Measuring and modelling the effects of systematic non adherence to mass drug administration

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warwick.ac.uk/lib-publications

Original citation:

Dyson, Louise, Stolk, Wilma, Farrell, Sam H. and Hollingsworth, T. Déirdre. (2017) Measuring

and modelling the effects of systematic non-adherence to mass drug administration.

Epidemics, 18. pp. 56-66.

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ContentslistsavailableatScienceDirect

Epidemics

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

Measuring

and

modelling

the

effects

of

systematic

non-adherence

to

mass

drug

administration

Louise

Dyson

a,b,∗

_,

_Wilma

_A.

_Stolk

c

_,

_Sam

_H.

_Farrell

d

_,

_T.

_Déirdre

_{Hollingsworth}

a,b

a_Mathematics_Institute,_University_of_Warwick,_Coventry,_UK b_School_of_Life_Sciences,_University_of_Warwick,_Coventry,_UK

c_Department_of_Public_Health,_Erasmus_MC,_University_Medical_Center_Rotterdam,_Rotterdam,_The_Netherlands

d_London_Centre_for_Neglected_Tropical_Disease_Research,_Department_of_Infectious_Disease_{Epidemiology,}_St_Mary’s_Campus,_Imperial_College_London, LondonWC21PG,UK

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received21December2016

Receivedinrevisedform1February2017 Accepted2February2017

Keywords:

Neglectedtropicaldiseases Coverage

Systematicnon-compliance Systematic

non-adherence Modelling

a

b

s

t

r

a

c

t

Itiswellunderstoodthatthesuccessorfailureofamassdrugadministrationcampaigncriticallydepends onthelevelofcoverageachieved.Tothatendcoveragelevelsareoftencloselyscrutinisedduring cam-paignsandtheresponsetounderperformingcampaignsistoattempttoimprovecoverage.Modelling workhasindicated,however,thatthequalityofthecoverageachievedmayalsohaveasigniﬁcantimpact ontheoutcome.Ifthecoverageachievedislikelytomisssimilarpeopleeveryroundthenthiscanhave aseriousdetrimentaleffectonthecampaignoutcome.Webeginbyreviewingthecurrentmodelling descriptionsofthiseffectandintroduceanewmodellingframeworkthatcanbeusedtosimulateagiven levelofsystematicnon-adherence.Weformalisethelikelihoodthatpeoplemaymissseveralroundsof treatmentusingthecorrelationintheattendanceofdifferentrounds.Usingtwoverysimpliﬁed mod-elsoftheinfectionofhelminthsandnon-helminths,respectively,wedemonstratethatthemodelling descriptionusedandthecorrelationincludedbetweentreatmentroundscanhaveaprofoundeffecton thetimetoeliminationofdiseaseinapopulation.Itisthereforeclearthatmoredetailedcoveragedata isrequiredtoaccuratelypredictthetimetodiseaseelimination.Wereviewpublishedcoveragedatain whichindividualsareaskedhowmanypreviousroundstheyhaveattended,andshowhowthis informa-tionmaybeusedtoassessthelevelofsystematicnon-adherence.Wenotethatwhilethecoveragesin thedatafoundrangefrom40.5%to95.5%,stillthecorrelationsfoundlieinafairlynarrowrange(between 0.2806and0.5351).Thisindicatesthatthelevelofsystematicnon-adherencemaybesimilarevenindata fromdifferentyears,countries,diseasesandadministereddrugs.

1. Background

Massdrugadministration(MDA)isthecornerstoneofanumber ofcontrolprograms,particularlyhelminthcontrolandtrachoma programs,andalsoformsapartofthesuiteofinterventionsfor diseasessuchasmalariaand yaws (World HealthOrganization, 2013).Theseprogramsarebasedontheuseofdrugswithagood safetyproﬁlewhichcanbedistributedwithoutcloseclinical super-vision,and areusually prioritisedbecausetheyaremuch more cost-effectivethanscreeningandtreatingonlyinfectedindividuals duetothelogisticcostsinvolved(Brookeretal.,2008;Hollandetal., 1996).Forneglectedtropicaldiseases(NTDs),billionsofindividuals

∗Correspondingauthorat:MathematicsInstitute,UniversityofWarwick, Coven-try,UK.

E-mailaddress:l.dyson@warwick.ac.uk(L.Dyson).

havebeentreatedinMDAprograms.Insomeoftheseprogrammes keydiseasecontrolgoalshave beenmetsothat MDAcouldbe stopped(e.g.MDAprogrammesforlymphaticﬁlariasisinEgypt, Yemen,SriLanka,etc.WorldHealthOrganization,2015).However, otherprogramsarenotachievingtheexpectedgoals,andsoweare facingthequestionofwhythese“failures”areoccurringandhow bettertomeasuretheeffectivenessofcontrolprograms.

Mathematicalmodellingplaysanimportantroleinthedesign ofMDAprograms—whototreat,whentotreat(Andersonetal.,

2012, 2015; Coffeng et al., 2014, 2015; Gambhir and Pinsent,

2015;Gurarieetal.,2015;Irvineetal.,2015;Jambulingametal.,

2016;Liuetal.,2015;SinghandMichael,2015;Stolketal.,2015;

Truscott et al., 2015; Winnen et al., 2002)—and in setting the

‘expected’prevalenceaftera certainnumber ofrounds, particu-larlyforonchocerciasis(Tekleetal.,2016).Modellingstudieshave highlightedtheimportanceofcoverage(theproportionofthe tar-getpopulationwhoare treated),withhighcoverageleadingto

http://dx.doi.org/10.1016/j.epidem.2017.02.002

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morerapid declinesin prevalenceand sustainedhighcoverage leadingtothepossibilityofelimination(Okelletal.,2011;Slater etal.,2014).Empiricalstudies(Krenteletal.,2013;Briegeretal.,

2012;Kingetal.,2011;Boydetal.,2010)havehighlightedthat

someindividualsdonotreceivetreatmentnotthroughchance,but throughasystematiclackofaccesstothetreatments(suchas work-erswhoareawayduringthedaytimetreatments,Rocketal.,2015;

Mpanyaetal.,2012)orlackofacceptanceofthetreatment.These

studies,amongothers,investigatehowtreatmentcampaignsand interventionsareaffectedbytheculturalandsocio-economic con-textsinwhichtheyoccur(Krenteletal.,2016;ParkerandAllen,

2013a,2013b;Royetal.,2013;Shufordetal.,2016).Inaddition,

manyinvestigationsintotreatmentcampaigncoveragehighlight theunreliabilityofreportedcoveragedata,furthercomplicating modellingefforts(Briegeretal.,2011;Cromwelletal.,2009).

Earlymodellingworkforlymphaticﬁlariasishighlightedhow thesetypesofsystematicnon-adherencetoaprogramcan under-minethesuccessofthatprogramand,dependingonthesizeofthe untreatedgroup,actasanimportantreservoirforinfection, lead-ingtoonwardtransmissiontotherestofthepopulation(Plaisier etal.,2000).Thedecisiontoproceedwithposttreatment surveil-lancemaybebasedonthereportedcoveragelevelscombinedwith modellingpredictions(forexampleinlymphaticﬁlariasis,where achievingaround7yearsofhighcoverageisseenasatriggerto begintransmissionassessmentsurveys).Itisimportanttomeasure andunderstandtheseeffectstopreventthedangerofstoppingtoo soonorcontinuingcostlyinterventionsaftertheyarenolonger needed.Ifuntreatedindividualsaregeographicallyclustered,then thistypeofnon-adherence,orlackofaccess,canleadtohotspots ofongoingtransmission.Amorerecentstudyappliedthemethod

byPlaisieretal.(2000)(whichwaspreviouslyusedina

determin-isticsetting)tostudytheeffectofdifferentmodelsofsystematic non-adherenceinanindividual-basedmodelofhelminthinfections (Farrelletal.,2017).

Differentmodellinggroupshaveapproachedmodelling system-aticnon-adherence(whichweshalluseasacatch-alltermforthe situationwhensomeparts ofthepopulationrepeatedlydo not receivetreatments)indifferentways,butthesedifferentmethods haveneverbeenexplicitlycomparedwithrespecttotheresulting simulatedcoveragepatternsortheresultingpredictedtrendsin infection.Hereweaimtoformaliseanewmodelforthisbehaviour whichisﬂexibleenoughtocapturethedifferentmethodologiesand allowmoredirectcomparisonwithempiricaldata.Weinvestigate theimpactofdifferentassumptionsforsystematicnon-adherence usinga simple susceptible-infected-susceptible (SIS)model and ahelminthmodel. Weuseexamplesfromthesmallnumber of publishedempiricalstudieswhichmeasurethesephenomenato evaluatethesizeoftheeffect,anddiscussthevalueoffurther sur-veystoinformfuturemodellingwork.Wenotethatourworkisan attempttocaptureeffectsthatmaybegeneralacrossmultiple dif-ferentdiseasesandtoapplythistoanyparticulardiseaseorcountry wouldrequiremorein-depthstudyofthespeciﬁcsituation.

2. Overview

Wewillbeginbyreviewinghowvariousmodelsinclude sys-tematicnon-adherenceandintroducinganewwayofmodelling treatmentthatallowstheusertospecifythelevelofsystematic non-adherenceinadditiontothecoverage(Section3).Thenwe willconsidertheconsequences of systematic non-adherencein MDAcampaignsbyimplementingthevariousschemesintoa(very simpliﬁed)modelofSISdynamicsandoneforhelminthinfections, demonstratingthatthelevelofsystematicnon-adherencehasa sig-niﬁcantimpactontheoutcomeofinterventions(Section4).Finally, wewillconsiderwhatdataisrequired(andhowtoanalyseit)to

assessthelevelofsystematicnon-adherenceandwillshowthatfor thelimiteddataintheliteraturethecorrelationbetweenroundsof treatmentliesinanarrowrangeofvalues(Section5).

3. Modellingdescriptionsofsystematicnon-adherence

Manymodellingdescriptionsofsystematicnon-adherencehave beenusedina varietyofmodelsofdifferentdiseases. Herewe review and compare thedifferentschemes and propose a new method.

3.1. Listofschemes

1.Random–eachroundarandomlyselectedgroupofindividuals aretreated.(1parameter–coverage)

2.Populationpartitioning:

(a)Fullysystematic–twogroupsthataretreated:everyround; ornevertreated(1parameter–coverage)

(b)Deterministicapproximationtoasemi-systematicscheme (numberofparametersdependsonthescheme)

3.Semi-systematic–eachindividualhasaprobabilitypi(thesame

foreveryround)ofbeingtreatedineachround.(1parameter– coverage)

4.Variablecorrelationscheme–treatedindividualsaredistributed witha givenexpectation whilecorrelationis controlledbya givenparameter.(2parameters–coverageandcorrelation) (a)SchemebyGrifﬁnetal.(2010)andIrvineetal.(2015) (b)Controlledcorrelationschemeintroducedinthispaper

Wediscusseachschemeindetailbelow.

3.1.1. Random

Themajorityofmodellingpredictionsfortheoutcomeofmass drugadministrationcampaignsassumerandomcoverage(Truscott

etal.,2015;GambhirandPinsent,2015;Liuetal.,2015;Bloketal.,

2015;Pandeyetal.,2015;SinghandMichael,2015;Gurarieetal.,

2015;Andersonetal.,2015).Inthisscheme,eachindividualineach

roundhasthesameprobability,c,ofreceivingtreatment,wherec isthecoverageachievedbythecampaign.Ifthecampaign con-tinuesrunningforenoughroundstheneventuallyallindividuals willhavereceivedatleastonetreatment.Sinceeachindividualhas thesameprobabilityofbeingtreatedineachround,the propor-tionofthepopulationthatisnevertreateddropsoffveryquickly asthenumberofroundsincreases.Toensureaprobabilityofat mostTthatarandomlyselectedindividualhasneverreceived treat-ment,atagivencoveragec,requiresgreaterthanlog(T)/log(1−c) roundsofMDA.Thedistributionofnumberofroundsattendedin thepopulationafter10roundsat70%coverageisshowninFig.2(a), demonstrating that the proportion ofthe populationthat have neverattendedaroundisverysmall.Thedistributionisclustered around7roundsattended,sincethiswouldbethemeannumberof roundsattendedafter10roundsat70%coverageunderthisscheme.

3.1.2. Populationpartitioning

Asimplewayofincorporatingsystematicnon-adherenceinto anymodel(deterministicorindividual-based)istopartitionthe populationintosubpopulations thatreceive differenttreatment regimes.

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Fig.1. Aschematictorepresentthedifferentschemesusedtomodeltreatmentcampaigns.Foreachschemewegivetworoundsoftreatment.Individualsreceivingtreatment inthatroundarecolouredred,whereasthosenotreceivingtreatmentareinblack.Ineachdiagramthebackgroundcolourrepresentstheprobabilitythatapersonwill receivetreatmentinthatround,fromwhite(neverreceivetreatment)todarkblue(alwaysreceivetreatment).Thecontrolledcorrelationschemeisnotshownexplicitlyin thisdiagrambutcangivedifferentlevelsofsystematicnessdependingonthecorrelationparameterused.

2003;Plaisieretal., 1998),and isalsostudiedin a

determinis-ticmodelforonchocerciasis(Turneretal.,2015,2014a,b).Plaisier

etal.(2000)assessedthecomparativeeffectiveness ofMDAfor

lymphaticﬁlariasiswithrandom,systematicorsemi-systematic coverageschemes.TheschemeisshowninFig.2(c).

Anotherpartitionwouldﬁrstassumeasubpopulationthatnever attendsscreeningandthenuseanothermodel ofchoiceforthe remaining population. For example, incorporating a randomly-participating population and a never-participating population (HAT:Rocketal.,2015,hookworm:WORMSIM:Coffengetal.,2015) or a never-par-ticipating population and a semi-systematically participating population (see Section 3.1.3) (onchocerciasis: ONCHOSIM:Plaisieretal.,1990).Toapproximateasemi-systematic schemeinadeterministicmodel,onecanpartitionthepopulation intogroupsthatreceivetreatmentatdifferentrates.Forexample,a PDEmodelofonchocerciasis(EPIONCHO:BasánezandBoussinesq,

1999;Turneret al.,2013)theauthorssplitthepopulationinto

fourgroups:oneinwhichindividualsparticipateeveryround;one wheretheyparticipateinevenrounds;oneparticipatinginodd rounds;andonegroupthatneverparticipates.Thisschemeisvery distinctivewhenweconsiderthenumberofroundsattendedby thedifferentpopulations(Fig.2(f)),anditwouldbeverysurprising ifthiswasseeninrealdata.Howeveritisimportantto remem-berthat thisschemeis not intended asa direct representation oftherealworld,butasanattempttomake asemi-systematic schemeinadeterministicsetting.Inaddition,thisschemecould beextendedbyaddingfurthersubgroupsthataretreatedevery1, 2,3,...roundsorindeedincludingaseparatesubpopulationfor eachpossiblecombinationofroundsattended.

3.1.3. Semi-systematic

Under the semi-systematic scheme the ith individual has a probabilitypiofattendingaroundoftreatment.Toachievea

cov-eragec,eachindividualmusthaveprobabilitypi=u(1i −c)/c,where

uiisauniformlydistributedrandomnumberontheinterval[0,

1].Notethatthisschemediffersfromtherandomscheme,since theprobabilitydiffersbetweenindividuals (butisthesame for

each round), whereas in therandom schemetheprobability is thesameforallindividuals(andisalsothesameforallrounds). Thiscanbeextendedtoincludesex-and age-related participa-tionrates. The differencebetweenthesemi-systematic scheme andtherandomschememaybeeasilyseeninFig.2(b)whereit isclearthatthesemi-systematicschemeresultsinalarger pro-portionof thepopulationreceivingzeroor very fewrounds of treatment, evenat 70%coverage levels,thus havingthe poten-tialtoseriouslyundermineMDAcampaigns.Thesemi-systematic schemehasbeenconsideredinmodelsoflymphaticﬁlariasis (LYM-FASIM:Jambulingametal.,2016;Plaisieretal.,1998,2000;Stolk

etal., 2003;Subramanianet al.,2004), hookworm(WORMSIM:

Coffengetal.,2015),onchocerciasis(ONCHOSIM:Coffengetal.,

2014;Plaisieretal.,1990;Stolketal.,2015),andschistosomiasis

(SCHISTOSIM:deVlasetal.,1996).

3.1.4. Variablecorrelationschemes

Itispossibletofitmanyoftheprecedingschemesintoageneral frameworkinwhichthecorrelationbetweenroundsattended(i.e. ifanindividualattendsoneroundtowhatextenttheyaremore likelytoattendothers)issetbytheuserinadditiontosettingthe coverageachieved.ThiswasfirstattemptedbyGriffinetal.(2010) andtheirschemewassubsequentlyusedbyIrvineetal.(2015) (detailsinthesupplementaryinformation).However,whiletheir schemegivesawayofincreasingthecorrelationbetweenrounds, itdoesnotallowtheusertodirectlysetthecorrelationexactly. Inaddition,thereisnowayofreproducingthesemi-systematic schemedescribedabove,sincehighercorrelationsareachievedby includingalargernumberofpeoplethatalwaysorneverattend treatment(seeFig.2(d)and(e)).

Weproposeanewscheme(usingamethodbyQaqish,2003)in whichboththecoverage,c,andthecorrelationbetweenrounds,, maybecontrolledexactly.Wecallthisschemethecontrolled cor-relationscheme.Theprocedureisasfollows:intheﬁrstround,each personattendstreatmentwithprobabilityc.Inroundk,individual iattendstreatmentwithprobability(c(1₋)+Ri)/(1+(k−2)),

whereRiisthenumberofroundsattendedbypersonisofar.Itis

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Fig.2. Distributionofthennumberofroundsoftreatmentexperiencedbythepopulationfordifferentschemesat70%coverage.

morelikelytheyaretoattendsubsequentrounds,andthestrength ofthiseffectiscontrolledby.If=0thenthisreducestothe randomscheme(Section3.1.1,Fig.2(a)and(g)),andif=1then eachpersonwillattendroundkif,andonlyif,theyattendedthe ﬁrstround,thusreducingtothesystematicschemeinSection3.1.2 (Fig.2(c)and(i)).Infactthisschemeisequivalenttogivingeach per-sonaparameterthatgivestheirprobabilityofattendinganyround (whichisﬁxedforthatperson),asinthesemi-systematicscheme, butdrawingthatparameterfromaBetadistributionwith param-eters˛=y(1−)/)andˇ=(1−y)(1−)/(seesupplementary

information).

Asforpreviousschemes,thevariablecorrelationschememaybe straightforwardlyappliedtosubpopulationswithdifferent atten-danceparameters(forexampledifferentagegroups)bygenerating attendancesseparatelyforeachsubpopulation.Itisalsopossible toextendthisscheme(seesupplementaryinformation)toinclude additional correlated variables to model correlations between adherencetodifferenttypesofinterventionsorbetweenriskand adherencetointerventions.Forexample,itmightbethatpeople whoarelikelytoreceivedrugtreatmentsarealsomorelikelyto

receiveindoorresidualspraying(IRS)ortoreceiveandusebednets (Grifﬁnetal.,2010).

4. Whataretheconsequencesofsystematic non-adherence?

ToassesstheimpactoftheschemesdiscussedinSection3.1, weusetwoverysimplifiedmodelsofinfectiondynamics:an‘SIS’ model;andasimplifiedhelminthinfectionmodel,beforebriefly consideringtheeffectofcorrelationsbetweentreatmentand infec-tionrisk.

4.1. SISdynamics

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prevalenceofinfectionafter5years.Foreachprevalencemeasure, wegivetheprevalencescaledbytheprevalenceachievedbymost effectivescheme:afullyrandomtreatmentcampaign.Forexample, inFig.3(a)weseethattheprevalenceafter5yearscanbeupto180 timesgreaterforasystematicschemethanforrandomcoverage.

4.1.1. Impactoftheintervention

Werunthemodeltosteadystate(200years,givingastarting prevalenceof0.08forˇ=0.2and0.25forˇ=0.8)beforebeginning theplottedsimulationswithamassdrugtreatmentatyearzero. Codeforthesimulationsmaybefoundassupplementary infor-mation.Atthesecond roundthedifferentschemes willhave a differentlevel ofoverlapwithpreviouslycured individuals.The more‘systematic’schemeswilltendtore-treatindividuals who werepreviouslytreatedattimezero,sothatthiswillonlydecrease theprevalenceifthoseindividualshavesincebeenreinfected.Over repeatedtreatments,thedifferencebetweenthemoreandless sys-tematicschemesbecomesprogressivelygreater(Fig.3(a)and(b)). Varyingthecoveragelevelsandconsideringtheprevalenceafter5 yearsdemonstratesthattheeffectofsystematicnon-adherenceis greaterathighercoverages

Wemayalsoinvestigatedifferentendemicsettings,inwhich infectionhappensatdifferentrates.Systematicnon-adherencehas amuchgreatereffectwheninfectionratesareslower(Fig.3(a)and (b)),sinceatlowerinfectionratestheindividualsthatare repeat-edlytreatedinthemoresystematicschemesareunlikelytohave becomereinfectedbetweentreatments.Attheextreme,ifthe infec-tionrateissohighthatallindividualsarereinfectedbytheendof ayear,itisclearthatthedifferentschemeswouldhaveexactlythe sameimpact,sincethecoverageisthesameinalltheschemes.

4.1.2. Prevalenceafter5years

Toinvestigatemorehowthedifferentschemesvarywith cover-agerates,weconsidertheprevalenceafter5yearsforvaryinglevels ofcoverageanddifferentinfectionrates(Fig.3(c)and(d)).These ﬁg-uresdisplayanevenmorecleardistinctionbetweenthedifferent schemes,withmoresystematic schemesdisplayinghuge differ-encesinprevalences. Theeffectofsystematic non-adherenceis morepronouncedathighercoverages,sincethedifferencebetween thepopulationstreated isgreater whenmore peoplearebeing treated in general. At very high coverages the less systematic schemescaneliminatethediseasefromthepopulation,andforthis reasonwedonotgivedataforgreaterthan70%coverage(sincewe scalebytheprevalencefromtherandomscheme,whichisoften zeroafter5yearsathighcoverages).

4.2. Helminthdynamics

Theimpactofsystematicallyre-treatingindividualsislessclear inamodelofhelminthinfections,sinceindividualsarenotregarded tobesimplyinfectedorsusceptible.Insteadtheyareinfectedwith anumberofworms(whichmaybezero).Inthismodelthe preva-lenceofthediseaseinthepopulationisgivenbytheproportionof thepopulationthathaveanon-zeronumberofworms.When indi-vidualsaretreatedtheyarenotnecessarilyfullycured,butinstead aproportionoftheirwormsarekilled.Inthesemodels,therefore, individualsthataretreatedmultipletimesaremorelikelytobe curedthanthosethatonlyreceiveonetreatment.Henceitis pos-siblethatadegreeof’systematicness’couldreducetheprevalence inthepopulation,particularlyatlowcoverages,byconcentrating thosetreatmentssothatalowersubpopulationistreated,butthey aremorelikelytobefullycured.

Weagaintakeaverysimpliﬁedmodeltohighlightthe differ-encesin thetreatmentschemes withoutincluding muchdetail abouttheinfectiondynamics.Inparticular,wearenotmodelling anyparticulartypeofhelminth,andtheparameterswe useare

notinformedbyrealworlddata.Wedonotincludeanydetailsof wormreplicationwhichinreality,dependingonthespecies,can besexualorasexual,andweonlyconsideradultworms, neglect-inglarvaestagesandvectorsofinfection,suchasinsectsorsnails. Insteadweuseamodelinwhichindividualsareinfectedwitha numberofworms,whichdieatarate.Anindividualigainsworms increasethroughcontactwithanotherinfectedindividual,j,ata rate(ˇ

jWj/N)C/(C+Wi),whereˇistheinfectivity,Nisthe

popu-lationsizeandCgivesdensitydependence,sothatasthenumberof wormsinasingleindividualincreases,the‘space’fornewworms decreases.Wealsoincludedeathoftheindividual,whichispaired withnewbirthssothattheneteffectofapersondyingisthatthey arereplacedbyacompletelyuninfectedperson.

4.2.1. Plottingtheprevalenceduringamassdrugcampaign Asbeforeweplottheprevalenceinthepopulationovertime duringamassdrugcampaign,scaledbythatattainedbyarandom coveragemodel.Werunthemodeltosteadystate(200years, giv-ingastartingprevalenceof0.15forˇ=0.2and0.25forˇ=0.25) beforeadministeringatreatmentroundattime=0years(Fig.4). Codeforthesimulationsmaybefoundassupplementary informa-tion.Wepreviouslymentionedthepossibilitythatconcentrating treatmentsinasubpopulationmayleadtoalowerprevalence(i.e. proportionofthepopulationthatisinfected)whilestill increas-ingtheaveragenumberofworms.We notehere,however,that thisis neverobservedin ourmodel simulations.Anincrease in ‘systematicness’alwaysleadstohigherprevalencesinourmodel simulations(Fig.4),aswasobservedintheSISsystem.AsintheSIS model,theeffectofsystematicnon-adherenceismorepronounced atlowinfectionrates(Fig.4(a)and(c)).Wenotethattheeffect issomewhatreducedcomparedtotheSISmodelwithsystematic treatmentproducingprevalencesupto70timesthatforrandom treatmentinthehelminthmodel,comparedto180timesintheSIS model.Howeverthismaybeinﬂuencedbytheparametervalues chosen.

4.3. Correlationsbetweentreatmentandinfectionrisk

Anothertypeofsystematiceffectthatcanhavealargeinﬂuence onthesystemdynamicsisacorrelationbetweenadherenceand infectionrisk.Inthissituationindividualsthatareunlikelytobe treatedalsohaveahigherriskofbeinginfected.Wewouldexpect thistohavenegativeconsequencesforatreatmentcampaign,since thepopulationthatismostlikelytobeinfectedisalsotheleast likelytobetreatedforthatinfection.

Wemaystudythisusingaverysimplemodel,inwhicheach individualihassomeprobability Ti ofreceivingtreatment, and

acquiresdiseaseatsomerateˇi,thentheirprobabilityPi(t)ofbeing

infectedattimetisgivenby

dPi

dt =ˇi(1−Pi)−TiPi, (1)

Pi(Ti,ˇi,t)=ˇi+Tie

−t(T_i+ˇ_i)

ˇi+Ti .

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Fig.3.Theimpactofdifferenttypesmassdrugadministrationcoverageon:(a)and(b)theprevalence;(c)and(d)theprevalenceafter5years;ofanSISmodelovermultiple roundsoftreatment,fordifferentinfectionrates,ˇwhenusingacoverageof70%.Ineachplottheschemesweexpecttohavehighsystematicnon-coverageareshownin red,thosethataremorerandomareinblue,andthosewithsomesystematicnessareshowningreen.Wetaketherateofrecovery,=0.15.Linesareaveragedover1000 simulationsandarescaledbytheprevalenceattainedwhenusingtherandomcoveragescheme.Forreference,therandomcoverageschemeattainsaprevalence0.0003for ˇ=0.2andof0.03forˇ=0.25.

5. Usingdatatoassesstheextentofsystematic non-adherence

Theprecedingsectionshavedemonstratedtheimpactof sys-tematicnon-adherenceontheprevalenceofdisease.Inaddition, theformofthenon-adherencealsohasanimpactonelimination timeanddiseaseburdenovertime.Whilethecoverageis gener-allyacknowledgedtohaveafundamentalimpactonthesuccess ofacampaign,theformthatcoveragemighttakeislesswidely studied.Forthisreasongoodqualitydataonthelevelandformof non-adherenceisrelativelysparse.Itisimportanttonote, how-ever,thatevenifthecoverageandcorrelationsareknown,this doesnot fullyspecifythedistributionofattendance.Inspiteof this,wewillarguethatdataaboutnon-adherenceshouldbe rou-tinelycollectedduringamassdrugadministrationcampaign,inthe samewaythatdataaboutcoverageiscommonlytakenand stud-ied.Thiswouldrepresentasigniﬁcantstepforwardinquantifying systematicnon-adherence.

5.1. Existingdata

Forhelminthinfections,asystematicreviewwasundertakenby

Shufordetal.(2016).Manyofthestudiesincludedinthisreview

reportedcoveragedata,orwereinvestigationsintothereasonsfor non-compliance.Thesepapersgiveinsightintofactorsassociated withnon-compliance,butnottheextenttowhichanindividualis likelytoreceivemultipleroundsoftreatment.Discoveringthe rea-sonsfornon-complianceisinvaluablewhenattemptingtoincrease coverage,butformodellingpurposesamoresimplemeasureof thelevelofcorrelationbetweentreatmentroundswould signiﬁ-cantlyincreasetheaccuracyofpredictions.Somepublishedarticles

(Kingetal.,2011;Briegeretal.,2012)hintataccesstodatathat

wouldgivethisinformation,butcorrelationmeasuresarenot gen-erallycalculatedorpublished.Afewarticlesdoincludedataofthe formplottedinFig.2(Newell,1997;Plaisieretal.,2000;Brieger

etal.,2011;Mathieuetal.,2006;El-Setouhyetal.,2007).Notably

Plaisieretal.(2000)alsoincludeacomparisonofthedistributionof

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Fig.4.Theimpactofdifferenttypesmassdrugadministrationcoverageon:(a)and(b)theprevalence;(c)and(d)theprevalenceafter5years;ofasimpliﬁedhelminthmodel overmultipleroundsoftreatment,fordifferentinfectionrates,ˇwhenusingacoverageof70%.Ineachplottheschemesweexpecttohavehighsystematicnon-coverage areshowninred,thosethataremorerandomareinblue,andthosewithsomesystematicnessareshowningreen.Wetaketherateofdeathofwormstobe=0.1,the birth/deathrateofpeopletobe0.1,thedensitydependenceparametertobeC=50andassumethateachtreatmentkills70%ofthatperson’sworms.Linesareaveraged over1000simulationsandarescaledbytheprevalenceattainedwhenusingtherandomcoveragescheme.Forreference,therandomcoverageschemeattainsaprevalence 0.0005forˇ=0.2andof0.016forˇ=0.8.

attendance,andconcludethatsemi-systematicattendanceisthe mostrealistic ofthethree schemes.Sincenumericaldataisnot giveninPlaisieretal.(2000),wewillconsideronlyNewell(1997),

Briegeretal.(2011),Mathieuetal.(2006)andEl-Setouhyetal.

(2007).BothBriegeretal.(2011)andNewell(1997)investigate

treatmentforonchocerciasiswithivermectin.Newell(1997)report 4roundsoftreatmentinBurundi,whileBriegeretal.(2011) inves-tigatetheAfricanProgrammeforOnchocerciasisControl(APOC), studyingprojectsinNigeriaandCameroon.Mathieuetal.(2006)

andEl-Setouhyetal.(2007)examineparticipationinmassdrug

administrationoflymphaticﬁlariasiswithDECandalbendazolein Leogane,HaitiandEgypt,respectively.

5.2. Dataanalysis

OnlyMathieuetal.(2006)givesthenumbersattendingall

differ-entcombinationsofrounds(e.g.thepercentageofthepopulation attendingonly rounds1 and 2,say). Fromthecombinations of

roundsinMathieuetal.(2006)itisstraightforwardtocalculate thecoveragesofdifferentrounds(round1=60%,round2=62% andround3=68%)andthecorrelationsbetweendifferentrounds (corr12=0.5351,corr13=0.2979andcorr23=0.5247).

Howeveritisalsopossibletousethedistributionofnumberof roundsattended,bymakingtheassumptionthatallroundsare sim-ilar.Thisisasimplifyingassumption,thatisnotgenerallyentirely satisﬁed,butgivesanindicationoftherequiredcorrelations.Touse thedistributionofnumberofroundsattended,wedeﬁneXitobe

avectoroflengthgivenbythepopulationsize,whichisoneifthat individualattendedthedrugadministrationinroundi,andzero otherwise.ThenZ=

iXigiveshowmanyroundseachindividual

attended.Wewishtoknowthecorrelationscorr(Xi,Xj)fori=/ j.To

determinethisweusetherelationship:

var

i

Xi

=

i

var(Xi)+2

i i

j=1

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Fig.5. Existingdata(bluebars)withcontrolledcorrelationschemedistributionsusingtheestimatedcorrelationsandcoverages(redlines).

HenceiftheXiareidenticallydistributedthenvar(Xi)=var(X)for

alliandcov(Xi1,Xj1)=cov(Xi2,Xj2)foralli1,j1,i2,j2,and

corr(Xi,Xj)=

cov(Xi,Xj)

var(X) , (4)

= var(Z)

M(M+1)var(X)− 1

M+1, (5)

whereMisthenumberofrounds.Wemayalsocalculatevar(X) fromZviatheformula

E(Z)=E

Xi

=

E(Xi), (6)

hence

E(X)= 1

ME(Z), (7)

and,sinceXisaBernoullirandomvariablewithmean_E(X),then var(X)=E(X)(1−E(X)). For each dataset we calculate the esti-matedcoverageperyearandestimatedcorrelation.Weplotthe data(bluebarsinFig.5)alongwithdistributionobtainedbyusing thesewiththecontrolledcorrelationscheme(redlinesinFig.5).

ApplyingthistothedatainMathieuetal.(2006)weobtainan estimatedcoverageperyearof66%andanestimatedcorrelation of0.4152betweenyears.Thisseemslikeareasonable estimate ofboththecoveragesandthecorrelations,whileclearlynot cap-turingthelowercorrelationbetweenrounds1and3seeninthe individual-leveldata.Thislimitationcanalsobeseenwhen plot-tingthedistributions(Fig.5)sincethelowproportionattending exactlyoneroundisnotwellcaptured.

BothBriegeretal.(2011)andNewell(1997)giveonlythe

num-berof rounds attended. Using our technique onthe datafrom

Newell(1997)givesanestimatedcoverageof60%andacorrelation

of0.3268,contrastingwithreportedcoveragesofbetween40.5% and49.0%(Newell,1997).However,theﬁtobtainedbyusingthe estimatedcoverageandcorrelationisgood,onlyshowingasmall overestimateforthepercentageattendingoneround(Fig.5(b)).

Briegeretal.(2011)presentalargernumberoftreatmentrounds

(Fig.5(c)),fromwhichweestimateacoverageof57%anda corre-lationof0.3108.Thisdatasethighlightstheissueofassumingall roundsareapproximatelythesame,sincewewouldexpect cov-eragestovaryoverthelargenumberofrounds.Meancoverage rateswereonlyreportedforthreeyears:70%in2003;70%in2004

and74%in2005(Briegeretal.,2011).Giventhesedrawbacksitis

perhapssurprisingthatthisdatasetseemstoshowthebestfitso far(Fig.5(c)).Thismaybeduetothelargeramountofdatathat canbefitandthesmallerimpactofthefluctuationsinindividual yearsontheoverallfit.Inaddition,theattendancesinthisdataset weretakenfromvillageregisterstoavoidreportingbias,whichmay

improvethequalityofthedataset,whilealsoindicatingthatmore detailedindividual-leveldatamaybeavailable.Thediscrepancies foundbyBriegeretal.(2011)betweenthevillageregistersandthe reportedcoveragelevelsisindicativeoftheneedtoexaminethe accuracyofcoveragereportingandassessment.

Finally,El-Setouhyetal.(2007)reportedthenumberofrounds attended(assessedbyasamplesurvey)aftereachroundofMDAup toatotalof5(Fig.6).Thisgivesustheopportunitytocalculateour statisticalmeasuresovermultiplerounds,testingtheassumption thatthedifferentroundsareroughlythesame.Themean cover-agesfoundaftereachyearwere82.41%,88.24%,83.74%,69.26%and 74.51%,whichwerealittlelowerthanthosereported(86.7%,95.5%, 90.1%and88.8%forrounds1–4,whilecoveragewasnotreported forround5).Notethatthetwovaluesarenotexactlycomparable foreach roundsince,forexample, inround4,themean cover-ageisaveragedoverrounds1–4,whereasthereportedcoverage isjustforthatyear.Itshouldalsobenotedthat,sincethe peo-plesurveyedweredifferentaftereachround,thereporteddatais infactinconsistent,withthepercentageofpeoplereceivingzero roundsoftreatmentincreasingovertime.Theestimatedaverage correlationbetweenroundswasfound(usingequation(5))tobe 0.2806,0.3957,0.3446and0.4467afterrounds2,3,4and5, respec-tively.Thiswouldimplythatthelevelofsystematicnoncompliance increasesovertime,whichissomewhatintuitive:onemightexpect that aftermultiplerounds ofMDApeopleget intothehabitof attendingornotattending.

TherangeofvaluestakenbythedataisshowninFig.7with cal-culatedaveragecoveragesandcorrelations(colouredcircles)and reportedcoverages(colouredtriangles).Wecanseefromthisthat ourcalculatedcoveragescanbesystematicallyhigherorlowerthan thereportedcoveragesbut,withtheexceptionoftheEl-Setouhy

etal.(2007)data,arenotalargedeviation.Inaddition,whilethe

coveragesinourdatarangefrom40.5%to95.5%,therangeof cor-relationsfoundisquitenarrow(between0.2806and0.5351).Thus thereissomeevidencethatcorrelationsmaybeapproximatelythe same,evenin datafromdifferentyears,countries, diseasesand administereddrugs.Weshowthedistributionofnumberofrounds attendedforourcontrolledcorrelationmodelwithcorrelation0.4 inFig.2(h)forcomparisonwiththeotherschemes.

6. Discussion

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Fig.6. Data(bluebars)fromEl-Setouhyetal.,2007withcontrolledcorrelationschemedistributionsusingtheestimatedcorrelationuptothatround(redlines)andusing theestimatedcorrelationfromalltherounds(redstars).

differentways ofmodellingsystematic non-adherence,showing therangeof differentassumptionsthat havebeen madeinthe modellingliterature.Individual-basedmodellersweretheﬁrstto introducesystematicnon-adherence,makinguseoftheirmodel’s ﬂexibilityin characterisingindividualbehaviours(Plaisieretal.,

1990,1998; de Vlas et al.,1996).More recently,

compartmen-tal,deterministicmodelshavebeenadapted tousea varietyof methodologiesforrepresentingthisbehaviour,eachofwhichhave particularlimitations(BasánezandBoussinesq,1999;Turneretal., 2013).Herewehaveintroducedanew,moreflexiblewayof includ-ing this effect in mathematical models. Our proposed variable correlationschemeallowstheexplicitinclusion ofacorrelation betweenrounds,buttheschemeasproposedrequiresthecoverage levelstoremainthesameovermultipleroundsandthecorrelations betweenanytworoundstobethesame.Wenotethatthescheme mayeasilybeextendedusingtechniquesbyQaqish(2003)to pro-ducespecifiedcoveragelevelsand/oraspecifiedcorrelationmatrix betweenrounds.

Usingsimpliﬁedmodelsofinfection,weinvestigatedtheimpact ofdifferentassumptionsoninfectionrates,coverageand system-atic non-adherence and conclude that the effect of systematic non-adherence is more extreme at lower rates of infection. It appearsthattheeffectsareslightlylowerinhelminthmodels com-paredtotheSISmodel,howeverthismaybeduetotheparameter valueschosen.Wenotethatmorecomplicatedmodelsofhelminth dynamics,inwhichdifferentassumptionsaretakenforeach

spe-ciﬁchelminthspeciesmayaffectthisresult.Moreworkisneeded tofullyunderstandhowtheimpactoftreatmentsonthe proba-bilityofdiseasetransmissionmaychangetheeffectsofsystematic non-adherence.

In thecase where non-adherenceto treatmentis correlated withinfectionrisk,suchasinsub-populationswithpoor sanita-tionandpooraccesstohealth-care,thenthisgenerallyleadsto higherprevalencesinthelongrun.Howeverinthissituation, sur-prisingly,itisbettertofocusontreating peoplewhoarenotat riskofinfectionearlyintheprogram,sincetheyaremorelikelyto remainuninfectedafterbeingcured.

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Fig.7.Anoverviewofthedatasetsobtainedwithcalculatedaveragecoveragesand correlations(colouredcircles)andreportedcoverages(colouredtriangles).Forthe El-Setouhyetal.(2007)datasetthecoloursrefertotheroundthatthedataistaken from,sothatthetrianglesgivethereportedcoverageforrounds1–4,whilethe cir-clesrepresentthecalculatedaveragecoveragesandcorrelationsafter2–5rounds. Thehorizontallinesdemonstratewhichreportedcoveragesrefertowhich calcu-latedvalues,whiletheverticallineforMathieuetal.(2006)showstherangeof correlationsfoundwhenusingthefulldataset(whichreportswhichroundspeople attended,ratherthanjusthowmanyrounds).

onageneraldescription,itisimportanttoidentifyandquantifythe socialandlogisticaldriversinordertoovercomethem.Itis impor-tanttonotethatcorrelationsindifferentgeographicalareasmay beawayofpredictionwherehotspotsaremostlikelytooccur.

7. Conclusions

Overall this study highlights the importance of careful considerationofthedriversandcharacteristicsofsystematic non-adherence,andofmodelcomparison,sothatdifferentpredictions canbeevaluatedintermsoftheirparameterandstructural assump-tions. Further workshould focus in two main areas: gathering dataandextendinganalyticaltoolstoquantifytheextentof sys-tematicnon-adherence;andexpandingcurrentandfuturemodels toincludeandanalysetheseeffects.Wedonotmakeclaimsfor anyparticulardiseasesinthiswork,butinsteaddemonstratethat systematic non-adherence can have a large effect and encour-ageotherstoinvestigatetheseeffectsintheirowndisease-and country-speciﬁccircumstances.

Acknowledgements

LD,WASandTDHgratefullyacknowledgefundingoftheNTD ModellingConsortiumbytheBillandMelindaGatesFoundationin partnershipwiththeTaskForceforGlobalHealth.Theviews, opin-ions,assumptionsoranyotherinformationsetoutinthisarticle shouldnotbeattributedtotheBill&MelindaGatesFoundation andTheTaskForceforGlobalHealthoranypersonconnectedwith them.

AppendixA. SupplementaryData

Supplementarydataassociatedwiththisarticlecanbefound,in theonlineversion,athttp://dx.doi.org/10.1016/j.epidem.2017.02. 002.

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