1-s2.0-S1755436517300051-main.pdf

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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

Gender

asymmetry

in

concurrent

partnerships

and

HIV

prevalence

Ka

Yin

Leung

a,b,∗

_,

_Kimberly

_A.

_Powers

c

_,

_Mirjam

_Kretzschmar

b,d

a_Utrecht_University,_PO_Box_80010,₃₅₀₈_TA_Utrecht,_The_Netherlands

b_University_Medical_Center_Utrecht,_PO_Box_85500,₃₅₀₈_GA_Utrecht,_The_Netherlands

c_The_University_of_North_Carolina_at_Chapel_Hill,_2105D_{McGavran-Greenberg}_Hall,_Campus_Box_7435,_Chapel_Hill,_NC_27599-7435,_USA d_National_Institute_of_Public_Health_and_the_Environment,_PO_Box_1,₃₇₂₀_BA_Bilthoven,_The_Netherlands

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received30June2016

Receivedinrevisedform8January2017

Accepted15January2017

Availableonline20January2017

Keywords:

Concurrency

Genderasymmetry

Polygyny

HIVprevalence

Mathematicalmodel

a

b

s

t

r

a

c

t

ThestructureofthesexualnetworkofapopulationplaysanessentialroleinthetransmissionofHIV. Concurrentpartnerships,i.e.partnershipsthatoverlapintime,areimportantindeterminingthisnetwork structure.Menandwomenmaydifferintheirconcurrentbehavior,e.g.inthecaseofpolygynywhere womenaremonogamouswhilemenmayhaveconcurrentpartnerships.Polygynyhasbeenshown empir-icallytobenegativelyassociatedwithHIVprevalence,buttheepidemiologicalimpactsofotherforms ofgender-asymmetricconcurrencyhavenotbeenformallyexplored.Hereweinvestigatehowgender asymmetryinconcurrency,includingpolygyny,canaffectthediseasedynamics.Weuseamodelfora dynamicnetworkwhereindividualsmayhaveconcurrentpartners.Themaximumpossiblenumberof simultaneouspartnershipscandifferformenandwomen,e.g.inthecaseofpolygyny.Wecontrolfor meanpartnershipduration,meanlifetimenumberofpartners,meandegree,andsexuallyactivelifespan. Weassesstheeffectsofgenderasymmetryinconcurrencyontwoepidemicphasequantities(R0andthe

contributionoftheacuteHIVstagetoR0)andontheendemicHIVprevalence.Weﬁndthatgender

asym-metryinconcurrentpartnershipsisassociatedwithlowerlevelsofallthreeepidemiologicalquantities, especiallyinthepolygynouscase.Thiseffectondiseasetransmissioncanbeattributedtochangesin networkstructure,whereincreasingasymmetryleadstodecreasingnetworkconnectivity.

1. Introduction

Sexualbehaviorplaysanessentialroleinthetransmissionof HIVandothersexuallytransmittedinfections(STIs)ina popula-tion.Thereislargediversityinsexualbehavior,leadingtovarying risksofacquiringor transmittinginfection.Somebehavior pre-dominantlyinﬂuencestransmissibilityandsusceptibility,suchas condomuse.Otherbehavior,suchasconcurrency,inﬂuencesthe populationnetworkstructure.

WhetherconcurrentpartnershipsaredrivingHIVepidemicsin sub-SaharanAfricahasbeendiscussedanddebatedforalmost20 years(e.g.MorrisandKretzschmar,2000;HalperinandEpstein, 2004;LurieandRosenthal,2010;MahandHalperin,2010;Sawers and Stillwaggon, 2010; Kretzschmar et al., 2010; Epstein and Morris, 2011; Goodreau, 2011; Boily et al., 2011; Reniers and Watkins, 2010; Kretzschmar and Caraël, 2012). The biological

∗_{Corresponding}_author._Present_address:_Department_of_Mathematics,_Stockholm University,10691Stockholm,Sweden.

E-mailaddresses:[email protected](K.Y.Leung),[email protected]

(K.A.Powers),[email protected](M.Kretzschmar).

explanationbehindanimportantroleforconcurrencyis plausi-ble:inthesettingofserialmonogamy,anewlyinfectedindividual istypicallystillinapartnershiponlywithhis/herinfectorduring theacutephaseofHIV,i.e.theelevatedinfectiousnessintheﬁrst fewweeks/monthsimmediatelyafterinfection.Incontrast,when partnershipsoverlapintime,newlyinfectedindividualscan trans-mitinfectiontotheirsusceptiblepartnersinthisacutephase.It hasbeenhypothesizedandshownthroughmathematicalmodeling (Eatonetal.,2011)thattheinteractionbetweenacute infectious-nessand concurrentpartnershipscouldpotentially enhancethe spreadofHIVinthepopulation.

Someformsofconcurrencysuchaspolygyny(menmayhave multiple partners at a time while women are monogamous) naturallyleadtogenderasymmetry inconcurrentbehavior. An ecologicalstudyof34countriesinsub-SaharanAfricasuggested that thisformof concurrencyis negativelycorrelated withHIV prevalence in the population (Reniers and Watkins, 2010). A recentmodelingstudy(Reniersetal.,2015)comparingpolygyny togender-symmetricconcurrencyalsoconcludedthatpolygyny is associated withlower HIV prevalence. Besidespolygyny and gender-symmetricconcurrency,however,thereisabroadrange ofconcurrencypatternsexistinginpopulations.Inparticular,men

http://dx.doi.org/10.1016/j.epidem.2017.01.003

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areoftenfoundtoreportmoreconcurrentpartnersthanwomen, butwomenmayalsohavemorethanonepartneratatime(Morris andKretzschmar,2000;Glynnetal.,2012;Maughan-Brown,2013). Itisnotstraightforwardtopredicthowthis‘intermediate’typeof asymmetricconcurrencymightaffectpopulation-levelHIV trans-mission. We therefore sought to assess through mathematical modelinghowvaryingtypesofgenderasymmetryinconcurrent partnerships–includingpolygyny–affectHIVdiseasedynamicsin apopulation.

AsmostHIVepidemicsinsub-SaharanAfricaarenowendemic, our main aim is to study the relationship between gender-asymmetricconcurrencyandendemicprevalence.Moreover,we studythebasicreproductionnumberR0andtherelative contribu-tionoftheacutestageofHIVtoR0.Howdoesgenderasymmetry affectthepopulationprevalence; aremoreasymmetric popula-tionsassociatedwithlowerlevelsofHIVprevalence(asfoundin

ReniersandWatkins,2010)?Inthisstudyweareinterestedinthe qualitativebehaviorandwedonotattempttomakeany quantita-tivepredictionsaboutdiseaseprevalence.Weuseamathematical modelforthespreadofHIVonadynamicsexualnetworkfora het-erosexualpopulationwherewecontrolforsexuallyactivelifetime, meanpartnershipduration,meandegree,andmeanlifetime num-berofpartners.Toparameterizethemodelweuseexistingdataon sexualbehaviorfromastudypopulationinMalawiandpublished estimatesofHIVinfectivityandnaturalhistoryparameters. Differ-entscenariosinthemaximumnumberofsimultaneouspartners areinvestigated.

2. Methods

2.1. Model

Weusea mathematical modeltostudytheeffect ofgender asymmetryinconcurrencyontransmissiondynamicsinthe pop-ulation.Tothis end, we usea previously introducedmodel for infectiondynamicsonadynamicnetworkthattakesintoaccount partnership formation and separation as well as demographic turnover(Leungetal.,2012,2015).Generalanalyticalresultsfor thespreadofinfectionsonnetworkshavemainlybeenforstatic networks(e.g.Diekmannetal.,1998;BallandNeal,2008;House andKeeling,2011;Miller etal.,2012).Early exceptionsaree.g.

Altmann(1995)andVolzandMeyers(2007)whoconsiderdisease transmissionondynamicnetworkswithoutdemographicturnover. Ourmodel,whichtakesintoaccountbothpartnershipchangesand demographicchanges,canbeseenasageneralizationofpair forma-tionmodelsthatdescribesequentiallymonogamouspopulations (whichwereﬁrstintroducedinepidemiologybyDietzandHadeler, 1988).

Inthispaper,weextendedthemodelofLeungetal.(2012,2015)

toaheterosexualpopulationwitha1:1sexratioandatwo-stage infectiousdisease.Concurrentpartnershipsaremodeledby allow-ingamaximumnumberofpartnersthatanindividualcanhaveat thesametime;wecallthisnumberthepartnershipcapacity.We assumethattherearetwopartnershipcapacitiesnmandnfformen

andwomeninthepopulation,respectively,andwevariedthese valuesacrossmodelscenarios(whileholdingthevaluesofnmand

nfﬁxedacrossindividualswithinascenario).Newlyinfected

indi-vidualsareintheﬁrstphaseofinfectionandthenprogresstothe chronicphaseafteranexponentiallydistributedamountoftime. Aninfectiousindividualremainsinfectiousfortherestofhisorher sexuallyactivelife.Werefertotheﬁrstandsecondphaseof infec-tionastheacuteandchronicphase,respectively.Theacutephaseis characterizedbyahighertransmissionratethanthechronicphase. Thenetworkstructureisstableovertime,whichleadsto sta-bledegree distributionsfor menand women(the degreeof an individualisthenumberofpartnersthatheorshehas).Themean

degreewaskeptconstantthroughoutthisstudy.However,the spe-ciﬁclinkswithinthenetworkaredynamicintimeduetoindividuals formingnewpartnershipsandseparatingfromexistingpartners duringthecourseoftheirlife.Furthermore,thereisdemographic ﬂowdue toindividuals enteringandleavingthesexuallyactive population.Infectionisassumedtohavenoimpactonpartnership formationorseparationnoronmortality.

Themodelisdeterministicinnature.Itisfullycharacterized byasetofparameterscomprisingsexualbehaviorandinfection parameters.Individualsarecharacterizedbysex-speciﬁc behav-ioralparameters;seeSections2.2and 2.3.We assumethatthe infectionparameters(i.e.transmissionratesanddisease progres-sion)donotdependongender(Hughesetal.,2012;Powersetal., 2008;Quinnetal.,2000).

Concurrencyismeasuredusingthepartnership-based concur-rencyindex(MorrisandKretzschmar,1997;Leung etal.,2012; LeungandKretzschmar,2015).Itmeasuresthemeannumberof additionalpartnersofanindividualinarandomlychosen partner-shipandcanbeexpressedintermsofthemeanandvarianceof adegreedistribution.Herewedistinguishbetweenconcurrency indicesmP andfPformenandwomen,respectively,andthe

con-currencyindexpopul_P forthepopulationasawhole.Wevarythe gendersymmetryofconcurrencyinthepopulationbyvaryingnf

andnm.

Wehavepreviouslyderivedexplicitexpressionsforthedegree distributions,partnership-basedconcurrencyindices,basic repro-ductionnumberR0,and relativecontributionof theacutestage toR0intermsofmodelparameters(Leungetal.,2012,2015).The endemicprevalenceischaracterizedimplicitlyassolutionofaﬁxed pointproblemthatcanbecomputednumerically.

Amoredetailedandtechnicaldescriptionofthemodelcanbe foundinAppendixAofthesupplementarydata.

2.2. Scenarios

ThroughoutthisstudywekeepmeanpartnershipdurationdP, sexuallyactivelifespan ¯L,andmeanlifetimenumberofpartners ¯

inthepopulationfixed.Thisalsomeansthatmeandegreeinthe populationisfixed(Eq.(A.2)inAppendixAofthesupplementary data).Thescenariosaredefinedbyvaryingthesex-specific quan-titiesnm,nf,Lf,andLm(whichalsovariesfandm,Eq.(A.1)in

AppendixA.1ofthesupplementarydata).

Wefocusongenderasymmetryinthemaximumnumberof partnersmenandwomenmayhaveatthesametime,i.e.in partner-shipcapacitiesnfandnm.Meanpartnershipcapacityisﬁxedacross

scenariosat ¯n=(nf+nm)/2=4.Theextremesaretheasymmetric

polygynous situation (nf, nm)=(1, 7) (i.e. women are

monoga-mous)andthesymmetricsituation(nf,nm)=(4,4).Notethat,since

infectionparametersaregenderindependent,asymmetryinthe oppositedirection((nf,nm)=(7,1)etc.)isautomaticallyincluded

byswitchingtheroleofmalesandfemalesintheresults.

MathematicalmodelsofheterosexualHIVtransmissionin sub-Saharan African settings commonly assume a sexually active populationbetween theages of 15 and 49 years (Eatonet al., 2012),correspondingtoanaveragesexuallyactivelifespan ¯Lof 35years.Differencesinthedurationofsexualactivitybysexare notwelldeﬁned.Giventheuncertaintyinsexuallifespansformen andwomen,wemodeledthreescenarios:oneinwhichbothmen andwomenhadameansexuallifespanofLf=Lm=35years;onein

whichthesexuallyactivelifespanwasLf=33yearsforwomenand Lm=37yearsformen;andoneinwhichthesexuallyactive lifes-panwasLm=33yearsformenandLf=37yearsforwomen.Larger

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Table1

The12differentscenarios.Correspondinglifetimenumberofpartnersaref=3.3

andm=3.7forLf=33years,Lm=37years,f=3.5=mforLf=35=Lmyears,and f=3.7andm=3.3forLf=37years,Lm=33years,seealsoSection2.3.

Lf=33years Lf=35years Lf=37years,

Lm=37years Lm=35years Lm=33years

(nf,nm)=(1,7) A1 A2 A3

(nf,nm)=(2,6) B1 B2 B3

(nf,nm)=(3,5) C1 C2 C3

(nf,nm)=(4,4) D1 D2 D3

Together,genderasymmetryinpartnershipcapacitiesand sex-uallyactivelifespansgiverisetotwelvedifferentscenarios;these aresummarizedinTable1.

2.3. Parameterizationofthemodel

Meanpartnershipdurationisbasedonself-reporteddataon steadypartnershipdurationcollectedinastudyofacuteHIV detec-tionstrategiesandlongitudinalHIVviraldynamicsatanSTIclinicin Lilongwe,Malawi.Samplingstrategiesandstudyprocedureshave beendescribedpreviously(Fiscusetal.,2007;Pilcheretal.,2007; Powersetal.,2007,2011a,b).Asourmodelassumesaconstant part-nershipdissolutionrate,wecalculatedtheslopeofalinearcurve ﬁttedtothe reportedduration ofsteady partnerships(indays) onthex-axisandthe–ln(partnershipsurvival)onthey-axis,see AppendixBofthesupplementarydata.Weestimatedaweighted meanpartnershipdurationdPof3.92yearsacrossmenandwomen astheinverseofthisslope.

Sinceweareinterestedinqualitativebehaviorandwedonot aimtomakeanyquantitativepredictionsaboutdiseaseprevalence inthisstudy,themeanlifetimenumberofpartners ¯iscalibratedto anendemicHIVprevalenceofaround13%forscenariosB2andC2, correspondingtoestimatedHIVprevalenceinMalawiin2010(DHS Program,2010).Thisleadstoameanlifetimenumberofpartners ¯

=3.5.Consistencybetweenmeanlifetimenumberofpartners andsexuallifespanthendeterminesthemeanlifetimenumberof partnersm andffor menandwomen, thesearefoundbelow Table1(seealsoEq.(A.1)inAppendixA.1ofthesupplementary data).

Notethatourchoiceofkeepingmeansexualbehavior param-eters constant also leads toa constant mean degree acrossall scenarios, which isthe samefor menand women (Eq.(A.2)in Appendix A.1 ofthe supplementarydata). Variance, and there-foretheconcurrencyindex,doesdifferacrossthescenarios;see Section3.1fordetailsandadiscussion.

Infectionischaracterizedbythetransmissionratesˇ1,ˇ2and duration oftheacute phase dA (Table2).We letˇ1=2.76/year,

dA=2.9months,andˇ2=0.106/year(Hollingsworthetal.,2008). InAppendixC.3ofthesupplementarydataweconducted sensitiv-ityanalysisusinginfectivityparameterestimatesfromadifferent,

Table2

Modelparametersandvalues.Variationaroundtheparametervaluesareaddressed inthesensitivityanalysisinAppendixCofthesupplementarydata.

Parameter Description Estimate

nm Partnershipcapacityformen Varied(seeTable1)

nf Partnershipcapacityforwomen Varied(seeTable1)

dP Meanpartnershipduration 3.92years

Lf Sexuallyactivelifespanforwomen Varied(seeTable1)

Lm Sexuallyactivelifespanformen Varied(seeTable1)

¯

L=1

2(Lf+Lm) Sexuallyactivelifespan 35years

f Meanlifetimenumberofpartnersfor

women

Varied(seeTable1)

m Meanlifetimenumberofpartnersfor

men

Varied(seeTable1)

¯

=1

2(f+m) Meanlifetimenumberofpartners 3.5

ˇ1 Transmissionrateacutephase 2.76/year

dA Durationacutephase 2.9months

ˇ2 Transmissionratechronicphase 0.106/year

morerecentlypublishedstudy(Bellanetal.,2015).InBellanetal. (2015)amuchlowertransmissionratefortheacutestageof infec-tionwasestimatedbutahighertransmissionrateforthechronic stageofinfection(yieldingacomparabletotalinfectivity). Qualita-tively,thisdidnotchangeourconclusions.

ParametervaluesaresummarizedinTable2.Variationaround theparametervaluesareaddressedinthesensitivityanalysisin AppendixCofthesupplementarydata.

3. Results

ThethreedifferentsexuallyactivelifespancasesLf<Lm,Lf=Lm,

andLf>Lmleadtoverysimilarresultsfornetworkstructure,R0, andtherelativecontributionoftheacutephasetoR0.Therefore,in bothSections3.1and3.2,wefocusonthesexuallyactivelifespan caseLf=Lm,i.e.scenariosA2,B2,C2,andD2.InSection3.3all12

scenarios(deﬁnedinTable1)areconsidered.Asensitivityanalysis wascarriedoutinAppendixCofthesupplementarydata,which showsthattheresultspresentedinthissectionarequalitatively robusttovariationsinparametervalues.

3.1. Networkstructure

Astheoveralldegreedistributions(acrosswomenandmen)are verysimilaracrossscenariosB2,C2,andD2,weshowonlyscenarios A2andD2inFig.1.Ourdeﬁnitionofscenariosleadstoaconstant meandegreeformenandwomenacrossall12scenariosanditis givenby ¯/_Ld¯ _P₌0.39(notethatthisisindependentofgenderand partnershipcapacity).Thevarianceformenandwomendoesdiffer acrossthescenarios.VariancevaluesforscenariosA2,B2,C2,and D2aresummarizedinTable3(variancevaluesfortheothereight scenariosaresimilar).

Weﬁndthatthedegreedistributionformenissimilaracross allfourscenarios,illustratedbythetwomostextremescenarios

men women

7 6 5 4 3 2 1 0

number of partners

0 0.2 0.4 0.6

probability

Scenario A2

men women

7 6 5 4 3 2 1 0

number of partners

0 0.2 0.4 0.6

probability

Scenario D2

(a) (b)

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Table3

Thedegreevarianceforwomen(f)andmen(m)forthescenariosA2,B2,C2,andD2 (Table1).Meandegreeis0.39forbothmenandwomenacrossallscenarios.

A2 B2 C2 D2

f m f m f m f m

0.24 0.38 0.32 0.37 0.35 0.37 0.36 0.36

A2andD2andthedegreevarianceinTable3.Thekeydifference acrossscenariosisinthedegreedistributionsforwomen:inA2, womenaremonogamousandcanhaveeitherzerooronepartner, whichisdistinctlydifferentfromscenariosB2,C2,andD2,where therearefractionsofwomenwithmorethanonepartneratatime. Thedegreedistributionformenandwomentogetherare impor-tantinmakingupthenetworkstructure,thisisillustratedbyFig.2. InFig.2networksforscenariosA2andD2arevisualizedby simu-latingcross-sectionsofthenetworksusingdegreedistributionsof

Fig.1.Networkstructureissigniﬁcantlydifferentinthetwo scenar-iosA2andD2.InscenarioD2,longer‘chains’ofpartnershipscanbe found,e.g.achainconsistingofsixindividuals.Thisisabsentin sce-narioA2wherewomenhaveatmostonepartneratatime(sochains consistofatmostthreeindividuals).NotethatFig.2represents snapshotsofonerandomlychosenpointintimeasthenetworks areevolvingintimewithpartnershipformationanddissolutionas wellasindividualsenteringandleavingthesexuallyactive popula-tion.Furthermore,singleindividualsareomittedfromthenetwork ﬁgurestobettershowthepartnerships.Finally,although partner-shipcapacitychoicesinprincipleallowformuchlargerconnected components,ourparametervaluesinthisstudyaresuchthatthis israrelythecase;thereisonlyaverysmallfractionofthe popula-tionwithmorethantwopartners(Fig.1).Nevertheless,thesmall proportionoflongerchainsintheD2networkresultin substan-tiallyhigherratesofdiseasetransmissioninthenon-polygynous scenarios(Sections3.2and3.3).

Next,weconsiderthelevelofconcurrencyforthescenariosA2, B2,C2,andD2.Differentaspectsofconcurrency,suchasthemean numberofconcurrentpartnershipsandthemeandurationof over-lap,arecapturedbythedegreedistributioninthepopulation(Fig.1

andnetworkparameterssuchasthemeanpartnershipduration andlifetimenumberofpartners).Fromthedegreedistributions inFig.1,weobservethatonlysmallfractionsofmenandwomen haveconcurrentpartnerships,i.e.morethantwopartners,acrossall scenarios,andalargeproportionofthepopulationiswithoutany partner.Wecombinetheseaspectsofconcurrencyandmeasure thelevelofconcurrencywiththepartnership-basedconcurrency index(cf.MorrisandKretzschmar,1997;Leungetal.,2012;Leung

Fig.3. Theconcurrencyindexm P formen,

f

Pforwomen,and

popul

P forthe

popula-tion,inscenariosA2,B2,C2,andD2.

andKretzschmar,2015,seealsoAppendixAofthesupplementary data).TheresultsarepresentedinFig.3.

Notethatfemaleconcurrencyiszerofornf=1(monogamy)and

increasesfornf=2,3,4.Maleconcurrencyishighestinthemost

asymmetricscenarioA2anddecreaseswithmoresymmetric sce-narios.Similartothecaseforwomen,thisdecreaseinconcurrency iscausedbyadecreaseinthemaximumnumberofsimultaneous partnerships(nm=7,6, 5,4).Thedecreasein maleconcurrency isless thantheincreaseinfemaleconcurrency.Therefore, with increasingequalityofmenandwomeninthenumbersofpartners theymayhaveatthesametime,theoverallpopulation-level con-currencyindexpopul_P increases,evenifaveragelifetimenumbers ofpartnersremainconstant.

3.2. R0andthecontributionoftheacutephasetoR0

ThebeginningoftheepidemicisstudiedbyconsideringR0and therelativecontributionoftheacutephasetoR0forscenariosA2, B2,C2,andD2inFig.4.R0 increaseswithincreasingsymmetry inthepopulation.InallfourscenariosR0isabovetheepidemic thresholdvalueone.

Transmissionofdiseaseduringtheprimaryphaseisstrongly determinedbyconcurrency(combineFig.3withFig.4).Onlyif concurrentpartnersareavailableduringtheprimaryphasecanthe infectionbetransmittedinthatphase,becauseseparationand ﬁnd-inganewpartnerrequiresamuchlongertime.Inthepolygynous scenariothenetworkstructureismadeupofshortchainsof part-nerships(involvingamaximumofthreeindividuals,seealsoFig.2). Therefore,itisreasonablethatprimaryinfectionwillhavelittle impactinthepolygynousscenarioA2.Theacutephasedoesplaya

Scenario D2 Scenario A2

men women partnerships

) b ( )

a (

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chronic phase acute phase

1.0% 10.9% 12.4% 12.7%

Fig.4.R0inthefourscenariosA2,B2,C2,andD2.Thedashedlinerepresentsthe

epidemicthresholdvalueofone.Thelightanddarkgreenindicatethefractionof

R0thatiscausedintheacuteandchronicphaseofinfection,respectively,whilethe

heightofthebarrepresentsR0.(Forinterpretationofthereferencestocolorinthis

ﬁgurelegend,thereaderisreferredtothewebversionofthisarticle.)

(n ,n )=(1,7)f m

(n ,n )=(2,6)f m

(n ,n )=(3,5)f m

(n ,n )=(4,4)f m

A1 B1 C1 D1

A2 B2 C2 D2

A3 B3 C3 D3

Fig.5.Theendemicprevalenceinthepopulationforthe12differentscenarios plot-tedasafunctionofsexuallyactivelifespanwithpartnershipcapacitiesﬁxed.The correspondingscenario(seeTable1)isdenotedforeachpointinthegraph.

signiﬁcantroleintheotherthreescenarioswithacontributionof upto_∼13%toR0.

3.3. Endemicprevalence

InFig.5populationHIVprevalenceisconsideredasafunction oftheconcurrencyandsexuallyactivelifespanscenarios.

SimilartothequantitiesexploredinSections3.1and3.2,from theresultsinFig.5weconcludethatgenderasymmetryinsexually activelifespanshavelittleeffectonpopulationprevalence (com-paree.g.scenariosB1,B2,andB3toeachother,thesescenarioshave thesamegenderasymmetryinconcurrencybutdifferentsexually activelifespanscenarios).Moreover,weseethatgenderasymmetry inconcurrentpartnershipsdoeshaveanimportanteffecton popu-lationprevalence(comparee.g.scenariosA1,B1,C1,andD1toeach other,wheregenderasymmetryinconcurrentpartnershipis var-iedbutsexuallyactivelifespansarethesame).Inparticular,weﬁnd

thatthepolygynoussettingAisassociatedwithmuchlowerlevels ofdiseaseprevalencecomparedtotheotherconcurrencysettings B,C,andD.Thisisconsistentwiththenetworkstructurein set-tingAcomparedtotheothersettings(seealsoFig.2).InsettingA, thelongestchainsofpartnershipsinvolveatmostthree individu-als(menhavingmorethanonepartner).Potentiallymuchlonger chainsofpartnershipscanbefoundintheothersettings(seeFig.2

fora simulatedcross-sectionofanetwork structurein scenario D2;scenariosB2andC2areinbetweenA2andD2intermsoftheir networkstructure).

Weinvestigatethegenderasymmetryfurtherbyconsidering endemicprevalenceseparatelyformenandwomenasafunction ofpartnershipcapacityscenariosinFig.6.Thefactthatendemic prevalenceisinareasonablerange(DHSProgram,2010)isa con-sequenceofourparametervaluechoicefor ¯(recallSection2.3: wecalibrated¯toanendemicdiseaseprevalenceofaround13%in thepopulationforthescenariosB2andC2;seeFig.6).Ourinterest wasinthequalitativebehavior,inparticularweareinterestedin comparingtheendemicprevalencesofscenariosA2,B2,C2,and D2relativetoeachother.Weﬁndthat,qualitatively,gender asym-metryinpartnershipcapacityisassociated withlowerendemic prevalenceinthepopulation.Again,thisimpliesthatmoreequally distributedconcurrencybehaviorcanleadtohigher epidemiolog-icaloutcomes.

Next,thereisalsogenderasymmetryindiseaseprevalence.In

Fig.6(b)and(c)femaleprevalenceishigherthanmaleprevalence forallfourpartnershipcapacitycombinations.Weﬁndthatthe female-to-maleratioofHIVprevalencerangesfrom1inscenarioD2 (symmetricsexuallifespansandpartnershipcapacities)to1.22in scenarioA3(Lf>Lmandgreatestasymmetryinpartnership

capac-ities).ThisgenderratioislargestinFig.6(c).However,inFig.6(a) for(nf,nm)=(3,5),and(4,4),femaleprevalenceisslightlylower

thanmaleprevalence.Sensitivityanalysisshowsthatthisisa gen-eralresultforpartnershipcapacitycombination(nf,nm)=(4,4)and Lm>Lf(Fig.C.6inAppendixCofthesupplementarydata).Onthe

otherhand,thesameanalysisshowsthatfor(nf,nm)=(1,7),female

prevalenceisalwayslargerthanmaleprevalenceregardlessof gen-derdifferencesinsexuallyactivelifespans(so(nf,nm)=(2,6),(3,5)

liesomewhereinbetween).Thefemale-to-maleratioofHIV preva-lenceisonlyslightlyskewedtowardswomeninFig.6.However,we consideredonlyrelativelysmalldifferencesinlifespansbetween menandwomen. Weﬁndthat thefemale-to-male ratioofHIV prevalencebecomesmuchmoreskewedtowardswomenwhenthe female-to-maleratioofsexuallyactivelifespansalsoincreases(Fig. C.6inAppendixCofthesupplementarydata).

We conclude thatnot only is gender asymmetryin concur-rentpartnershipsassociatedwithlowerdiseaseprevalence,italso affectsthefemale-to-maleratioofprevalence.Genderasymmetry inconcurrentpartnershipsisassociatedwithanincreasedgender asymmetryindiseaseprevalence.Whilegenderasymmetryin sex-uallyactivelifespanshaslittleeffectontotalpopulationprevalence, itdoesaffectthegenderratioindiseaseprevalence.

(a) (b) (c)

_>

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(a)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

popu

lat ion concurr

ency

0.00 0.05 0.10 0.15 0.20

fra

c

ti

o

n

case A case D

(b)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

popu

lat ion concurr

ency

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

e

nd

em

ic

p

rev

al

en

c

e

case A

case D

(c)

Fig.7.Correlationbetweenpopulationconcurrencyand(a)R0,(b)therelativecontributionoftheacutephasetoR0,and(c)populationprevalence,whenvaryingmodel

parametersinthesensitivityanalysis(seeAppendixCofthesupplementarydata).CaseAcorrespondstotheasymmetricpolygynoussettingandcaseDcorrespondstothe symmetricsetting.

3.4. Concurrencyandepidemiologicaloutcomes

Takingtheresultsof Sections3.1–3.3together,weconclude thattheepidemiologicalquantitiesR0,therelativecontributionof theacutephase toR0,andtheendemicprevalenceare increas-ingwithincreasingconcurrencyinthepopulation(combineFig.3

witheitherFig.4,orFig.6).Theeffectofgender-asymmetric con-currencyonepidemiologicalparametersthusappearstobeinpart mediatedthroughincreasedoverallconcurrency.Moreover,inour sensitivityanalysiswhereweallowforvariationinallparameter valuessimultaneously,weﬁndaverystrongcorrelationbetween population-levelconcurrencyandR0andpopulationdisease preva-lence;seeFig.7(a)and(c)andAppendixCofthesupplementary datafordetails.Thissuggeststhat,althoughallsexualbehavior parametersseparatelyhaveanimpactonprevalence,the syner-gisticcombinationoftheseparametersintheconcurrencyindex

popul_P yieldsamuchmorepronouncedcorrelation.Thiscorrelation

ismuchlessstrong(butclearlystillexistent)whenconsideringthe relativecontributionoftheacutephasetoR0;seeFig.7(b).

4. Conclusionanddiscussion

Inthisstudyweinvestigatedtheeffectofgenderasymmetry inthemaximumnumberofsimultaneouspartnershipsan individ-ualcanhave,theso-calledpartnershipcapacity,onHIVdisease dynamics.Ourstudywasqualitativeinnatureandcannotbeused tomake anyquantitativestatements.Motivatedbytheﬁndings ofReniersandWatkins(2010),wewereespeciallyinterestedin

theextremecaseofpolygynousunions,i.e.(nf,nm)=(1,7),with

additionalinterestinothertypesofasymmetries.Wefoundthat allthreeepidemiologicaloutcomesR0,therelativecontributionof theacutephase toR0,andendemicprevalencearemuchlower inthepolygynousscenariothaninmoregendersymmetriccases. Theseﬁndingshighlightpolygynyasaspecialcaseofasymmetric concurrencythatmaybeparticularlyprotectiveatthepopulation level.However,theepidemiologicaloutcomeswerealsolowerin theothertwogenderasymmetricscenarios (nf, nm)=(2,6)and

(3,5)compared tothesymmetricscenario (nf,nm)=(4,4).Our

modelresultsshowthat anylevel ofasymmetryleadstolower endemicprevalencethanapopulationwithfullsymmetry. There-fore,consistentwiththeﬁndingsofReniersandWatkins(2010), ourresultssupportthatpolygyny,evenifpracticedonlybyapart ofthepopulation,canindeedreduceHIVprevalenceascompared toafullymixedpopulationwherebothsexeshaveequalnumbers ofconcurrentpartners.However,basedonourstudy,wherewe controlledformanynetworkstatisticssuchasmeanpartnership duration,itisnotpossibletosayhowtheeffectofpolygynyinthe modelextendstoacausalexplanationfortheecologicalcorrelation betweenpolygynyandprevalenceinReniersandWatkins(2010). We also investigated the inﬂuence of gender asymme-try in sexually active lifespans Lf and Lm for women and

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inthesexuallyactivepopulation, weconsideredthreedifferent cases(but seeLindauand Gavrilova,2010 for anestimation of sexuallyactivelifespansforaheterosexualpopulationintheUSA). Nevertheless,genderasymmetryinsexuallyactivelifespansdid notqualitativelychangeourﬁndingsaboutgenderasymmetryin partnershipcapacitiesandtheassociationwiththeepidemiological quantitiesunderconsideration.Additionally,wefoundthatincase ofgendersymmetricsexuallyactive lifespans(Lf=Lm)orlonger

sexuallyactivelifespansforwomenthanmen(Lf>Lm),the

female-to-maleratioofHIV prevalenceisskewedtowardswomen.The sensitivityofHIVprevalenceratiostodifferencesinsexuallifespans whenweconsideredabroaderrange(Fig.C.6inAppendixCofthe supplementarydata)indicatesthatmoreempiricalinformation– andattentioninmodeling–onsexuallyactivelifespansandtheir effectsisneededtobetterunderstandthesedurations.

Consistent with our ﬁndings, HIV prevalence data indeed showsthat morewomen thanmenareliving withHIVin sub-SaharanAfrica,e.g.inMalawi(UNAIDS,2013;DHSProgram,2010). Empiricalestimatesofthefemale-to-maleratioofinfectionsin sub-SaharanAfricanpopulationsrangefrom1.31to2.21(Reniersetal., 2015).Manyfactorscanleadtothesedifferences,andoneofthem couldbethedifferenceinconcurrentpartnershipsbetweenmen andwomen.Theratiosinourstudyaresmallerthantheempirical estimates,mostlikelybecauseweonlyinvestigatedtheeffectof genderasymmetryinconcurrentpartnerships.

Ourmainfocuswasontheeffectofgenderasymmetryin part-nershipcapacitiesonHIVprevalencewhilecontrollingformean partnership duration, mean lifetime number of partners, mean degree,sexuallyactivelifespan,andinfectionparameters.Wewere nottryingtoisolatetheeffectoftheseasymmetriesindependently ofoverallconcurrency,ratherwe wantedtoseethetotaleffect oftheseasymmetries(evenifpartiallymediatedthroughchanges inoverallconcurrency).Wefoundthatanyofthesexualbehavior parameters,suchaslifetimenumberofpartnersandpartnership durationcorrelatepositivelywiththeepidemiologicalquantities. Thereisaparticularlystrongcorrelationbetweenpopulation-level concurrency with R0 and population disease prevalence (Sec-tion3.4).Similarquestionsongenderasymmetryinconcurrency andHIVprevalencewereaddressedinReniersetal.(2015).Their focuswasoncomparingpolygynyandgender-symmetric concur-rencyandtheireffectonthefemale-to-maleratioofHIVprevalence usinganagent-basedmodelingapproach.Inourstudywelooked atothertypesofgender-asymmetricconcurrencybeyondpolygny. Moreover,inourstudy,thelevelsofconcurrencyinthepopulation weredeterminedbysexualbehaviorparametersthatwerekept constantthroughoutdifferentscenarios.Asaconsequence,gender asymmetryledtoaslightlylowerlevelofoverallconcurrencyinthe populationcomparedtogendersymmetry.Therefore,resultsfrom ourstudyarenotdirectlycomparablewithReniersetal.(2015). However,consistentwithourﬁndings,Reniersetal.(2015)found thathigherlevelsofconcurrencyareassociatedwithhigherdisease prevalence.

Thereareofcoursemanyfactorsthatcanplayarolein deter-miningdiseasedynamics.Forexample,agemixing(whereyoung womentendtoformpartnershipswitholdermen)hasbeen pro-posedasanexplanationincreatingthedifferenceinHIVprevalence in men and women (Kellyet al., 2003; Leclerc-Madlala, 2008; Maughan-Brownetal.,2014).Butnotethattherearealso longitu-dinalcohortstudiesfindingnoincreaseinHIVacquisitionamong youngwomenwitholderpartnersinSouthAfrica(Jewkersetal., 2012;Harlingetal.,2014;Balkusetal.,2015).Inourmodel,partner acquisition(orseparation)doesnotdependontheageof individu-alsatall.Changesinbehavioroverthecourseoftheepidemicand heterogeneityinbehavioracrossindividualsofagivensexarealso notconsidered.Obviouslythesefactorsmayalsobeofinfluence, althoughnon-linearitiesinepidemicsystemsmakeitdifficultto

predicttheexactnatureoftheseinfluences(butseeReniersetal., 2015wheretheyfindthatbehaviorchangeamongHIV-positive womencanenlargethegenderasymmetryindiseaseprevalence). Ourmodelalsodidnotdifferentiatebetweenlong-and short-termpartnerships. Withrespecttopartnershipduration,wedid performsensitivityanalysisthatshowedthatendemicprevalence andR0increasewithincreasingpartnershipduration.Ina popu-lationwithonlyshorttermpartnerships, endemictransmission cannotbemaintained(withoutincreasingnumbersofpartners). However,wehypothesizethatinapopulationwitha combina-tionofshortandlongpartnerships,there couldbea synergistic effectofthetwo,withshort-termpartnershipsenabling transmis-sionduringprimaryinfectionandlong-termpartnershipsensuring a sufficientlyhighreproductionnumber.Futuremodeling work that includes different types of partnerships will allow analy-sesofgender-asymmetricconcurrencyeffectsonepidemiological parameterswithinthistypeofmorerealisticsystem.

Anothermodel aspectworth mentioningis that we didnot include anydisease-relatedmortality,whichwould shortenthe infectiousperiod.Asaconsequence,infectiousindividualswould remaininthesexuallyactivepopulationforashortertime,causing fewersecondarycasesthanifthereisnodisease-relatedmortality. Inclusionofthisphenomenoncouldthenleadtolowerlevelsof diseaseprevalence.However,assumingthatdisease-related mor-talityaffectsbothgendersinthesameway,webelievethatitwill notinﬂuencetherelationshipsdemonstratedinthispaper.

Inconclusion,genderasymmetryinthemaximumnumberof concurrentpartnershipscanaffecttheepidemicandendemicstates ofHIVtransmissiondynamics,withtheseeffectsbeingpartially mediatedthroughchangesinoverallconcurrencylevels.Our find-ingssuggestthatincreasedasymmetryisassociatedwithdecreased endemicprevalence,R0,andtheacute-phasecontributiontoR0, particularlyinthespecialcaseofpolygyny.Thisprovidespotential mechanisticexplanationsforassociationsbetweenpolygynyand keyepidemiologicalmeasuresofdiseasetransmission.Thefindings inthisstudyhelptoroundoutpriorempiricalandmodelingstudies byaddingconsiderationsofscenariosbetweenthetwoextremes ofsymmetricconcurrencyandpolygyny.Wealsofindthatgender imbalancesinconcurrencymaypartiallyexplainhigherobserved HIVprevalenceinfemalesvs.malesincontextswheresexual lifes-pansforfemalesareatleastaslongasformales.Takentogether,our findingsindicatethatimprovedempiricalunderstandingof male-vs-female concurrency patternsand sexuallifespans,combined withourmodelinginsightsabouttheeffectsofthesequantities onHIVtransmissiondynamics,couldimprove understandingof observedepidemiologicalpatternsacrossHIV-endemicsettings.

Funding

K.Y.L.issupportedbytheNetherlandsOrganisationfor Scien-tiﬁcResearch(NWO)[grantMozaïek017.009.082]andtheSwedish ResearchCouncil(VR)[grant2015-050153].K.A.P.issupportedby theNationalInstitutesofHealth(NIH)[grantKL2TR001109].

Acknowledgements

We thanktwoanonymousreviewersforcarefulreading and usefulsuggestionsthathelpedimprovedthearticle.

AppendixA. Supplementarydata

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