorldHealthOrganization(WHO)report (ThompsonandFoege,1968)describestheoutbreak,with infor-mationon notonly the time seriesof case detectionsbut also theirplaceof dwelling(compound),vaccinationstatus,andFTC membership.Theoutbreak hasinherenthistorical interestas it occurredduringtheWHOsmallpoxeradicationprogramme initi-atedin1959.Althoughsmallpoxwasdeclarederadicatedin1980, it regainedattentionasa potentialbioterrorism weaponinthe early2000s(seee.g.GaniandLeach,2001;Meltzeretal.,2001; Halloranetal.,2002)andcontinuestobeofinterestduetoconcerns aboutitsre-emergenceorsynthesis(seee.g.HendersonandArita, 2014;Etoetal.,2015;WorldHealthOrganisation,2015and refer-encestherein).Publichealthplanningforpotentialfuturesmallpox outbreaksrequiresestimatesoftheparametersgoverningdisease transmission,andthusbeingabletoaccuratelyobtainsuch quan-titiesfromavailabledataisofconsiderableimportance.

Withinmathematicalinfectiousdiseasemodelling,the Abaka-likismallpoxdatasethasbeenfrequentlycited,theﬁrstappearance being(BaileyandThomas,1971).Thedataarealmostalwaysused

∗Correspondingauthor.

E-mailaddress:[email protected](P.D.O’Neill).

toillustratenewdataanalysismethodology,butin virtually all casesmostaspectsofthedataareignoredapartfromthe popu-lationof120FTCindividualsandthecasedetectiontimes,while themodelsusedarenotparticularlyappropriateforsmallpox(see forexampleBecker,1976;Yip,1989;O’NeillandRoberts,1999; O’Neilland Becker, 2001; Huggins etal., 2004; Boysand Giles, 2007;LauandYip,2008;ClancyandO’Neill,2008;Kypraios,2009; Shanmugan,2011;XiangandNeal,2014;McKinleyetal.,2014; Golightlyet al.,2014; Oh,2014; Xuet al.,2016and references therein). In Ray and Marzouk(2008) a more realistic smallpox modelisusedandaccounttakenofthecompoundswhere indi-vidualslived,butagainallnon-FTCindividualsareignored.

ThemainobjectiveofthispaperistopresentaBayesiananalysis ofthefulldataset.Toourknowledge,theonlyprevious analy-sisofthefullAbakalikidataisthatofEichnerandDietz(2003), wheretheauthorsdefineastochasticindividual-based transmis-sionmodelthatconsiders notonlythecasedetectiontimesbut alsotheotheraspectsofthedata.Theirmodeltakesaccountof thepopulationstructure,thediseaseprogressionforsmallpox,the vaccinationstatusofindividualsandtheintroductionofcontrol measuresduringtheoutbreak.Themodelparametersarethen esti-matedbyconstructingandmaximisingalikelihoodfunctionwhich isitselfconstructedusingvariousapproximations.Specifically,the truelikelihoodoftheobserveddatagiventhemodelparameters ispracticallyintractable,sinceitinvolvesintegratingoverall pos-sibleunobservedevents,suchasthetimesatwhichindividuals becomeinfected. Eichnerand Dietztackle thisproblembyfirst usingaback-calculationmethodtoapproximatethedistribution

http://dx.doi.org/10.1016/j.epidem.2016.11.005

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ofunobservedeventtimesforagivenindividual,andthenby mak-ingvariousassumptionsaboutindependencebetweenindividuals inordertoconstructanapproximatelikelihoodfunction.

Analternativesolutiontotheintractablelikelihoodproblem istousedata-augmentationmethodstoproduceananalytically tractable(andcorrect)likelihood,whichcanthenbeincorporated inaBayesianestimationframeworkbyusingMarkovchainMonte Carlo(MCMC)methodsalong thelinesdescribedin O’Neilland Roberts(1999)andGibsonand Renshaw(1998).Weadopt this approachtocarryoutafullBayesiananalysisoftheAbakaliki small-poxdata,whilst alsoassessinghow welltheEichnerand Dietz approximationmethodworksinthissetting.Ourapproach pro-videsresultswhichcanbedirectlycomparedwiththoseofEichner andDietz,speciﬁcallyestimatesofmodelparameters,estimates of associated quantities of interest suchas reproduction num-bers,andthesensitivityoftheresultstothediseaseprogression assumptions.Inaddition,wealsoestimatequantitiesderivedvia data-augmentation,suchaswho-infected-whomandthetimeof infectionfor each individual, carry out various forms of model assessmenttoseehowwellthemodelﬁtsthedata,andexplore par-ticularaspectsofthemodelviasimulation.Noneoftheseadditional elementsfeatureintheEichnerandDietzanalysis.

Thepaperis structuredasfollows. InSection 2we describe thedata,stochastictransmissionmodelandmethodofinference. Section3containsresultsand detailsofsensitivityanalysisand model-checkingprocedures.WeﬁnishwithdiscussioninSection4. Thesupplementarymaterialcontainsdetailsof somelikelihood calculationsandtheMCMCalgorithm.

2. Data,modelandinferencemethods

Theoutbreak is described in detail in Thompson and Foege (1968)andEichnerandDietz(2003).Therewere32casesintotal, 30ofwhichwereFTCmembers.Alloftheinfectedindividualslived incompounds, which weretypically one-storeydwellingsbuilt aroundacentralcourtyard,andcapableofhousingseveral fam-ilies.TheFTCmembersfrequentlyvisitedoneanotherandwere somewhatisolatedfromtherestofthecommunity,whichisone reasonwhymostpreviousdataanalysesonlyconsiderFTC mem-bers.AlthoughFTCmembersrefusedvaccination,manyofthem hadbeenvaccinatedpriortojoiningFTCasdescribedbelow.

2.1. Abakalikismallpoxdata

Table1containsdetailsofthe32casesofsmallpoxrecorded duringtheoutbreak,specifyingthedateofonsetofrash,compound identiﬁer,FTCmembershipstatusandvaccinationstatus.Notethat wesetatimescalebysettingdayzerooftheoutbreaktobetheﬁrst onsetofrashdate.Thecompositionoftheaffectedcompoundsis providedinTable2,wherethetotalnumbersofvaccinated and non-vaccinatedFTCandnon-FTCmemberswithineachcompound arelisted.Notethatonday25,fourFTCindividualsfromcompound 1(threevaccinatedandonenon-vaccinated)movedtocompound 2.Inaddition,quarantinemeasureswereputinplaceinAbakaliki, butnotuntilpartwaythroughtheoutbreak.Theexacttimethese measureswereintroducedwasnotrecorded.

2.2. Stochastictransmissionmodel

WesupposethattheresidentsofAbakalikiformaclosed popula-tionwithN=31,200individuals,labelled0,1,...,N−1.Individuals 0,1,...,ncom−1arethoseinsidethecompounds,wherencom=251 isthenumberofpeoplewithinthecompounds.Anyindividualk=0, ...,N₋1maybecategorisedastype(c_k,f_k),where(i)c_k=1,...,9 isthecompoundofk,withck=0ifkisoutsidethecompounds,and (ii)fk isk’sconfession;FTCornon-FTC.Thesetypesmayleadto

Table1

SmallpoxcasesinAbakaliki,Nigeriaduring1967,takenfromThompsonandFoege (1968).Compoundslistedarethosebeforethemoveofcases7and8,and2other uninfectedindividuals,onday25fromcompound1tocompound2.

Casenumber Dayofonsetofrash Compound Confession Vaccination

0 0 1 FTC No

1 13 1 FTC No

2 20 1 FTC No

3 22 1 FTC No

4 25 1 FTC No

5 25 1 FTC No

6 25 1 FTC No

7 26 2 FTC Yes

8 30 2 FTC Yes

9 35 1 FTC No

10 38 4 FTC No

11 40 5 FTC No

12 40 1 FTC No

13 42 1 FTC No

14 42 1 FTC No

15 47 1 FTC No

16 50 5 FTC No

17 51 2 FTC No

18 55 1 FTC No

19 55 2 FTC No

20 56 6 Non Yes

21 56 5 FTC Yes

22 57 2 FTC Yes

23 58 7 FTC No

24 60 4 FTC No

25 60 2 FTC No

26 61 2 FTC No

27 63 8 Non Yes

28 66 3 FTC No

29 66 9 FTC No

30 71 5 FTC No

31 76 2 FTC Yes

differencesinthemixingbehaviourofindividuals,butotherwise

individualsareconsideredtobeidenticalintheirsusceptibilityto smallpoxandtheirabilitytoinfectothers.

Wenowdescribeastochasticdisease-transmissionmodelfor

thespread ofsmallpox throughout thepopulationof Abakaliki.

ThismodelisessentiallythesameasthatdescribedinEichnerand

Dietz(2003),andisavariantofa Susceptible-Exposed-Infective-Removed (SEIR) model. At any given time t each individual in thepopulationwillbeinanyoneofsixstates,namely suscepti-ble,exposed,withfever,withrash,quarantinedorremoved.For

j=0,...,N−1,letej,ij,rj,qj,jdenote,respectively,thetimesof exposure,fever,rash,quarantineandrecoveryforindividualj.Any susceptibleindividualmaybecomeexposed,asdescribedbelow,at whichpointtheyenteranincubation(orlatent)period.Theynext enterthefeverstageofthedisease,atwhichpointtheybecome infectiousandmayhenceinfectothers.Duringtherashstagewhich follows,theindividualremainsinfectiousbutwithapotentially dif-ferentlevelofinfectivity.Wedeﬁnetheinfectiousperiodtobethe combinedtimespentinthefeverandrashstages.Infectious indi-vidualswilleitherbecomeremoved(namelyrecoveryordeath; wedonotdistinguishthese)orisolated,inwhichtheindividualis quarantinedandhenceforthunabletoinfectothers.Control meas-ures,inwhichcasesareplacedintoisolationsoonafterdetection, areintroducedpartwaythroughtheoutbreakattimetq.Wedonot allowre-infection,sothatindividualswhohavebeeninfected can-notbecomesusceptibleagain.Theepidemiccontinuesuntilthere arenoinfectiousorexposedindividualsremaininginthe popula-tion,atwhichpointeachpersonwilleitherstillbesusceptible,or willhavebeenquarantined/removed.

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Table2

CompositionofthecompoundsaffectedbysmallpoxinAbakaliki,Nigeriaduring1967,asdescribedinThompsonandFoege(1968).Sincewedonothavecompletevaccination statusforallcompounds,weusei4,i5,i7toallowfordifferentpossibleconﬁgurations.Thetotalnumberofvaccinatedindividualsisknownandsoi4+i5+i7=4.Itisalso knownthati4∈{0,1},i5∈{0,1,2}andi7∈{1,2,3}.Notethatthistabledisplaysthecompoundcompositionafterthemoveofthe4individualsfromcompound1to compound2onday25.Numbersoutsidethecompoundsarealsogiven,whereweassumevaccinationcoverageforFTCandnon-FTCindividualsisthesameasthatinside thecompound.

Compound FTC Non-FTC

Vaccinated Nonvaccinated nc,FTC Vaccinated Nonvaccinated nc,non

1 18 15 33 0 0 0

2 9 5 14 1 0 1

3 2 8 10 0 0 0

4 2−i4 2+i4 4 28+i4 1−i4 29

5 4−i5 3+i5 7 13+i5 2−i5 15

6 0 0 0 40 3 43

7 4−i7 1+i7 5 12+i7 3−i7 15

8 0 0 0 37 5 42

9 0 1 1 26 6 32

Sum1–9 35 39 74 161 16 177

Outside 46×35/74 46×39/74 46 30903×161/177 30903×16/177 30903

Total 120 31080

Table3

Durationsofdiseasestagesinthesmallpoxmodel.Thetimeuntilquarantinechanges aftertheintroductionofcontrolmeasuresasdescribedinthetext.

Mean(days) Standarddeviation (days)

Periodbeforefever I=11.6 I=1.9

Fromfevertorash F=2.49 F=0.88

Fromrashuntilrecovery R=16.0 R=2.83

Fromrashtoquarantineorfromtqto

quarantine

Q=2.0 Q=2.00

EichnerandDietzmodel,sothattheincubationperiod,feverperiod

andrashperiodallhaveGammadistributionswithvaluesasshown

inTable3.Ifquarantinemeasureshavebeenintroduced,thenan individualmaybeputintoisolationafterarandomdelayfollowing theirrashonsetdate.Speciﬁcally,wedeﬁnethequarantinetime ofindividualjasqj=max(rj,tq)+(2,2),where(,)denotes agammadistributedrandomvariablewithmeanandstandard deviation.Thismeansthatnoquarantiningoccurspriortotimetq, afterwhichittakesanaverageoftwodaysforadetectedindividual tobeplacedinisolation.Notethatbothremovalandquarantining ofanindividualareequivalentinthesensethatboth meanthe individualcannolongerinfectothers,butweincludebothinthe modelforclarity,andalsoforcomparisonwiththeEichnerand Dietzmodel.Notealsothataninfectedindividualwillhavebotha removaltimeandaquarantinetime,andbothquantitiesappearin thelikelihoodfunctionasexplainedlater.

Weassumethattheepidemicisinitiatedbyasingleexposed individual,whomwelabel.Theepidemicthusbeginsattimee withtheexposureoftheinitialinfective.Recallthatthe infec-tiousperiodisdeﬁnedintwoparts:thefeverperiodandtherash period,duringeachofwhichanindividualwillbeinfectious,but atpotentiallydifferentrates.Duringtheirrashperiod,an individ-ualjwillhavecontactswithothermembersoftheircompound whoareofthesameconfessionattimesgivenbythepointsofa Poissonprocessofratehperday.Individualsoutsideofthenine compoundsdonothavesuchcontacts.Inaddition,FTCindividuals willhavecontactsatratefperdaywithotherFTCindividualsand contactsatrateaperdaywithanybody(includingFTC individ-uals)inthepopulation.Non-FTCindividualsareassumedtohave contactswithanybodyinthepopulationatratea+f perday. Thisassumptionismadetoensurethatallindividualshavethe sameaveragenumberofcontactsperdayoutsideoftheirown com-pound.Duringthefeverperiod,contactsoccurinexactlythesame mannerexceptthatallratesaremultipliedbyafactorb,toaccount forapotentialdifferenceininfectivityduringthefeverperiod.In

eachcase, theindividualactuallycontactedischosenuniformly atrandomfromthepoolofpotentialindividualsinquestion.For example,contactsmadebyanindividualwiththeentirepopulation aredrawnfromtheN−1otherindividuals.Notethatthismeans thattheindividual-to-individualcontactrateforsuchcontactsis a/(N−1).Anycontactfromaninfectivetoasusceptibleresultsin immediateexposureofthesusceptible.AllofthePoissonprocesses describingcontactsareassumedtobemutuallyindependent.

Inaddition,a proportionofthepopulationisvaccinated.The vaccinationstatusofallbutafewindividualswithinthecompounds isknown,andtheproportionsofFTCandnon-FTCvaccinated indi-vidualsoutside thecompounds areassumed tobethesameas forthecorrespondingFTCandnon-FTCindividualsinside. How-ever,vaccinationisnotnecessarilyeffective:eachrecipientofthe vaccine is completely protected with probability

v

, or remains completelysusceptiblewithprobability1−

v

.Althoughthetotal numberofvaccinatedindividualsisknown,wedonothave com-pleteinformationonthecompositionofindividualswithrespect tovaccinationstatusandFTCmembership(seeTable2).Thereare ﬁvepotentialconﬁgurationsoftwelveindividualswithunknown detailstoconsider.Foreachindividualinthepopulationwewill haveavaccinationstatus,whichisassumedknownformost indi-viduals,andaprotectionstatus,whichisunknown.

2.3. Reproductionnumbers

Withinmathematicalepidemicmodelling,theso-called basic reproductionnumberisofprimaryimportance.Itcanbedefined astheaveragenumberofcasescausedbyatypicalindexcasein alargepopulation,anditsvaluetypicallygovernscertainaspects oftheepidemicsuchasthefinal number ofcasesor the prob-ability of an epidemic dying out rapidly.From a mathematical viewpoint,reproductionnumbersforstochasticepidemicsare typ-icallydefinedbyallowingthepopulationsizetobecomeinfinite inanappropriatesense.Formodelsfeaturingstructured popula-tionswithdifferentkindsofcontactrates,reproductionnumbers canbedefined invariousways.EichnerandDietzconsidertwo such reproduction numbers, namely R0=(R+bF)(a+f+h) and RF=bF(a+f+h).Here,R0 is areproduction numberfor aninfectedFTCmemberwithinthecompounds,andcanbe inter-pretedastheaveragenumberofnewinfectionssuchanindividual wouldcause,undertheassumptionthattheFTCpopulation, com-poundpopulationsandentirepopulationarealllarge.Similarly,

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Table4

Principalnotationusedinthepaper. Parameter Meaning

N Populationsize

ncom Numberofindividualsinthecompounds a Globalinfectionrate

f FTCinfectionrate

h Compoundsame-confessioninfectionrate

b Infectivityfactorduringfever

v Vaccineefﬁcacy

tq Timequarantinemeasuresintroduced Fixedparametersfordiseaseprogression Identityofinitialinfective

e Exposuretimeof

s Vectorofvaccinationstatuses(allindividuals)

p Vectorofprotectionstatuses(allindividuals)

˚ (,e,tq,b,v,a,f,h,,s)

e Vectorofexposuretimesotherthane

i Vectoroffever-periodstarttimes

r Vectorofrashtimes(data)

Vectorofremovaltimes

q Vectorofquarantinetimes

(e,i,q,,p)

Ninf(t) Setofindividualswhoarecurrentlyinfectiveattimet

Ninf Setofever-infectedindividuals

fortheAbakalikidataisthatthecompoundsthemselvesarenot

particularlylarge.However,forcomparisonpurposesweshallalso

considerR0andRFinouranalysis.

2.4. BayesianinferenceandMCMC

OuraimistoperformBayesianinferencefortheunknownmodel

parametersgiventhedata,whichconsistofrashtimesforall

infec-tives,knowledgeofthepopulationstructureandvaccinationstatus

ofindividuals.WewilluseanMCMCalgorithmtoobtain

approxi-matesamplesfromtheposteriordensityofthemodelparameters,

namelytheinfectionrates a,f and h,thevaccineefﬁcacy

v

,

theinfectivityfactor bandthetime quarantinemeasureswere

introduced,tq.Foreaseofreference,theseparametersandother

notationusedinthesequel aregiven inTable4.Our approach

involvesdataaugmentation,speciﬁcallyinvolvingtheexposure, fever,removalandquarantinetimesofeachinfectedindividual, andalsotheprotectionstatusesandunknownvaccinationstatuses. Firstwederiveanexpressionforthelikelihoodoftheobservedand augmenteddata.Let

=(,e,tq,b,

v

,a,f,h,,s)

where

=(I,I,F,F,R,R,Q,Q),

sothat the components of are parameters that areassumed known,andwheres=(s0,s1,...,sN−1),wherefori=0,1,...,N−1,

siisequalto1ifindividualiisvaccinatedand0ifnot.These vacci-nationstatusesareassumedknownforthemajorityofindividuals, withasmallnumberofexceptions,asshowninTable2.

We define e, i, q and ␶ as the unknown sets of exposure (notincludingtheinitialexposuree),infection,quarantineand removaltimes,respectively.Similarly,wedefinerastheknown setofrashtimesforallinfectives.Thenwedefinetheaugmented dataas

=(e,i,q,,p),

wherep=(p0,p1,...,pN−1)containstheunknownprotectionstatus ofeachindividual,speciﬁcallywithpi=1ifindividualiisprotected, andpi=0iftheyarenot.

Foranindividualj=0,...,N−1whoissusceptibleattimetwe deﬁnej(t)astheinfectiouspressureactinguponthemattimet, sothat

P(jisexposedin(t,t+ıt])=j(t)ıt+o(ıt),

whilstforanindividualjwhoisnolongersusceptibleattimetwe setj(t)=0.Fromthemodeldeﬁnition,ifjissusceptibleattimet thenj(t)canbewrittenas

j(t)=

k∈Ninf (t) m(k,t)×

⎧

⎨

⎩

a

N−1+ fıf(k,j)

n−1 +

hıc(k,j;t)

n_c,fk(t)₋1 iffk=FTC, a+f

N−1 +

hıc(k,j;t)

n_c,fk(t)−1 otherwise,

where

m(k,t)=

b ifik≤t <rk,

1 ifrk≤t < min(k,qk), 0 otherwise,

and(i)ıf(j,k)=1ifbothkandjareFTCandzerootherwise;(ii) ıc(j,k;t)=1ifbothkandjliveinthesamecompoundattimet

andareofthesameconfession,andzerootherwise;(iii)n=120, thenumberofFTCindividualsinthepopulation;(iv)nc,fj(t)isthe numberofindividualsinj’scompoundofthesameconfessionasjat timet,includingjthemselves,and(v)Ninf(t)isthesetofindividuals infectiveattimet.

Wedenotethelikelihoodofthedatargiventhemodel parame-tersas (r|).Thisispracticallyintractablesinceitsevaluation involves integrating over all possible unobserved events. We insteadproceedbyaugmentingthedatarwith␥toobtainthe tractablelikelihood

(r,␥|)=

⎛

⎝

j∈Ninf j(ej−)

⎞

⎠

_×e−

T e(t)dt

× j∈Ninf

fI(ij−ej)fF(rj−ij)fR(j−rj)fQ(qj−max(rj,tq))

×

v

N−1

r=0 prsr

(1−

v

) N−1

r=0 (1−pr)sr

, (1)

where(i)fort≥e,

(t)= N−1

j=0 j(t)

denotes thetotal pressure acting onall susceptible individuals attimet;(ii)j(ej−)=lim_t↑e

j

j(t)is thepressureonjjust before theirexposure;(iii)Ninfisthesetofindividualswhoeverbecome infected;(iv)Tistheendoftheepidemic,i.e.theﬁrsttimeatwhich noinfectivesorexposedindividualsremaininthepopulation(in practicewesetTequaltotheﬁnalrashtime);and(v)fA,forA=(I,F,

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Table5

Parameterestimatesandequal-tailed95%posteriorcredibleintervalsfortheAbakalikismallpoxoutbreakfromthetruelikelihoodapproach,withresultsfromEichnerand Dietz(2003)forcomparison(ED).100,000MCMCsampleswereused.HereR0=(R+bF)(a+f+h)andRF=bF(a+f+h).

Parameter Posteriormean Posteriormedian Credibleinterval EDestimate EDconﬁdenceinterval

a 0.041 0.035 (0,0.093) 0.0281 (0.00447,0.101)

f 0.063 0.059 (0.009,0.010) 0.0562 (0.0187,0.127)

h 0.358 0.349 (0.150,0.565) 0.335 (0.192,0.527)

v 0.808 0.817 (0.668,0.947) 0.816 (0.644,0.922)

b 0.522 0.374 (0,1.500) 0.157 (0,1.89)

tq 50.4 50.2 (42.4,58.4) 51.5 (44.7,59.6)

R0 7.96 7.79 (4.33,11.56) 6.87 (4.52,10.1)

RF 0.531 0.431 (0,1.364) 0.164 (0,1.31)

Onepracticaldrawbackwithourdataaugmentationschemeas

itstandsisthatitincludesprotectionstatusespiforallN=31,200

individualsinthepopulation.However,itispossibletointegrate

outtheseparametersforallindividualsoutsidethecompounds,

essentiallybecausethenumberofprotectedindividualsfollowsa

Binomialdistribution.Thecalculationsarefairlylengthyandsoare

providedinthesupplementarymaterial.

ByBayes’Theorem,theposteriordensityofinterestis

(,␥|r)∝ (r,␥|) (),

where ()denotestheprior densityof.We assumeapriori

independenceofthecomponentsofsothat

()= () (tq) (b) (

v

) (a) (f) (h) () (s).

Weseta,fandhtohave(103,106)priordistributionswhich

correspondstovaguepriorassumptionsfortheseparameters,set

v

,

bandtqtohaveuniformpriordistributionson(0,1),(0,∞)and(0,

∞)respectively,andsettohaveadiscreteuniformdistribution

overallinfectedindividuals.Sinceisassumedknown, ()isjust apointmass.Finally, (s)consistsofapointmassattheknown

vac-cinationstatuseswithauniformdistributionovertheﬁvepossible

conﬁgurationsoftwelveunknownvaccinationstatusesasshown

inTable2.

WeuseanMCMCalgorithmtoproducesamplesofthe param-etersofinterestfromthetargetposterior distribution,updating boththemodelparametersandtheunknowneventtimesaswell asprotectionstatusesandvaccinationcombinations.Thealgorithm isnon-standard,andalthoughitissimilarinprincipletothatin O’NeillandRoberts(1999),inpracticeitisfarmorecomplexand

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involvesconsiderablebook-keeping.Fulldetailsofthealgorithm areprovidedinthesupplementarymaterial.

3. Resultsandanalysis

3.1. Parameterestimatesandreproductionnumbers

Table5containsposteriorsummariesforthemodel parame-tersalongwiththecorrespondingmaximumlikelihoodestimates fromEichnerandDietz’approximatelikelihoodmethod,andFig.1 containscorrespondingdensityplots,scatterplotsandposterior correlationcoefficients.Ourestimatesarefairlysimilartothoseof EichnerandDietz,andinparticulartheposteriormodalvaluesfor thesixbasicmodelparametersarequiteclose.Althoughthey rep-resentdifferentquantities,ourposteriorcredibleintervalsandthe confidenceintervalsofEichnerandDietzarealsoquitesimilar.Our meanestimateofthebasicreproductionnumberR0is7.96,which isslightlyhigherthanEichnerandDietz’estimateof6.87.Similarly, ourestimateofthereproductionnumberforthefeverperiodRFis 0.53comparedtoEichnerandDietz’s0.164.Inthiscasethe differ-encecanbeexplainedbythehighlyskewedposteriordensityforb, sothatthemeanandmodeareclearlydifferent.Thescatterplots andcorrelationcoefficientssuggestthatthebasicmodel parame-terscanbeseparatelyestimatedfromthedataandthatthemodel isnotover-parameterised.

3.2. Whoinfectedwhom

Since theMCMC algorithm involvesimputation of all event times,itisstraightforwardtoobtainestimatesofthepathof infec-tion,i.e.whoinfectedwhom.Speciﬁcally,ifanindividualjissubject toinfectiouspressurej(t)=

m

k=1ak(t)atthetimeoftheir expo-sure,whereak(t)isthepressurefromthekthofminfectivesattime

t,thentheprobabilitythatindividualkactuallyinfectedjissimply

ak(t)/j(t).Fig.2showsthemostlikelyinfectorforeachobserved case and also a greyscale plot which illustrates the associated uncertainty.We seethat mostinfections occurredwithin com-pounds;notethatindividuals7and8movedfromcompound1to compound2duringtheoutbreakandsoinrealitymostofthe infec-tionscausedbyindividual8wereprobablyalsowithin-compound. Mostindividualsgiverisetoonesecondarycase,butindividuals 0and8bothcausemultiplesecondarycases.Thegreyscaleplot showsthat thereis a modestdegreeofuncertainty aroundthe identityofeachinfector.

3.3. Exposuretimes

Fig.3illustratestheposteriordistributionoftheinitial expo-suretime for each ofthe32 cases. Generallyspeaking there is relativelylittleuncertainty,andmostoftheexposuretimesfollow theorderingoftherashtimes,bothfeaturesthatarelikelytobe consequencesof thedistributions assumed fordisease progres-sion.Theﬁgurealsoshowsthetemporalaspectsoftheoutbreak intermsofgenerationsofinfection:theﬁrsttwogenerations(i.e. thoseinfectedbytheindexcase,andthosetheyinfect)areclearly discernible,whilethethirdandfourthgenerationsarelessdistinct fromeachother,although(accordingtoFig.2)thefourthgeneration onlycontainstwoindividuals.Weseesomegroupsofindividuals withverysimilarexposuretimesandwhoareallinfectedbythe sameperson,accordingtoFig.2,individuals4,5,6and10,11,12 beingtwoexamples.Suchclustering,moreakintoapoint-source outbreak,illustratesthehightransmissionpotentialforsmallpox inclose-contactsettings.Finally,we commentthatEichner and Dietz(2003)alsoprovideaplotshowinglikelyexposuretimes, alongwithothereventtimes,butthat thisisbased entirelyon back-calculationusingtheassumeddiseaseprogressionmodel.In

Fig.2. Estimationforwho-infected-whomduringtheAbakalikioutbreak.Thefirst figureshowsthemostlikelyinfectorforeachindividual,i.e.theinfectorshown hasthehighestposteriorprobabilityamongallpossibleinfectors.LabelsC1,...,C9 denotecompounds1,...,9,respectively.Individuals7and8movedintocompound 2duringtheoutbreak;thefigureshowstheinitialconfiguration.Thesecondfigure showstheposteriordistributionofpossibleinfectorsforeachinfectee.

particular,theirplottakesnoaccountofthetransmissionmodel itself.

3.4. Sensitivityanalysis

We now brieﬂy explore thesensitivity of ourresultstothe underlyingmodelassumptions,andinparticulartheassumed val-uesfortheperiodsoftimespentineachdiseasestate.Fig.4displays posterior densities for theparameters of interest over a range of values takenfrom Meltzer etal. (2001) and Gani and Leach (2001).Asmightbeexpected,whenR,themeanlengthofthe rashperiod beforeremoval,isreducedtomakeshorter average infectiousperiods,theestimatesoftheinfectionrateparameters increasetocompensate.Itisofinteresttonotethatestimationof

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Fig.3.Heatmapshowingtheposteriordistributionoftimeofexposureforeachofthe32casesintheAbakalikioutbreak.

halvingordoublingthemeantime,whilekeepingthecoefﬁcientof variationﬁxedatunity.Hereweseerelativelylittleimpact,which isreassuringsincewehaveverylittleinformationonwhichtobase ourmodellingassumptions,incontrasttothosewhichdependon moregenericfeaturesofsmallpox.

3.5. Modelassessment

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Fig.4. PosteriordensitiesofthemodelparametersandR0usingdifferentmean durationsofthediseasestages.Thesolidlinerepresentsthebaselinecaseasused inEichnerandDietzandourprimaryanalysis.

themodelforwardsintimeforeachsetofparametervalues.We thencomparevariousaspectsoftheobserveddatatotheensemble ofsimulationstoseeiftheformeralignsinsomesensewiththe latter.

Westartwiththefinalsizeoftheepidemic,i.e.thetotalnumber ofcases.Fig.6showsthatalthoughtheobservedfinalsize(32)is notuntypicalofthoseproducedbythemodel,itissomewayfrom themean(23.5)andmodeofthedistribution.Thisunderestimation appearstobelargelyduetothefactthatintheactualoutbreak,four individuals,ofwhomtwowereinfected,movedfromcompound1 tocompound2,leadingtonewcasesincompound2.Toaccount forthis,Fig.6alsoshowsahistogramofthefinalsizedistribution amongthosesimulatedepidemicsinwhichatleastoneofthefour movingindividualswasinfected(notethatinalloursimulations, theindividualsmoveasintherealdata).Itcanbeseenthatthis adjustmentgivesabetterfittotheobservedfinalsize.

Wenextconsiderepidemicduration,deﬁnedasthelengthof timebetweentheﬁrstcasedetection(rash)timeandthelast.Fig.7 showsahistogramofthedurationsof5000simulatedoutbreaks. Themeandurationis76.8days,whichisverysimilartothe Abaka-likioutbreak(76days).Includingonlythoseoutbreaksinwhich infectedindividualsmovedcompoundonlyincreasesthemeanby afewdays.

Wenextcomparethesimulatedcumulativenumberofcases withtheAbakalikidata.Thisiscomplicatedbythefactthat dif-ferentsimulatedoutbreaksusuallyhavedifferenttotalnumbersof

Fig.5. PosteriordensityofthemodelparametersandR0asthetimetakento quar-antineadetectedcaseisvaried.

Fig.6.Finalsizeof5000outbreakssimulatedfromposteriorestimates(mean23.5, dashedline),andthesubsetof848outbreaksinwhichatleastoneofthefour indi-vidualswhomovedcompoundwasinfectedpriortothemove(mean29.3,dotted line).Thesolidlineshowstheobservedﬁnalsize(32).

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Fig.7. Durationof5000outbreakssimulatedfromposteriorestimates,withthe solidlineshowingthedurationoftheAbakalikioutbreak.

Fig.8. 1000simulatedepidemicseachwithatotalof32cases.Thesolidlineisthe observedcumulativenumberofcases.

discrepancymeasure(seee.g.Gelmanetal.,1996),whichheretakes theform

D(r,)=

j

(rj−E(rj|))2 Var(rj|) ,

wheredenotesthemodelparametersandr=(r1,...,r32)denotes avectorofcase-detection(rash)times.Notethatneitherthemean norvariance termare available analytically, and soin practice theyareobtainedviasimulation:given,wesimulateepidemics repeatedlyuntilwehaveasampleofsizeM1,allwith32cases.The meanandvarianceofthejthrashtimeisthenestimateddirectly fromthissample.SupposenowthatwehaveMsamplesfromthe posteriordensityofthemodelparameters,labelled(1)_,_._._._,(M)_. Weusetheithsampletoobtainasimulatedepidemicwith32cases andrashtimesrrep_i byrepeatedlysimulatingepidemicsuntilwe obtainonewith32cases.Denotingatypicalsimulationreplicate rrep_and_letting_robs_denote_the_observed_rash_times,_the_posterior predictivep-valueisdeﬁnedas

ppp-value =P(D(rrep_,₎_≥_D₍_robs_,₎_|_robs₎ ≈ 1

M M

i=1 1_{D₍_rrep

i ,

(i)

)≥D(robs_,(i) )}.

Roughly speaking, if typical simulations are close to the observeddatathenwewouldexpecttheppp-valuetobearound 0.5,sincethenbothD(rrep_,₎_and_D₍_robs_,₎_should_have_similar distributions.ConverselyifoneoftheDvaluesisconsistentlylarger orsmallerthantheotherthentheppp-valuewillbeclosetozero orone,indicatingapoormodelfit.Wecarriedoutthisprocedure withM1=M=100andobtainedavalueof0.42,whichis sugges-tiveofagoodmodelfit.Amoreaccuratevaluecouldinprinciple beobtainedusinglargervaluesofMandM1,buttheprocedureis highlytime-consuminginpracticeduetothefactthatwerequire simulatedepidemicsofagivenfinalsize.

Asaﬁnalassessmentofmodelﬁt,from5000simulationswe foundthatonaverage91%ofthoseinfectedwereFTCmembers, comparedwith94%inthedata,althoughwealsofoundthataround 20%ofthoseinfectedwerefromoutsidethecompounds,compared tonosuchindividualsinthedata.

4. Discussion

4.1. Generalremarks

WehavepresentedananalysisoftheAbakalikismallpoxdata set.Theunderlyingstochastictransmission modelis essentially the same as that defined by Eichner and Dietz (2003). As dis-cussedbelow,someassumptionsofthismodelarequestionable, andthere isscopeforfurthermodellingstudieswhich incorpo-rateadditionalfeaturesintothemodel.However,ourmainfocus hasbeentoanalysetheoriginalmodelwhileavoiding the like-lihoodapproximationmethodsofEichnerandDietz(2003).The data-augmentedMCMCmethodsthatwehaveusedhavethe ben-efitsof(i)flexibility,sincetheycanbeappliedinmanycontexts; (ii)providingestimationandconsiderationofunobserved quan-tities,suchasinfectiontimes;(iii) providingestimatesnotonly ofmodelparameters,butfunctionsofmodelparameterssuchas reproductionnumbers,alongwithassociateduncertainty(in con-trast,maximumlikelihoodestimationusuallyrequiresasymptotic resultstogobeyondpointestimates,butgeneraltheorydoesnot usuallyapplytotransmissionmodels);and(iv)providingestimates ofbetween–parameterrelationships,whichcanindicateissuesof parameteridentifiabilityandinformmodelchoice.

Sinceouranalysisis basedonatransmission model,weare alsoabletoobtainsimulationsfromthemodelwhichenableusto assessmodelfitandalsoexplorewhat-ifscenarios.Thisisin con-trasttomethodswhichuseamodelconditionallyconstructedon theobserveddata,suchastree-reconstructionmethods(Wallinga andTeunis,2004).Onedrawbackwithourapproachto simulation-basedmodelfit assessmentisthatit canbetime-consumingto obtainsimulationswhichareinagreementwiththeobserveddata (inourcase,thefinalnumberofcases).Itwouldbeofbenefitto developmethodstoimprovetheefficiencyofthisapproach.

4.2. Transmission

Theposteriorestimatesofthebasicmodelparametersindicate clearly that the dominant mode of transmission was within-compound between individuals of the same confession. Our estimatesofwho-infected-whomalignwiththis;forinstance,from Fig.2,thevastmajorityoftransmissioneventsoccurredwithina compoundinthemost-likelyinfectionpathway.The epidemiolog-icalinvestigationreportedinThompsonandFoege(1968)found thatspreadwithincompounds,andwithinfamiliesinparticular, appearedtodrivetheepidemicandthatmembershipofFTCitself wasnottheprimarytransmissionroute.Thisisinagreementwith ourﬁndings.AsinEichnerandDietz(2003),wefoundthat infec-tiousnessismarkedlylessduringthefeverperiodthantherash period,althoughourmeanestimateofthereductionparameterb

islargerthantheirmaximumlikelihoodvalue,whichismostlikely duetotheskewedshapeofthemarginalposteriordensity.

4.3. Reproductionnumbers

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slightlylargerthan one,even allowingfor theinfectionrateto varywithtime(seee.g.O’NeillandRoberts,1999;Xuetal.,2016). Thishighlightstheimportanceofmodelswhichproperlytake pop-ulationstructure into account. Aspreviously stated, R0 can be interpretedastheaveragenumberofsecondarycasesproducedby asingleinfectiveindividualinalargesusceptiblepopulation.For theAbakalikidata,suchaninterpretationishardtoapplydirectly sincethecompounds,whereinmosttransmissionoccurs,aresmall enoughtoprovidea rapidsaturationeffectviathedepletionof availablesusceptibleindividuals.

4.4. Modeladequacy

Ourmodel appearstoﬁt thedata reasonablywell,withthe possiblecaveatthatthemodelinvariablypredictscasesoccurring outsideofthecompounds.Thefactthattheentirepopulationof Abakalikiisratherunrealistically modelledasa homogeneously mixingpopulationgoessomewaytoexplainingthis;in particu-lar,thepotentialforcontactsbetweenthoseinsideandoutsidethe compounds,andespeciallybetweenFTCmembersandthose out-sidethecompounds,couldwellhavebeenrather lessthanthat assumedinthemodel.AccordingtoThompsonandFoege(1968), theFTCcommunitywaslargelyisolatedfromthecommunityat large,althoughseveralofitsadultmemberswereinvolvedin trad-ingactivitiesinandaroundAbakaliki.Consequently,a modelin whichsomefractionofFTCmembershadcontactwiththe out-sidecommunity mightbemorerealistic, althoughthere areno datatodirectlyinformthis.Ourmodelalsotakesnoaccountofany priorimmunitywithinthepopulationatlarge,meaningthatsome fractionofthepopulationmighthavebeenpreviouslyexposedto smallpoxandnolongersusceptible.

Anotheraspectthatismissingfromourmodelisthatofage cat-egories;ThompsonandFoege(1968)statesthatthehighestattack rateswereamongchildren.However,theredonotappeartobe suf-ficientdataoncompoundcompositiontoaccuratelyincorporate agecategories,anditseemslikelythatamodelwithage-specific transmissionrates maybeover-parameterised. We also do not explicitlymodelpotentialtransmissionbetweenindividualsinthe samecompoundwhoareofadifferentconfession,otherthanvia thegeneralarate(recallthathappliestoindividualsinthesame compoundwhoarealsoofthesameconfession).However,Table2 showsthatmostcompoundsarealmostentirelymadeupofeither FTCornon-FTCindividuals,andsoincludinganadditionalterminto themodelisunlikelytohaveamaterialimpactontheresults.The compoundthatismostheterogeneousiscompound5, compris-ingsevenFTCandfifteennon-FTCindividuals,butheretherewere onlyfourcases,allofwhomwereFTC.Thisinturnsuggeststhat thereislittleinthedatatoinformestimationofacross-confession transmissionparameter.

4.5. Controlmeasuresandtheendoftheoutbreak

Itseemslikelythattheadventofcontrolmeasuresattimetq playeda crucial role in bringing theoutbreak toitsconclusion rapidly.Underthemodelassumptions,controlmeasuresreduce therashperiodfromanaverageof16daystojust2days,whichin turnreducesthenumberofnewinfections.Interestingly,the pos-teriormeanofR0aftertq(i.e.R0withR=Q=2.0)isaround1.5, butthisinitselfisinsufﬁcienttopermitfurtherlarge-scalespread duetothedepletionofsusceptibleswithinthecompounds,andthe factthattheepidemicinthepopulationoutsidethecompounds issub-critical(i.e.thebasicreproductionnumberislessthan1). Expandingthelatterpoint,ifwedeﬁnepre-andpost-control mea-surereproductionnumbersforspreadwithincompounds,FTCand thewider population(e.g.Ra=(R+bF)a,etc.),thenposterior meanestimates showthat (i)within compounds,theepidemic

issuper-critical beforeandaftertq; (ii)withintheFTC commu-nity,theepidemicswitchesfromsuper-tosub-critical;(iii)inthe widerpopulation,theepidemicisalwayssub-critical.Despitethis, increasingthevalueoftqinsimulationswasfoundtoincreasethe outbreaksize;forinstance,settingtqtobe50,100and200gave meanoutbreaksizesofaround24,44and64,respectively. How-ever,withnorestrictions,wefoundtheaverageoutbreaksizeto bearound86,which underlinesthefactthat theepidemicwas sub-criticalinthewiderpopulation.

4.6. AccuracyoftheEichnerandDietzlikelihoodapproximation

Itisofinteresttoseethatourresultsarefairlysimilartothose obtainedbyEichnerandDietz(2003).Themostplausible explana-tionforthisisthefactthatdistributionsusedforthelengthoftime ineachdiseasestagedonothaveparticularlylargevariances,which inturnmeansthatthemodelisnotallthatdifferenttooneinwhich alleventtimesareassumedknown.Forsuchamodel,the approxi-mationmethodusedbyEichnerandDietzgivesthetruelikelihood, essentiallybecausethedistributionsusedtoapproximate uncer-taineventtimescollapsetopointmassesaroundthetruevalues.A furtherpointofinterestisthattheEichnerandDietz approxima-tionproducesalikelihoodfunctionwhichisnumericallybutnot analyticallytractable,speciﬁcallybecauseitinvolvesintegralsthat mustbeevaluatednumerically.Althoughthisissufﬁcientfor opti-mizationpurposessuchasmaximumlikelihood,inpracticesuch likelihood functionscan becomputationally prohibitive for use withinMCMCalgorithmssincetheymustberepeatedlyevaluated. Itwouldthereforebeofinteresttodevelopanalyticallytractable approximatelikelihoodfunctions.

Acknowledgements

It is a pleasureto thankMartin Eichner for helpful discuss-ionsaboutthiswork,andtwoanonymousreviewersforhelpful comments that have improvedthe manuscript. This work was supportedbytheUKEngineeringandPhysicalSciencesResearch Council[grantnumberEP/M506588/1].

AppendixA. Supplementarydata

Supplementarydataassociatedwiththisarticlecanbefound,in theonlineversion,athttp://dx.doi.org/10.1016/j.epidem.2016.11. 005.

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