©CopyrightJASSS

NicholasGeard,JamesMMcCaw,AlanDorin,KevinBKorbandJodieMcVernon(2013)

SyntheticPopulationDynamics:AModelofHouseholdDemography

JournalofArtificialSocietiesandSocialSimulation 16(1)8<http://jasss.soc.surrey.ac.uk/16/1/8.html>

Received:05-Jun-2012Accepted:03-Oct-2012Published:31-Jan-2013

Abstract

Computer-simulatedsyntheticpopulationsareusedbyresearchersandpolicymakerstohelpunderstandandpredicttheaggregatebehaviouroflargenumbersofindividuals.Researchaimsincludeexplainingthestructuralanddynamiccharacteristicsofpopulations,andtheimplicationsofthesecharacteristicsfordynamicprocessessuchasthespreadofdisease,opinionsandsocialnorms.Policymakersplanningforthefutureeconomic,healthcareorinfrastructureneedsofapopulationwanttobeabletoevaluatethepossibleeffectsoftheirpolicies.Inbothcases,itisdesirablethatthestructureanddynamicbehaviourofsyntheticpopulationsbestatisticallycongruenttothatofrealpopulations.Here,wepresentaparsimoniousindividual-basedmodelforgeneratingsyntheticpopulationdynamicsthatfocusesontheeffectsthatdemographicchangehaveonthestructureandcompositionofhouseholds.

Keywords:Demography,SyntheticPopulations,HouseholdDynamics,Individual-BasedModels

Introduction

1.1 Thestructureanddynamicsofapopulationaretheoutcomeofmanyeventsoccurringtoitsindividualmembers.Understandingandpredictingtrendsinpopulationstructureanddynamicsischallenging,buthasthepotentialtobeofgreatutilitytoresearchers,plannersandpolicymakers.Syntheticpopulationscanbeusedasatestbedforavarietyofpurposes,fromsimulatingdiseaseoutbreaksandinterventions(Ajelli&Merler2009),toevaluatingtheimpactofland-useandtransportpolicies(Iaconoetal.2008;Spielauer2010).Toconductcorrespondingexperimentsinrealpopulationsmaybecostly,unethicalorotherwiseinfeasible.

1.2 Thekeyrequirementforausefulsyntheticpopulationmodelisthatitgeneratespopulationswhosestructureanddynamicsmatchthoseofrealpopulations(Gargiuloetal.2010).Itmightseemthatthemethodusedtosynthesiseamodelpopulationshouldberelativelyindependentofthepurposetowhichthatpopulationissubsequentlyput.However,nomodelisaperfectsimulacrumofrealityandamodel'sintendedpurposewillinfluencedecisionsmadeinitsdesignandconstruction.Pragmaticsdictatethat,foranyquestion,anappropriatemodelmustfocusonaccuratelyrepresentingthoseaspectsofapopulationmostrelevanttothatquestion,whileperhapstoleratingsomedeviationinthoseaspectsjudgedlessrelevant(Taperetal.2008).Assuch,anessentialstepinmodelbuildingisthespecificationofwhichdimensionsofapopulationareofgreatestimportance,andvalidationofmodelbehaviouragainstthesedimensions.

1.3 Inthispaper,weproposeaparsimoniousindividual-basedmodelofhouseholdcompositionanddynamics,developedforthepurposeofexploringinteractionbetweendemographicprocessesandpatternsofinfectionandimmunity.Tobegin,wedescribethespecificquestionsthathavemotivatedthedevelopmentofourmodel,andtherequirementsthatthesequestionsimpose.Wereviewhowthesequestionshavebeenexploredinavarietyofdifferentmodellingparadigms.Wethendescribeourownmodel,togetherwiththreecasestudiesthatdemonstrateitsabilitytogeneratepopulationsunderarangeofdemographicscenariosrelevanttoinfectiousdisease.Weconcludebydiscussingthestrengthsandlimitationsofourmodelanditspotentialdirectionsforitsfuturedevelopment.

Background

Motivation

2.1 Thissectionoutlinesthequestionsmotivatingthedevelopmentofourmodel,andtherequirementsthatthesequestionsimposeonthedesignofourmodel.Theprimarygoalofinfectiousdiseasemodellingistounderstandhowdiseasesspreadandhowtheycanbecontrolled,anendeavourinwhichmodellinghaslongplayedanimportantrole(Anderson&May1991).Akeycomponentofinfectiousdiseasemodelsistherepresentationofthepopulationthroughwhichadiseasespreads.Theagestructureofapopulationcaninfluencepatternsofsusceptibilitytodisease(Anderson&May1991).Thesocialstructureofapopulationgivesrisetocontactnetworksthataffecthowinfectionistransmittedthroughapopulation(Danonetal.2011).Understandingthedemographicprocessesthatunderlieobservedpopulations,andhowtheymightshapefuturepopulations,canhelpusexplaincurrentpatternsofdiseaseandpredicthowthesepatternswillevolveovertime,insightsthatwillaidinthedesignmoreeffectivestrategiesfordiseasecontrol(John1990).

2.2 Theassumptionsandapproximationsmadewhendesigningamodelconstrainthetypeofquestionsitcanbeusedtoaddress.Onecommonassumptionmadebyinfectiousdiseasemodelsisthatthecompositionandstructureofapopulationisstaticovertime.Fordiseaseoutbreaksthatoccuroveraperiodofweeksormonths,thisassumptionmaybeappropriate,aspopulationstructuretypicallychangesmuchmoreslowlythandiseasestate.However,whenmodellingendemicdiseasesthatmaypersistinapopulationforyearsordecades,thisassumptionbecomeslessappropriate,asdemographicprocessessuchasbirth,deathandtheformationanddissolutionofhouseholdswillhaveasignificanteffectonpopulationstructureandcomposition.Householdsareanorganisationalunitofparticularimportanceastheyareakeylocusofdiseasetransmission,particularlyforyounginfants(Jardineetal.2010).Householdsarealsoatargetfordiseasecontrolstrategiessuchascocoonvaccination,whichaimstoprovideaprotectiveenvironmentfornewbornsbyvaccinatingtheirparents(Coudevilleetal.2008).

2.3 Inadditiontothedynamicsarisingfromindividuallifeevents,theunderlyingstructureofglobalpopulationshasundergonedramaticchangesoverrecentcenturies.Thedemographictransitionmodelhasbeenproposedtodescribethelongtermchangestopopulationstructureassociatedwithindustrialisation,improvementsinpublichealth,education,andagriculture,andchangingsocialvalues(Kirk1996;Murphy2011).Asanexample,duringthetwentiethcentury,increasinglifeexpectancyandlowerfertilityratesinAustraliahaveincreasedtheproportionofpeopleagedover65from4%to14%ofthepopulation,whiletheproportionofpeopleagedunder15decreasedfrom35%to19%;thesizeoftheaveragehouseholdhasdecreasedfrom4.5to2.6peopleoverthesameperiod(Hayesetal.2010;AustralianBureauofStatistics2006b).Theeffectsofthistypeofpopulationchangeonpatternsofdiseasearenotyetwellunderstood.

2.4 Thequestionsthatwewouldliketousemodelstoaddressconcerntheeffectofdemographicchangesonobservedpatternsofinfectionandimmunity.Whateffectdoshrinkinghouseholdsizeshaveforthepatternsofinteractionrelevanttothespreadofchildhooddiseases?Howcanweassessthelong-termeffectivenessofvaccinationstrategiesinachangingpopulation?Whataretheimplicationsoftherapiddemographictransitionscurrentlyoccurringindevelopingcountries?Apopulationmodelcapableofaddressingsuchquestionsmustthereforecapture:

realisticpatternsofhouseholdcomposition,inparticularthehouseholdcontextofinfants;

thedynamiccharacteristicsofhouseholdsarisingfrompatternsofbirth,death,andhouseholdformationanddissolution;

thedifferencesinhouseholddynamicsacrossdifferentdemographicscenarios,correspondingtodevelopedanddevelopingcountries;and

thechangesthatoccurtohouseholddynamicsoverextendedperiodsoftimeunderchangingdemographicconditions.

2.5 Thispaperdescribesthedesignandvalidationofasyntheticpopulationmodelaimedatsatisfyingtheserequirements.Intheremainderofthissectionwebrieflyreviewhowpopulationstructure,householdsanddemographyhavebeenincorporatedintoexistinginfectiousdiseasemodels.

Previousapproaches

2.6 Methodsforgeneratingsyntheticpopulationsbasedonempiricaldatahavebeendevelopedinparallelinthefieldsofdemography,geographyandsocialscience.Despitesomerecentconvergence,thereisnosinglegeneralpurposeapproach(Mannionetal.2012;Birkin&Clarke2011).Thereareseveralrecentandcomprehensiveoverviewsofvariousmethodsforpopulationprojectionandmodelling(e.g.,Stillwell&Clarke2011;Wilson2011;Spielauer2010).Ratherthanattemptingtoreplicatetheseeffortsinthissection,wefocusspecificallyonapproachestakentorepresentingpopulationsininfectiousdiseasemodels.

Mathematicalmodels

2.7 Mathematicalmodelsofinfectiousdiseaserepresentpopulationsintermsoftheprevalenceofinfectionandimmunityatagivenpointintime.Apopulationisdividedinto'compartments'correspondingtoparticulardiseasestates(e.g.,susceptible,infectious,recovered).Themovementofindividualsbetweenthesecompartmentsismodelledbyspecifyingtransitionratesbetweenthem.Theentiresystemcanthenberepresentedasasetofordinarydifferentialequationsandsolved(analyticallyornumerically)topredictfuturepatternsofdisease(Hethcote2000;Grassly&Fraser2008).

2.8 Demographycanbeincludedinmathematicalmodelsbyfurthersubdividingcompartmentsaccordingtoageorsex,andbyintroducingadditionaltermsforthebirthanddeathofindividuals(Anderson&May1991).Mathematicalmodelsthatincorporatehouseholdstructurehavealsobeenproposed,adoptingtheoreticalorempiricaldistributionsforhouseholdsize(e.g.,Balletal.1997;Ball&Neal2002;Beckeretal.2005);however,thesemodelstypicallyassumestaticpopulations.Glassetal.(2011)doproposeadynamichouseholdmodel,howeverthedistributionofhouseholdsizesisheldfixedandthemodeldoesnotincludeage

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structure.Thesetrade-offsillustrateafundamentallimitationofmathematicalapproachestocapturingcomplexpopulationstructure.Theexplosionoftermsthatresultsfromsimultaneouslyincorporatingage,householdproperties,diseasestate,andotherfactorsofinterestresultsinmodelsthatareanalyticallyintractable.

Individual-basedmodels

2.9 Toovercomethelimitationsofmathematicalmodels,individual-basedmodels(IBMs)1havealsobeenappliedtotheproblemofunderstandingthedynamicsofinfectiousdisease.Byexplicitlymodelingeachindividual,togetherwiththeirage,location,diseasestatusandotherrelevantproperties,IBMsofferagreatdealofflexibilityinrepresentingheterogeneouspopulationstructures(Eubanketal.2004;Elvebacketal.1976;Fergusonetal.2005;Longinietal.2005).Mostofthesemodelshavebeenaimedatcapturingthedynamicsofasingleoutbreak.Hence,theirprimaryfocushasbeenonthedynamicsofindividualactivityoverthecourseofatypicalday,theresultingcontactnetworks,andthepatternsofdiseasetransmissionthatthesenetworkssupport.Thecompositionandstructureofthepopulationsthemselvesareusuallystatic.Comparedtotheproliferationofstaticpopulationmodels,fewerIBMshaveconsideredtherelationshipbetweenthelongtermdynamicsofpopulationstructureandthespreadofinfectiousdisease.

2.10 Thosemodelsthathaveconsideredlongertimeframeshaveadoptedheuristicapproachestocapturinghouseholdstructure.Ajelli&Merler(2009)andGuzzettaetal.(2011)investigatelongtermpatternsofhepatitisAandtuberculosisrespectively,usingsimulatedpopulationsthatincludebirth,deathandhouseholdformationprocesses.TheygenerateinitialhouseholddistributionsfromItaliancensusandsurveydatausingaMonteCarlosamplingmethod.Populationdemographicsareupdatedannually,usingempiricalmortalityratestodetermineindividualdeaths,andallocatingbirthsaccordingtohouseholdsizeandparentalage.Silhol&Boëlle(2011)modeltheoccurrenceofvaricellausingapopulationmodelthatincludesrealisticageandhouseholdstructureanddynamics.Theirapproachtoproducinganappropriatedistributionofhouseholdsizesistogenerateeachhouseholdasacompletedunitincludingallchildrenthatwilleventuallybebornthere.Thisfinalhouseholdisthen'rewound',resultingintheagesofyoungerchildrenbecomingnegative,andbirthsoccuroverthecourseofasimulationwhenachild'sagebecomespositive.Anadvantageofthisapproachisthathouseholdstructureisguaranteedtoaccuratelyreflectthatoftherealpopulation;however,itdoesnotgeneraliseinstraightforwardfashionbeyondavailabledata.

2.11 WhilewehavedrawnuponaspectsofthesepreviousmodelsinthedesignofthemodeldescribedinSection3,wenotethattheydonotfulfilourrequirementofcapturingthechangesinhouseholddynamicsarisingfromdemographicshifts.

Microsimulationmodels

2.12 Microsimulationisanapproachtoexperimentingwithvirtualsocietiesbymodellingtheactionsofindividualsinapopulationand(potentially)theinteractionsbetweenthemthathasitsoriginsineconomics(Orcutt1957).Thesemodelsareusedtoexplorethepotentialimpactsofpolicydecisions,wherethechoicesmadebyindividualsmaydependonpoliciesinanonlinearandcontext-dependentfashion.Dynamicmicrosimulationmodelsaddinatemporalelement,incorporatingdemographicprocessesandindividuallifecourses(Mannionetal.2012;Birkin&Clarke2011;Spielauer2010).MicrosimulationmodelsandIBMssharemanysimilarities.Theirdifferencesariseprimarilyfromhavingbeendevelopedandappliedtodifferentproblemsbydifferentresearchcommunities.Broadlyspeaking,microsimulationmodelshavebeenmoreconcernedwithempiricaldataandpredictiveaccuracy,whileIBMstendtobemoretheoreticallyoriented.Whilemanymicrosimulationmodelsaredesignedforthepurposeofforecastingfuturepopulationtrendsandpolicyinteractions,theyhavealsobeenappliedtothechallengeofelucidatinghistoricaldemographicpatterns(e.g.,Hammel2005;Murphy2011).

2.13 Giventherequirementofdiseasemodelsfordemographicallyplausiblepopulations,itseemsatfirstsurprisingthat,sofarasweareaware,onlyoneinfectiousdiseasemodel(EpiSimS,Eubanketal.2004)hasexplicitlymadeuseofapre-existingmicrosimulationmodel(TRANSIMS,Barrettetal.2000),albeitonedevelopedbythesameresearchgroup.Possiblereasonsforthisinclude:availabilityofsoftware—manymicrosimulationmodelsarenotreleasedpublicly;tightcouplingbetweenmodeldesignanddatarequirements,meaningthatifthedatarequiredtoinitialiseapopulationisnotpubliclyavailable,ornotavailableinthecorrectformforthepopulationofinterest,themodelmaynotbesuitable;limitedextensibility,makingitdifficulttoextendanexistingpopulationmodeltoincludediseasedynamics;andmodeldevelopmenteffort—dynamicmicrosimulationmodelsarenotoriouslytime-consumingandexpensivetobuild(Harding2007).

Summary

2.14 Asignificantimpedimenttomodellingthedemographicdynamicsofpopulationsisthelimitedavailabilityofdatawithwhichtoparameterisemodels.Hypothetically,ifsufficientdatawereavailabletoestimateaprobabilityforeachpossibletransitionthatanindividualorhouseholdmightundergo,usingademographicmicrosimulationmodeltoprojectapopulationforwardwouldbeatrivialexercise.Inpractice,thenumberofpossibletransitionsexplodescombinatoriallywithattributesofinterest,andavailabledatawillalwaysbeinsufficientforthisapproachtobepracticable.

2.15 Attheotherextreme,anidealindividual-basedmodelmightsimulatetheunderlyingcognitiveandsocialbehaviourofindividualsinapopulation,perhapsusingsomeinternalrepresentationofindividualutilitytogetherwithaforecastingmodelthatpredictsanindividual'sbehaviouronthebasisofitshistoryandcurrentcontext.Thisapproachhasthepotentialtofreedemographicmodelsfromtherelianceonthemassivequantitiesofdatarequiredbytransition-basedmicrosimulationmodels(Silvermanetal.2011).However,muchrestsuponthemodelusedtorepresentindividualdecisionmaking.Successfulindividual-basedmodelsofdemographicprocesseshavethusfartypicallybeenconstrainedtospecificaspectsofpopulations.Forexample,Billarietal.(2007)useanIBMofmarriagebasedonsocialinteractiontoexploretheemergenceofobservedtrendsinmarriageage.Extendingsuchanapproachtoallfacetsofhumanbehaviourremainsanopenchallenge.

2.16 Recently,therehasbeenaconvergenceoftechniquesfrommicrosimulationandindividual-basedmodelling,as(some)IBMsencounteraneedformoreempiricallyplausiblepopulations,andmicrosimulationmodulesbegintoincludemorebehaviouralaspects(Wu&Birkin2012).Theresultinghybridmodelsuseavailabledemographicdatatocalibratedistributionsofindividualattributes,andstochasticbehaviouralmodelsofindividualdecisionstogenerateparticularevents.

Ourmodel

3.1 Thebasicunitofdescriptioninourmodel2istheindividual.Individualsarecharacterisedbytheirage,sexandthehouseholdtowhichtheycurrentlybelong.Apopulationconsistsofthesetofcurrentlyaliveindividuals,structuredbyanetworkofinterpersonaltiesthatmapscouple,parent–childandhouseholdco-membershiprelations.Thefollowingsectionsdescribehowapopulationisinitialised,howitisupdatedovertime,andhowitcanbeparameterisedusingavailabledataonrealpopulations.

Initialisation

3.2 Aninitial'bootstrap'populationiscreatedbyrandomlygeneratingindividualswithagesdrawnfromaspecifiedagedistribution.Theseindividualsareassignedtohouseholdsatrandomaccordingtoaspecifiedhouseholdsizedistribution.Householdsofsizeoneortwoareassignedoneortwoadultsrespectively,whilehouseholdsofsizethreeorgreaterareassignedtwoadultsandoneormorechildren.Thisinitialpopulationstructurewilldivergeinseveralwaysfromarealpopulation;forexample,constraintsonbirthintervalandinter-generationalagedifferencewillnotberespected.Ifmoredetaileddataonhouseholdstructureisavailable,morecomplexmethods(Gargiuloetal.2010;Ajelli&Merler2009)couldbeusedtospecifyamorerealisticinitialpopulationstructure.Alternatively,theapproachweadopthereistoupdatethestateofthepopulationuntilsuchtimeasallofthesebootstrapindividualsandhouseholdshavebeenreplaced(i.e.,foratleast100years),atwhichstageinternalconstraintsonpopulationstructurewillberespected.

Updating

3.3 Thestateofapopulationisupdatedindiscretetimesteps,whereeachstepcorrespondstoaspecifiednumberofdays.Allparameterscontrollingtheoccurrenceofdemographiceventsarespecifiedasannualprobabilitiesandrates,thereforethesearescaledappropriately.

3.4 Fivetypesofdemographiceventscanoccurtoanindividual:

Death:Ageandsexspecificmortalityratesareusedtodetermineanindividual'sprobabilityofdeathduringeachtimeunit(e.g.,Figure1(a)).Ifadeathresultsinahouseholdcontainingonlychildrenthenthehouseholdisdissolved.Anyadultchildren(i.e.,aged18yearsorover)leavehomeandcreatenewsingle-personhouseholds,whileanychildrenunder18yearsarerandomlyrelocated(fostered)tootherhouseholdscontainingatleastonechild.

Birth:Age(andoptionally,parity3)specificfertilityratesareusedtodeterminetheprobabilitythatabirthoccurstoagivenindividual.Asfertilityisnotanindependentprocess,ratesaretransformedintorelativeprobabilitiesusedtodesignatethesubsetofthepopulationfromwhichtheactualmotheristhenchosen(e.g.,Figure1(b)).Upongivingbirth,amotherisexcludedfrombeingacandidateforfuturebirthsforanumberofdaysdrawnfromatruncatednormaldistribution,withaminimumdurationof270days.

Coupleformation:Anindividualwithinagivenagerangewhoiscurrentlysinglehasafixedprobabilitypertimeunitofforminganewcouple.Thenewpartnerischosenfromthepoolofindividualswhoarecurrentlysingle,oftheoppositesex,andwhoseagediffersbyanormallydistributedvalue.Thenewlycoupledindividualsmoveintoanewhousehold,togetherwithanydependents(e.g.,childrenfrompreviouscouples).

Coupledissolution:Anycurrentlycoupledindividualwithinagivenagerangehasafixedprobabilitypertimeunitofdissolvingthecouple.Upondissolution,onememberofthecouplemovesintoanewsinglepersonhousehold,whiletheotherremainsintheoriginalhouseholdtogetherwithanychildren.

Leavinghome:Individualsleavehomeautomaticallywhentheyformacouple,otherwiseanyindividualaboveaspecifiedageleaveshomewithafixedprobabilitypertimeunitandformsanewsinglepersonhousehold.

3.5 Themodelcanbeusedtosimulatepopulationsoffixedsize(i.e.,withreplacementlevelfertility),orpopulationsthatareincreasingordecreasinginsize.Changeinpopulationsizecanresultfromanimbalancebetweenthenumberofbirthsanddeaths,orbetweenthenumberofimmigrantsandemigrants.Inourmodel,growthduetoexcessbirthsisimplementedbytriggeringadditionalbirtheventsateachtimestep.Growthduetoimmigrationisimplementedbyintroducingnewindividualsandhouseholdsintothepopulationaccordingtoaspecifieddistributionofageandhouseholdstructure.Specifyingtheageandfamilycompositionofmigrantsisacomplexissue,dependingasitdoesonthecircumstancesofmigration,countryoforigin,andotherfactors.Thedefaultassumptionmadebythemodelisthatthemigrantpopulationisdemographicallysimilartothetargetpopulation.Calibratingmigrationtoreflectthedemographiccharacteristicsofspecificeventswouldberelativelystraightforwardprovidedthatsuitabledatawereavailable.

Parameterisation

3.6 TheinputparametersthatmustbespecifiedtoinitialiseandupdateapopulationaresummarisedinTable1.Ageneralprinciplefollowedindesigningthemodelwastoparameterisethemodelintermsofeventsoccurringtoindividuals,andtoallowthesizesandtypesofhouseholdsinapopulationtovaryasaconsequenceoftheseindividual-levelevents.Forexample,individualbirthswereallocatedtoparentsusingage-andparity-specificfertilityrates,butnottohouseholdsofaspecificsizeorcomposition;thus,incontrasttoexistingmodels(Silhol&Boëlle2011;Ajelli&Merler2009),householdsizedistributionswerenot

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controlled,butratherresultfromtheinteractionbetweenpatternsofhouseholdformationanddissolutionarisingfromindividual-levelevents.Furthercalibrationagainsthousehold-leveldatawouldofcoursebepossible,butreservingavailablehousehold-leveldataforvalidationpurposesenablesustobetterevaluatetheeffectivenessofoursimpleevent-basedmodel.

Table1:Modelparameters

Parameter DescriptionGeneral: Initialpopulationsize Numberofindividualsinpopulationatstartofsimulation.Populationgrowthrate Annualrateofchangeinpopulationsizeduetonaturalincrease.Immigrationrate Annualrateofchangeinpopulationsizeduetoimmigration.Death: Age-andsex-specificmortalityprobabilities

Annualprobabilitiesofdeathbysexandyearofage.

Birth: Age-specificrelativefertilityprobabilities

Relativeprobabilities,giventhebirthofachild,thatthemotherisofaspecifiedage.

Parity-specificrelativefertilityprobabilities

(optional)Relativeprobability,giventhebirthofachild,thatitistoawomanwithaspecifiednumberofpreviouschildren.

Birthgap(meanandSD) Parametersgoverningtheminimuminter-birthinterval.Coupleformation: Coupleformationparameters Agerangethatacurrentlysingleindividualiseligibletoformacouple,andannualprobabilitythatthiswilloccur.Partneragedifference(meanandSD)

Parametersgoverningthesex-dependentagedifferencebetweenpartnersduringcoupleformation.

Coupledissolution: Coupledissolutionparameters Agerangethatacurrentlycoupledindividualiseligibledissolvethatcouple,andannualprobabilitythatthiswilloccur.Leavinghome:

Leavinghomeparameters Minimumageatwhichanindividualcurrentlylivingwithaparent/guardianwillformanewsingle-personhousehold,andannualprobabilitythatthiswilloccur.

3.7 Mortalitydataisoftenavailableintheformoflifetables,ademographictoolthatshows,foreachage,theprobabilityofanindividualofthatagedyinginthenextyear.Fertilityratesbyagearealsoavailableformostcountries.Thesefertilityratescanbeusedintwodifferentways.Oneoptionistousethemtodeterminetheprobabilitythatawomanofaparticularagewillgivebirthinagivenyear.Thenumberofbirthsthatoccurinayearwillthenemergefromtheaggregateapplicationoftheseprobabilitiesacrossthefemalepopulation.Thealternative(whichweadoptedhere)istospecifythenumberofbirthsthatoccurinayear,andthenuseage-specificfertilityratestodeterminetheagesofthewomentowhichthosebirthsoccur.Effectively,ratherthanasking“whatistheprobabilityofawomanofagexgivingbirththisyear?”,weask“giventhatabirthoccurred,whatistheprobabilitythatthemotherwasagedx?”Anadvantageofthelatterapproachwasthatitenabledustocontrolthetotalsizeofapopulation.Forexample,populationsizecouldbeheldconstantbytriggeringsufficientbirthseachyeartoreplacetheindividualsdyinginthatyear,orconstrainedtogrow(orshrink)atagivenrate.

3.8 Coupleformationanddissolution,andthedepartureofchildrenfromtheirparents'householdareallgovernedbysimplemodelsthatassumeaconstantprobabilityofaparticulareventoccurringperunitoftime.Thus,theseprobabilitiesareindependentofage,durationofrelationship,previousmaritalstatus,andotherpotentiallyrelevantfactors.Thisapproachisconsiderablysimplerthanthattakenbydynamicmicrosimulationmodels.Thesemorecomplexmodelsrelyheavilyontheavailabilityofappropriatestatisticaldatainordertoparameterisetheeffectsthatdiversefactorshaveon,forexample,marriageanddivorce.Ourprimarymotivationforadoptingthissimplerapproachwastobeabletodealwiththeavailabilityofdataacrossavarietyofdifferentcountriesandhistorictimeperiods.

3.9 Thespecificparametervalues,datasources,andestimationprocessesusedtospecifytheprobabilityofindividualeventsaredetailedatthebeginningofeachcasestudyinthefollowingsection.

Results

4.1 Asdescribedabove,theprimarymotivationforthedevelopmentofourmodelwastouseitasademographicallyplausibletestbedinwhichtoexploreinteractionsbetweenpopulationdynamicsandpatternsofdisease.Inturn,thisaimimposedrequirementsonmodelbehaviour,asdescribedinSection2.1.Inthissection,weusethreedemographicscenariostoevaluatetheextenttowhichourmodelmeetstheserequirements.Thefirsttwoscenariosconsiderpopulationscorrespondingtothoseofadevelopedandadevelopingcountry.Thethirdscenarioconsidersapopulationundergoingademographictransitionfromhightolowfertility.

4.2 Quantitativecomparisonbetweenmodeloutputandempiricaldataisanimportantcomponentofvalidation.However,suitabledataforcomparisonisnotalwaysavailable,andwealsorelyoncomparisonofmorequalitativeaspectsofmodelbehaviourtobuildconfidenceinthevalidityofamodel(Korbetal.2013;Grimmetal.2005).

Casestudy1:Adevelopedcountrypopulation

4.3 OurfirstcasestudyexploredtheabilityofourmodeltocapturepatternsofhouseholdcompositionanddynamicscomparabletothoseexhibitedbyAustralianpopulationatthebeginningofthe21stcentury.Realpopulationsareever-changing:fertilityratesthatdeviatefromreplacementandimmigrationacttoincreaseordecreasethesizeofapopulation,whilechangesinlongevityreshapeitsagestructure.Forthiscasestudy,weexploreasimplerscenarioinwhichpopulationsize,mortalityandfertilityratesandothereventprobabilitiesarefixed,andthereisnomigration.Theresultingsteadystatepopulationhasaconstantsizeand,overtime,approachesastableagestructure.Whilesuchascenarioisobviouslyofonlylimiteduseforforecastingpurposes,itdoeshavethesignificantbenefitofprovidingastablebackgroundfortheoreticalinvestigationofinteractionsbetweenthedynamicsofdemographicandepidemicprocesses(e.g.,asdescribedinGlassetal.2011).Inparticular,itallowsustoevaluatethemodelfromtheperspectiveofthefirsttworequirementsdescribedinSection2.1.Amorerealisticscenariowithtime-varyingdemographicparametersisexploredinCaseStudy3.

4.4 TheinitialpopulationwasparameterisedusingrecentAustraliandataonagestructureandhouseholds(deVaus2004;AustralianBureauofStatistics2006b).ValuesforotherparameterwereestimatedonthebasisofcensusdatareportedindeVaus(2004).Coupleformationparameterswerecalibratedagainstdataonthepercentageofindividualsbyagewhohadnevermarriednorcohabited.Coupledissolutionparameterswerecalibratedagainstdataonpercentageofmarriagessurvivingbyduration.Leavinghomeparameterswerecalibratedagainstdataonthenumberofindividualsbyagelivingathome,accountingforthosewhohadlefttomarryorcohabit.Thepopulationwassimulatedfortwohundredyears,toensurethatasteadystatehadbeenachieved,beforecompositionanddynamicswereanalysed.

Table1:CaseStudy1:Modelparameters

Parameter Value/DatasourceInitialpopulationsize 20,000Populationgrowthrate 0%Immigrationrate 0%Mortalityprobabilities Australia,2006,byyearofage(AustralianBureauofStatistics2007)(seeFigure1(a))Fertilityprobabilities Australia,2006,byyearofage(AustralianBureauofStatistics2006a)(seeFigure1(b))Birthgap mean:365days;SD:90daysCoupleformationparameters agerange:18–60years;annualprobability:7.5%Partneragedifference mean:2years;SD:2years)Coupledissolutionparameters agerange:18–60;annualprobability:1.5%Leavinghomeparameters minimumage:18;annualprobability:0.8%

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Figure1:AustraliancensusdatausedtoparameterisethemodelinCaseStudy1.(a)Mortalityratesbyageandsex;(b)Relativeprobabilityofabirthoccurringtoawomanbyage.DatasourcesasinTable2.

Agedistribution

4.5 Figure2(a)showsthesimulatedagestructureofthefinalpopulationfortenindependentruns,togetherwiththestartingagedistribution(basedon2006censusdata).Intheabsenceofmigration,theagestructureofapopulationisdeterminedsolelybybirthsanddeaths.Whenage-specificratesoffertilityandmortalityremainconstantoveralongperiodoftime,andthereisnomigration,theresultingpopulationiscalled'stable'.Stablepopulationsarecharacterisedbyconstantagestructureandfixedgrowthrates.Astablepopulationwithagrowthrateofzeroiscalled'stationary'.Stableandstationarypopulationstructuresarelargelytheoreticalconstructsthatarerarelyobservedinrealpopulations.Overthelastcentury,lifeexpectancyinAustraliahasincreasedsteadily,whilefertilityandmigrationhavefluctuated.OneinterpretationofthefinalagestructuresshowninFigure2(a)isthattheyrepresentwhatAustralia'spopulationcouldlooklikeafteracenturywithreplacement-levelfertility,nomigration,andconstantage-specificbirthanddeathrates.

Figure2:(a)Thefinalagedistributionsoftensimulatedpopulations(grey),comparedtotheempiricalagedistributionusedtoinitialisethepopulation(black).(b)Thefinalhouseholdsizedistribution(white),averagedoverthetensimulatedpopulations(errorbarsindicatestandarddeviation),comparedtoempiricaldata(black).

Householdsizedistribution

4.6 Householdsizesandcompositionswereallowedtovaryinresponsetotheindividuallevelprocessesofleavinghome,forminganddissolvingcouples,birthanddeath.Thefinalhouseholdsizedistributionafter200years(meanandstandarddeviationacross10independentruns)isshowninFigure2(b).Someofthevariationbetweensimulatedandempiricalagestructurecanalsobeobservedinthefinalhouseholdsizedistribution.Olderindividuals,whoareover-representedinthesimulatedpopulation,aremorelikelytoliveinhouseholdsofsizeoneandtwo,whicharealsoover-representedinthesimulatedpopulation.

4.7 Asindividualhouseholdsareformedanddissolved,thenumberofhouseholdsofaparticularsizeincreasesordecreases;however,becauseweareholdingthedemographicratesfixed,theshapeofthehouseholdsizedistributionremainsrelativelyconstant,withsomestochasticfluctuation(Figure3(a)).

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Figure3:Theevolutionof(a)householdsizedistributionand(b)familyhouseholdtypedistributionover100yearsunderfixeddemographicconditions,forasinglesimulationrun.

4.8 Totesthowthefinalhouseholdsizedistributiondependeduponthethehouseholdsizedistributionthatwasusedtocreatetheinitialbootstrappopulation(asdescribedinSection3.3)wealsoransimulationsinwhichthepopulationwasinitialisedwithuniformlydistributedhouseholdsizes.Byyear200ofthesesimulations,thedistributionofhouseholdsizesonceagainapproximatedtheempiricaldistribution.Thus,weareconfidentthatthehouseholdsizedistributionisanemergentpropertyoftheunderlyingindividual-leveldemographicprocesses,ratherthanasimpleconsequenceoftheinitialconditions.

Householdtypedistribution

4.9 Beyondlookingsimplyathouseholdsize,wealsoinvestigatedthetypesofhouseholdthatindividualstendedtobelongtoatdifferentstagesoftheirlives.Figure4showstheproportionofindividualsthatlivinginvarioustypesofhouseholdsituation(couplewithchildren,couplewithoutchildren,singleparent,loneperson,andlivingwithparents),brokendownbyagecategory(meanandstandarddeviationacross10independentruns).ThesimulationmodelproducesaplausiblerepresentationofhouseholdtypeprevalenceobservedintheAustralianpopulation.Whilethereissomevariation,generaltrendsacrossagegroupsarewellcaptured;forexample,theproportionofindividualslivinginhouseholdswithoutchildreninitiallyincreasesasindividualsformcouples,thendecreasesasthesecoupleshavechildren,beforefinallyincreasingasthesechildrenleavethefamilyhousehold.

Figure4:Typeofhouseholdbyagegroupattheendofasimulationrun(white;errorbarsindicatestandarddeviation),comparedwithempiricaldata(deVaus2004)(black).

4.10 Figure3(b)indicatesthat,aswithhouseholdsizedistributions,theproportionoffamilyhouseholdsfluctuatesstochasticallyoverthetimeperiodreported,butisgenerallystable.

Changesinhouseholdstructure

4.11 Thedistributionofhouseholdsizes(Section4.1.2)andtheprevalenceofhouseholdtypes(Section4.1.3)bothremainstableoverthecourseofaparticularsimulationrun.However,thisstabilityhidesthefactthatthetypeofhouseholdthatanyoneindividualbelongstochangesmultipletimesoverthecourseoftheirlife(seeSection4.1.5).Anindividual'shouseholdtypecanchangewhentheyleavehome,whentheyformordissolveacouple,whentheirfirstchildisbornortheirlastchildleaveshome,orwhenanothermemberoftheirhouseholddies.Whilelongitudinaldataisnottypicallycapturedinacensus,theHousehold,IncomeandLabourDynamicsinAustralia(HILDA)Survey(Wilkinsetal.2011)reportsstatisticsontheproportionofindividualswhochangehouseholdtypeoverafiveyearperiod(Table3).Collatingoutputfrommultiplesimulationrunsrevealsasimilarpatternoftransitionsbetweenhouseholdtypes(Table4).

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Table3:Changesinhouseholdstructure,2003to2008(HILDA)(%)

Coupleonly Couplewithchildren

Singlewithchildren

Singleperson

Coupleonly 75.3 14.0 0.8 8.4Couplewithchildren 10.1 76.8 6.0 4.9Singlewithchildren 6.4 18.0 59.5 13.0Singleperson 11.1 8.8 3.5 74.9

Table4:Changesinhouseholdstructure,meanandstandarddeviationover10runs(%)

Coupleonly Couplewithchildren

Singlewithchildren

Singleperson

Coupleonly 78.2(0.6) 11.3(0.4) 0.6(0.1) 9.8(0.3)Couplewithchildren 14.8(1.3) 77.4(1.8) 5.2(0.8) 2.6(0.2)Singlewithchildren 9.5(0.6) 41.6(3.0) 42.6(2.4) 6.3(0.9)Singleperson 8.8(0.7) 11.6(0.9) 1.2(0.3) 78.3(1.0)

4.12 Giventhatmodelbehaviourwasnotexplicitlycalibratedagainstthesetransitionrates,thereisaremarkablelevelofagreementbetweenthedynamicsofrealandsimulatedhouseholds.Theprimarypointofdisagreementisthetransitionfrom'singlewithchildren'to'couplewithchildren',whichisover-representedinsimulatedpopulations,suggestingapossibledirectionforfuturerefinement.

Familylifecycle

4.13 Thefamilylifecycleisademographicpatternthatcapturesthelifeexperienceofalargeproportionofthedevelopedworld'spast,current,andmostprobablyfuturepopulation:“mostpersonswillgrowup,establishfamilies,rearandlaunchtheirchildren,experienceanemptynestperiodandeventuallyreachtheendoftheirlife.”(Glick1989,p.123).Figure5illustratestwodifferentviewsofthefamilylifecyclefromtheperspectiveofanindividual,showingtheagedistributionatwhichvarioussignificanteventsoccur.Exploringthesequencesofeventsthatconstituteasimulatedindividual'sfamilylifecycleatbothanindividualandaggregatelevelprovidesastraightforwardwaytoverifythattheselifecoursesappearplausible.

Figure5:Theagedistributionof(a)majorlifeevents(marriage,birthoffirstchildanddeath)and(b)birthoffirstandsubsequentchildrenoverthetotalsamplepopulation.

4.14 TheapproachtakeninFigure5toexploringthedistributionofsignificanteventsoveranindividual'slifespancanalsobeappliedtohouseholds.Anissuethatarisesishowbesttodefineahousehold's'age'.Theapproachtakenhereistodefinethecreationofahouseholdasoccurringwheneither(a)anindividualleavesherparents'householdtoformanewsinglepersonhousehold;(b)twoindividualsleavetheirparents'householdstoformanewcouplehousehold;or(c)anindividualleaveshisspouse'shouseholdafterdivorceandformsanewsinglepersonhousehold.Otherinter-householdmovementsdonotresultinthecreationofanewhousehold;forexample,theformationofacouplewhereatleastoneindividualcurrentlyresidesinasinglepersonhousehold.Inthesecases,onehouseholdmergeswithanother,butnonewhousehold(i.e.,ofagezero)iscreated.Ahouseholdisdissolvedwhenthelastindividualinthathouseholddiesorleaves;forexample,iftwoindividuals,eachlivinginasinglepersonhousehold,formacouple,thenoneofthesehouseholdswillcontinue,andtheotherwillbedissolved.

4.15 Assigninghouseholdsanageasdescribedabovethenallowsustolookatthedistributionofparticularindividualorhouseholdleveleventsbyhouseholdage.Figure6representsthefamilylife-cycleintermsofhouseholdsize:householdstypicallybegintheirlifewithoneortwoindividuals(e.g.,anewcouplewhohavejustlefttheirownparents'households).Householdsizeincreasessteadily,peakingbetweentwentyandtwenty-fiveyears(whenourexamplecoupleareintheirfortiesandhavehadallthechildrenthattheywillhave).Thereafter,householdsizedeclinesbacktoameanoftwobyaroundforty-fiveyears(withourexamplecouplebeing'emptynesters'intheirsixties).Theperiodofpeaksizerepresentsthestagewhenmostorallofthechildrenthatwillbebornintothathouseholdhavebeenborn,butnoneorfewofthemhaveyetlefthome.

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Figure6:Householdsizebyage.Boldlineindicatesmeanhouseholdsize.Boxplotsshowmediansize(redbars),inter-quartilerange(box),minimum/maximumwithin1.5oftheinter-quartilerange(whiskers)andoutliers.Thesamplepopulationofhouseholdsusedwasthesetof

householdsthatwerebothcreatedanddissolvedduringthetimeframeofthesimulationrun;thatis,householdscreatedatthebeginningandhouseholdsstillexistingattheendoftherunwereexcluded.

Householdcomposition

4.16 Wesimplifytheissueofdescribinghouseholdcompositionbydividingindividualsintodiscreteagecategoriesrelevantfordiseasetransmission:infantsunderfiveyears;school-childrenbetweenfiveandseventeenyears;adultsfromeighteentosixty-fouryears;andelderlyindividualssixty-fiveyearsandover.Moreordifferentcategoriescouldbechosendependingupontheagegroupsthatareofmostinterestinaparticularapplicationofthemodel.Figures7(a)and7(b)showthedistributionofindividualagesbyhouseholdageandhouseholdsizerespectivelyinasinglepopulationatagivenpointintime.Thesefiguresmatchobservedtrendsofthehouseholdlocationsofdifferenttypesofindividuals(deVaus2004).Elderlyindividualstendtobefoundprimarilyinsmallerandolderhouseholds.Childrenarefoundprimarilyinhouseholdsofintermediateageandintermediate-to-largesize.

Figure7:Agedistributionofindividualsgiven(a)householdageand(b)householdsize.Individualsaregroupedaccordingtoagecategories:<5years;5-17years;18-64years; 65years.

4.17 Figure7(b)depictshowindividualsofvariousagesareallocatedacrosshouseholdsofdifferentsizes,butprovidesnoinformationontheco-occurrenceofindividualsfromdifferentagegroupsinhouseholds.Figure8showsanovelapproachtovisualisingthetypesofhouseholdsthatappearinapopulationandtheirrelativefrequency.Asabove,thisfigurerepresentsasnapshotofapopulationatagivenpointintime,ratherthananaggregationacrosstime.Eachclusterofncirclesrepresentsauniquetypeofhouseholdcontainingnindividuals,withthecountofhouseholdsofthattypeappendedbelow.Eachcircleiscolouredaccordingtotheagecategoryoftheindividualinthathouseholdtype,andcirclesizereflectshouseholdtypefrequency,withlargercirclesindicatingmorecommonhouseholdtypes.4Householdtypesarefurtherarrangedbyhouseholdsize(increasinglefttoright)andfrequency(morefrequenthouseholdtypesappearatthetop).TherepresentationshowninFigure8providesaconvenientoverviewofthediversehouseholdstructuresthatariseinsimulatedpopulations.Atthesametime,itisstraightforwardtogainanimpressionofwhichhouseholdtypesaremostlikelytocontaininfantsandtheirfrequencyinthepopulation.

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Figure8:Achartofhouseholdtypesappearinginapopulationatagivenpointintime.Eachcirclerepresentsanindividual(colouredbyagecategory).Eachclusterofcirclesrepresentsahouseholdtype.Thefrequencywithwhichaparticularhouseholdtypeoccursinapopulationiswrittenbeneath,andrepresentedvisuallybysize(largercircles

indicatemorecommonhouseholdtypes).

Casestudy2:Adevelopingcountrypopulation

4.18 Thethirdrequirementofourmodelwasthatitbeabletoproduceplausiblepatternsofhouseholddynamicsacrossavarietyofdemographicscenarios.Inparticular,weareinterestedinexploringthedynamicsofinfectionandimmunityinadevelopingcountrysetting.Asdiscussedabove,thedemographictransitionmodeldescribesacountry'stransitionfromaphaseinwhichbothfertilityandmortalityarehigh,throughaphasewheremortalityfalls,butfertilityremainshigh,toaphasewherebothfertilityandmortalityhavefallen.Duringthefirstandlastphases,fertilityandmortalityarebalanced,andpopulationsizeisstable.Bycontrast,inthemiddlephase,fertilityexceedsmortality,andpopulationsizeincreases.WechoseZambiaasacountryrepresentativeofthismiddlephaseofdemographictransition:bothmortalityandfertilityratesarehigherthanAustralia's,andpopulationgrowthofapproximately2.5%perannumisalmostentirelyduetonaturalincrease(ratherthanimmigration).ComparedtoAustralia,lessdataisavailableontheZambianpopulation,restrictingtheamountofvalidationpossible.Here,wefocusonageandhouseholdsizedistribution.

4.19 Wesimulatedaninitialpopulationof500individuals,growingatanannualrateof2.5%(givingafinalpopulationsizeofapproximately70,000after200years).Age-specificmortalityrateswereobtainedfrom(Lopezetal.2000),andage-specificfertilityrateswereobtainedfromtheUNdatawebsite(data.un.org).Insufficientresourceswereavailabletoestimateprecisevaluesforcoupleformationanddissolutionparameters.WethereforeestimatedvaluesbasedoncomparisonwithvaluesusedinCaseStudy1.Namely,wespecifiedanearlierageforcoupleeligibility,alowerrateofdivorceandalowerrateofleavinghomeasasingleindividual(RepublicofZambiaCentralStatisticalOffice2000).Aswiththefirstcasestudy,wemodelthehypotheticalscenarioinwhichdemographicratesareconstantovertime.

Table5:CaseStudy2:Modelparameters

Parameter Value/DatasourceInitialpopulationsize 500Populationgrowthrate 2.5%Immigrationrate 0%Mortalityprobabilities Zambia,2000,5-yearagegroupsLopezetal.(2000)Fertilityprobabilities Zambia,2000,byyearofage(data.un.org)Birthgap mean:270days,SD:0days(i.e.,uniform)Coupleformationparameters agerange:15–60years;annualprobability:8%Partneragedifference mean:2years;SD:2yearsCoupledissolutionparameters agerange:18–60years;annualprobability:0.1%Leavinghomeparameters minimumage:18years;0.5%

4.20 AsdepictedinFigure9(a),theagestructureoftheZambianpopulationiscapturedwithahighdegreeofaccuracy.Householdsizedistribution(Figure9(b))isreproducedlessaccurately:smallhouseholdsizes(1–4people)areover-represented,whileverylargehouseholds(>8)peopleareunder-represented.Onepossibleexplanationisthatourparametervalueswerepoorlychosen.However,wealsonotethatZambiahasareasonablyhighlevelofmulti-generationalandmulti-nucleushouseholds(RepublicofZambiaCentralStatisticalOffice2000).Neitherofthesearecurrentlyrepresentedinourmodel,whichmayexplainsomeofthediscrepancy.ComparingFigures10and11withthecorrespondingfiguresfromthefirstcasestudy(Figures7and8)demonstratestheconsiderabledifferencesinpatternsofhouseholdcompositionthatexistbetweencountrieswithdifferentdemographicproperties.

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Figure9:(a)Thefinalagedistributionsoftensimulatedpopulations(grey),comparedtotheempiricalagedistributionusedtoinitialisethepopulation(black).(b)Thefinalhouseholdsizedistribution(white),averagedoverthetensimulatedpopulations(errorbarsindicatestandarddeviation),comparedtoempiricaldata(black).

Figure10:Agedistributionofindividualsgiven(a)householdageand(b)householdsize.Individualsaregroupedaccordingtoagecategories:<5years;5-17years;18-64years; 65years.

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Figure11:HouseholdcompositionforZambia.

CaseStudy3:Apopulationundergoingdemographicchange

4.21 Thetwopreviouscasestudiesdepictstabledemographicscenarios:thereisnochangetofertilityandmortalityratesoverthesimulatedperiod.Thefinalrequirementofourmodelwasthatitproducehouseholddynamicsofapopulationacrossaperiodofdemographicchange.AsdiscussedinSection2,patternsofinfectionandimmunityemergeoverlongtime-frames,duringwhichtimepopulationstructureisneitherstaticnorchanginginauniformfashion.Rather,underlyingdemographicrateschangeaslifeexpectancyincreasesduetoimprovementsinhealthandmedicine,andfertilitypatternschangeinresponsetoavailabilityofbirthcontrolandshiftingsocialnorms.Asatestcase,weparameterisedourmodelwith100yearsofAustraliancensusdata,coveringtheperiodfrom1910-2010.Duringthisperiod,Australia'spopulationincreasedfromalmost4.5milliontoover22million.Whilenaturalincrease(i.e.,resultingfrombirthratesbeinghigherthandeathrates)accountsforaroundtwo-thirdsofthisgrowth,immigrationhasalsoplayedasignificantrole.AttheconclusionofWorldWarII,Australiainitiatedalargeimmigrationprogrammeinanefforttoboostpopulationnumbers(DepartmentofImmigrationandMulticulturalAffairs2001).Toapproximatethisdemographichistory,weusedastartingpopulationsizeof500individuals,growingatarateof2.5%peryearduringtheinitial100yearinitialisationperiod,andatadecreasingrateduringthefollowing100years.Weusedanimmigrationrateof0%priorto1950and1%peryearafter1950.Theresultingfinalpopulationscontainedapproximately30,000peopleafter200yearsofsimulation.Weestimatedtime-varyingcoupleformationanddissolutionratesonthebasisofhistoricaltrendsofincreasingmarriageage,andincreasingdivorcerate(deVaus2004).

Table6:CaseStudy6:Modelparameters

Parameter Value/DatasourceInitialpopulationsize 500Populationgrowthrate 2.5%decreasingto0.5%over100yearsImmigrationrate 0%until1950,1%thereafterMortalityprobabilities Australia,1910-2008,varyingfrequency,byyearofage(AustralianBureauofStatistics2008)Fertilityprobabilities Australia,1910-2008,varyingfrequency,5yearagegroups(AustralianBureauofStatistics2008)Birthgap mean:365days;SD:90daysCoupleformationparameters agerange:(15increasingto18over100years)–60;annualprobability:7.5%Partneragedifference mean:2years;SD:2yearsCoupledissolutionparameters agerange:18–60;annualprobability:0.1%increasingto1.5%over100yearsLeavinghomeparameters minimumage:18;annualprobability:0.8%

4.22 Thekeydemographictrendsthatwerereproducedinthesimulatedpopulationsweretheeffectsthattime-varyingdemographicrateshaveonhouseholds.Figure12(a)showshowaveragehouseholdsizedecreasesataratecomparabletothatobservedintheempiricaldata.Figure12(b)showshowthe(rescaled)numberofhouseholdschangesoverthesametimeperiod,againcomparedtotheempiricaltrend.

4.23 Figure13showstwofurtherperspectivesonhowhouseholdsevolveoverthecourseofthesimulation:theprevalenceofhouseholdsofsizeoneandtwoincreasesrelativetolargerhouseholds(Figure13(a)),andtheprevalenceofhouseholdscontainingcoupleswithoutchildrenincreases(Figure13(b)).Onlylimiteddataisavailabletovalidatethesetrendsacrossthefulltimeperiod;however,ratesofchangeacrossthemostrecentdecadesareinagreementwithcensusdata.Empiricalvaluesfortwotimepoints(1981and2001)areoverlaidontheplotshowingtheevolutionofhouseholdsizedistributionovertime(Figure13(a)).Evenafterseventyyearsofsimulatedpopulationevolution,themodelreproducescomparableratesofincreaseanddecreaseintheoccurrenceofhouseholdsofagivensize.Withrespecttothedistributionofhouseholdtypes(Figure13(b)),inourtensimulatedpopulations,theproportionoffamilyhouseholdscontainingacouplewithchildrendecreasedbyanaverageof23%(SD1.5%)overthepenultimatetwodecades,whiletheproportioncontainingonlyacoupleincreasedbyanaverage28%(SD3.8%).ThecorrespondingchangesintheAustralianpopulationbetween1981and2001werea20%decreaseand28%increaserespectively(deVaus2004).

Figure12:(a)Averagehouseholdsizeover100years.(b)Numberofhouseholdsover100years(Numberofhouseholdsin1910=100).Bothfiguresshowtheoutputfrom10simulationrunscomparedwithhistoricalAustraliancensusdata(bold).

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Figure13:(a)Evolutionofhouseholdsizedistributionover100years.(b)Evolutionoffamilyhouseholdtypedistributionover100years.Familyhouseholds,inourmodel,includeanyhouseholdcontainingatleasttwopeople(i.e.,singlepersonhouseholdsareexcluded).Eachfigureshowstheoutputfromasinglerepresentativesimulationrun.

Evaluation

5.1 Manydifferentforcesactuponpopulationstoproducetheircharacteristicageandhouseholdstructure.Overthelastcentury,improvementstohealthcare,governmentpoliciesaroundmigrationandfertility,changingsocialnorms,aswellasunpredictableeventssuchaswarsandnaturaldisastershaveallplayedarole.Capturingallofthesecomplexandinteractingforcesinamodelischallenging,andwehavefavouredageneralandflexibleapproachtomechanismdesign.Thatsaid,webelievethat,forourresearchagenda,themodeldescribedinSection3representsagoodbalancebetweensimplicityandplausibility.Currently,thedatasourcesrequiredtoparameterisethemodelarerelativelymodest,henceitcanbeusedtomodelpopulationsforwhichonlylimiteddataareavailable(e.g.,theinternationalandhistoricalpopulationsconsideredinthesecondandthirdcasestudies)andhypotheticalscenarios.Undoubtedlyacloserfittoempiricaldatacouldbeachieved,butthiswouldinevitablycomeatthecostofintroducingfurtherdatadependenceintothemodel.

5.2 Initscurrentform,ourmodelmeetsthefourrequirementssetoutinSection2.1;namely,itcanproducerealisticpatternsofhouseholdcompositionandhouseholddynamics,whilebeingflexibleenoughtocaptureavarietyofdemographicscenariosandtransitions.Inendeavouringtodesignaparsimoniousmodel,wehavemadeseveralsimplifyingapproximations.Webrieflydiscussrefinementstobeconsideredinfuturemodeldevelopment:Firsttheabsenceofextendedfamilyhouseholds,asnotedabove,isanomissionthatmustbeaddressedinordertomoreaccuratelycapturehouseholdcompositionincertainpopulations.Theculturalandeconomicconditionsunderwhichextendedfamilyhouseholdsarisevarybycountryandacrosstime(Hammel&Laslett1974).AswiththemodeldescribedbyMurphy(2011),ourmodelcontainsagreatdealofinformationonkinshiprelationshipsbetweenindividualsthatcouldbeusedtoconstructmorecomplexhouseholdstructures.Fromanepidemiologicalperspective,multi-generationalmayhaveimplicationsforpatternsofsocialcontactanddiseasetransmission(Mossongetal.2008).Second,themodeldoesnotcurrentlydistinguishbetweenmarriageandcohabitation.Wejudgedthat,fromtheperspectiveofhouseholdstructuresrelevanttoinfectiousdiseasetransmission,marriageandcohabitationareeffectivelyindistinguishable;however,householdsinwhichpartnersaremarriedasopposedtocohabitingdoappeartohavedifferentcharacteristics(suchasdurationofcouplerelationship)thatmayinfluencehouseholddynamics(deVaus2004).Finally,amorerealisticmodelofimmigrationiscertainlypossible.Specifically,immigrantpopulationsarelikelytohavedifferentdemographiccharacteristicstothepopulationstheyjoin(Haugetal.2002),andtheirarrivalisthereforelikelytoinfluenceageandhouseholdstructureinnon-trivialways.Iannelli&Manfredi(2007)haveshownhowchangestotheagestructureofapopulationcanhaveimplicationsfordiseasedynamics.

5.3 Toconclude,mathematicalandnetworkmodelsthatworkwithstylisedpopulationstructureswillcontinuetoprovideimportantinsightsintothedynamicsandcontrolofinfectiousdiseases.However,manyopenquestionsininfectiousdiseaseepidemiologyconcerntherolesplayedbyspecifictypesofpopulationheterogeneity.Toanswerthesequestions,morerealisticpopulationmodels,suchasthatdescribedhere,willproveinvaluable.Asdiscussedatthebeginningofthispaper,syntheticpopulationmodelsfindapplicationinmanydomains,andthereisnoreasonwhytheutilityofourmodelcouldnotextendbeyondthedomainofinfectiousdiseases.Forexample,asimilarapproachisbeingusedtoexploreissuesaroundtheprovisionofsocialcareinageingpopulations(Silvermanetal.2012),anotherissueforwhichchangingpatternsofhouseholddemographyhaveimportantimplications.However,therequirementsguidingthedevelopmentofourmodelwerebasedonourparticularresearchagenda,anditiscriticaltoensurethat,forotherapplications,modelbehaviourissuitedtotheproblemathand.

Appendix

1. Createinitialpopulation,accordingtoinitialagedistributionandhouseholdsizedistribution.2. Scaleallannualprobabilitiesandratestogiveratespertimestep.3. Ateachtimestep,eachindividualhasthepossibilityofexperiencingalifeeventasfollows(individualattributesthataffecteventprobabilitiesarelistedinparentheses):

1. Testfordeath(age,sex).Ifdeathoccurs,thefollowingoccurs:1. Thedeadindividualisremovedfromthepopulation.Ifthisresultsinahouseholdcontainingorphanedchildren,anychildrenwhoareoldenough(e.g.,>18years)leavehome,asper

below,whileanyyoungerchildrenarerandomlyallocatedtoanotherfamilyhouseholdcontainingatleastonechild.2. Thebirthofareplacementindividualistriggered,andamotherischosen(age,sex,parity,timesincelastbirth).

2. Testforcoupleformation(age,sex).Ifcoupleformationistooccur,selectapartnerfromthepopulationwithanappropriateagedifferenceandupdatetheirhouseholdsasfollows:1. Ifbothindividualsliveathome,theymoveintoanewlycreatedhousehold.2. Ifeither(orboth)oftheindividualshavetheirownhousehold,theotherpartner(togetherwithanydependents)joinstheminthishousehold.

3. Testforleavinghome(age).Ifanindividualleaveshome,theyformasinglepersonhousehold.4. Testforcoupleseparation(age).Ifacoupleseparate,oneindividualremainsintheirformerhousehold,togetherwithanydependents,whiletheotherindividualleavestoformanewsingleperson

household.4. Calculatethenumberofadditionalbirthsduetonaturalincrease.Mothersarechosenforeachofthesenewindividualsasabove.5. Calculatethenumberofnewarrivalsduetoimmigration.Immigrantsarriveasahouseholdunit,withthesizeofthehouseholdandtheageofitsoccupantsdrawnfromthecurrentpopulationageand

householdsizedistributions.6. RepeatfromStep3.

Notes

1Theterms'agent-basedmodels'and'individual-basedmodels'arefrequentlyusedinterchangeablytodescribemodelsinwhicheachunitinapopulationisexplicitlyrepresented.Whereadistinctionismade,thetermagentsisusedtorefertoentitieswhosebehaviourisdeterminedonthebasisofcognitivefunctions,whilethebehaviourofindividualsisgovernedbylesscomplexbehaviouralrules(Parrottetal.2011).Wehavechosentouse'individual-basedmodels'throughout.

2AconciseoutlineofthemodelisprovidedintheAppendix.ThemodelisimplementedinPythonandsourcecodeisavailablefromhttp://github.com/nlgn/sim-demog.

3Parityreferstothenumberofchildrenborntoaparticularwoman.Forexample,awomanwhohasgivenbirthtotwopreviouschildrenhasaparityoftwo.

4Representinghouseholdfrequencybysizehelpstocounteractthevisualbiasthatotherwiseresultsfromthedominanceoflargerbutlesscommonhouseholdtypes.

References

AJELLI,M.&MERLER,S.(2009).Anindividual-basedmodelofhepatitisAtransmission.JournalofTheoreticalBiology259(3),478-88.[doi:10.1016/j.jtbi.2009.03.038]

ANDERSON,R.M.&MAY,R.M.(eds.)(1991).InfectiousDiseasesofHumans:DynamicsandControl.Oxford,UK:OxfordUniversityPress.

AUSTRALIANBUREAUOFSTATISTICS(2006a).Births,Australia,2006,cat.no.3301.0.

AUSTRALIANBUREAUOFSTATISTICS(2006b).PopulationbyAgeandSex,AustralianStatesandTerritories,Jun2006,Table9.EstimatedResidentPopulationBySingleYearofAge,Australia.

AUSTRALIANBUREAUOFSTATISTICS(2007).LifeTables,Australia,2005-2007,cat.no.3302.0.55.001.

AUSTRALIANBUREAUOFSTATISTICS(2008).AustralianHistoricalPopulationStatistics,2008,cat.no.3105.0.65.001.

BALL,F.,MOLLISON,D.&SCALIATOMBA,G.(1997).Epidemicswithtwolevelsofmixing.TheAnnalsofAppliedProbability7(1),46-89.[doi:10.1214/aoap/1034625252]

http://jasss.soc.surrey.ac.uk/16/1/8.html 11 14/10/2015

BALL,F.&NEAL,P.(2002).AgeneralmodelforstochasticSIRepidemicswithtwolevelsofmixing.MathematicalBiosciences180,73-102.[doi:10.1016/S0025-5564(02)00125-6]

BARRETT,C.L.,BECKMAN,R.J.,BERKBIGLER,K.P.,EUBANK,S.G.,HENSON,K.M.,ROMERO,P.R.&SMITH,J.P.(2000).TRANSIMS:Transportationanalysissimulation.Tech.Rep.00-1725,LosAlamosNationalLaboratory.

BECKER,N.G.,GLASS,K.,LI,Z.&ALDIS,G.K.(2005).ControllingemerginginfectiousdiseaseslikeSARS.MathematicalBiosciences193(2),205-21.[doi:10.1016/j.mbs.2004.07.006]

BILLARI,F.,FENT,T.,PRSKAWETZ,A.&APARICIODIAZ,B.(2007).The''wedding-ring'':anagent-basedmarriagemodelbasedonsocialinteraction.DemographicResearch17,59-82.[doi:10.4054/DemRes.2007.17.3]

BIRKIN,M.H.&CLARKE,M.(2011).Spatialmicrosimulationmodels:Areviewandaglimpseintothefuture.In:PopulationDynamicsandProjectionMethods.Springer,pp.193-208.[doi:10.1007/978-90-481-8930-4_9]

COUDEVILLE,L.,VANRIE,A.&ANDRE,P.(2008).Adultpertussisvaccinationstrategiesandtheirimpactonpertussisintheunitedstates:evaluationofroutineandtargeted(cocoon)strategies.Epidemiologyandinfection136(5),604-20.[doi:10.1017/S0950268807009041]

DANON,L.,FORD,A.P.,HOUSE,T.,JEWELL,C.P.,KEELING,M.J.,ROBERTS,G.O.,ROSS,J.V.&VERNON,M.C.(2011).Networksandtheepidemiologyofinfectiousdisease.InterdisciplinaryPerspectivesonInfectiousDiseases2011,284909.[doi:10.1155/2011/284909]

DEVAUS,D.(2004).DiversityandchangeinAustralianfamilies:Statisticalprofiles.Melbourne,Australia:AustralianInstituteofFamilyStudies.

DEPARTMENTOFIMMIGRATIONANDMULTICULTURALAFFAIRS(2001).Immigration:Federationtocentury'send.

ELVEBACK,L.R.,FOX,J.P.,ACKERMAN,E.,LANGWORTHY,A.,BOYD,M.&GATEWOOD,L.(1976).Aninfluenzasimulationmodelforimmunizationstudies.AmericanJournalofEpidemiology103(2),152-65.

EUBANK,S.,GUCLU,H.,KUMAR,V.S.A.,MARATHE,M.V.,SRINIVASAN,A.,TOROCZKAI,Z.&WANG,N.(2004).Modellingdiseaseoutbreaksinrealisticurbansocialnetworks.Nature429(6988),180-4.[doi:10.1038/nature02541]

FERGUSON,N.M.,CUMMINGS,D.A.T.,CAUCHEMEZ,S.,FRASER,C.,RILEY,S.,MEEYAI,A.,IAMSIRITHAWORN,S.&BURKE,D.S.(2005).StrategiesforcontaininganemerginginfluenzapandemicinSoutheastAsia.Nature437(7056),209-14.[doi:10.1038/nature04017]

GARGIULO,F.,TERNES,S.,HUET,S.&DEFFUANT,G.(2010).Aniterativeapproachforgeneratingstatisticallyrealisticpopulationsofhouseholds.PloSone5(1),e8828.[doi:10.1371/journal.pone.0008828]

GLASS,K.,MCCAW,J.&MCVERNON,J.(2011).Incorporatingpopulationdynamicsintohouseholdmodelsofinfectiousdiseasetransmission.Epidemics3,152-158.[doi:10.1016/j.epidem.2011.05.001]

GLICK,P.C.(1989).Thefamilylifecycleandsocialchange.FamilyRelations38,123-129.[doi:10.2307/583663]

GRASSLY,N.C.&FRASER,C.(2008).Mathematicalmodelsofinfectiousdiseasetransmission.NatureReviewsMicrobiology6(6),477-87.[doi:10.1038/nrmicro1845]

GRIMM,V.,REVILLA,E.,BERGER,U.,JELTSCH,F.,MOOIJ,W.M.,RAILSBACK,S.F.,THULKE,H.,WEINER,J.,WIEGAND,T.&DEANGELIS,D.L.(2005).Pattern-orientedmodelingofagent-basedcomplexsystems:Lessonsfromecology.Science310,987-991.[doi:10.1126/science.1116681]

GUZZETTA,G.,AJELLI,M.,YANG,Z.,MERLER,S.,FURLANELLO,C.&KIRSCHNER,D.(2011).Modelingsocio-demographytocapturetuberculosistransmissiondynamicsinalowburdensetting.JournalofTheoreticalBiology289,197-205.[doi:10.1016/j.jtbi.2011.08.032]

HAMMEL,E.A.(2005).Demographicdynamicsandkinshipinanthropologicalpopulations.ProceedingsoftheNationalAcademyofScience,USA102(6),2248-2253.[doi:10.1073/pnas.0409762102]

HAMMEL,E.A.&LASLETT,P.(1974).Comparinghouseholdstructureovertimeandbetweencultures.ComparativeStudiesinSocietyandHistory16(1),73-109.[doi:10.1017/S0010417500007362]

HARDING,A.(2007).ChallengesandOpportunitiesofDynamicMicrosimulationModelling,Plenarypaperpresentedtothe1stGeneralConferenceoftheInternationalMicrosimulationAssociation,Vienna,21August.

HAUG,W.,COMPTON,P.&COURBAGE,Y.(eds.)(2002).TheDemographicCharacteristicsofImmigrantPopulations.No.38inPopulationStudies.CouncilofEuropePublishing.

HAYES,A.,WESTON,R.,QU,L.&GRAY,M.(2010).Familiesthenandnow:1980-2010.Factsheet,AustralianInstituteofFamilyStudies.

HETHCOTE,H.W.(2000).Themathematicsofinfectiousdiseases.SIAMReview42(4),599-653.[doi:10.1137/S0036144500371907]

IACONO,M.,LEVINSON,D.&EL-GENEIDY,A.(2008).Modelsoftransportationandlandusechange:aguidetotheterritory.JournalofPlanningLiterature22(4),323-340.[doi:10.1177/0885412207314010]

IANNELLI,M.&MANFREDI,P.(2007).Demographicchangeandimmigrationinage-structuredepidemicmodels.MathematicalPopulationStudies14(3),161-191.[doi:10.1080/08898480701426241]

JARDINE,A.,CONATY,S.J.,LOWBRIDGE,C.,STAFF,M.&VALLY,H.(2010).Whogivespertussistoinfants?Sourceofinfectionforlaboratoryconfirmedcaseslessthan12monthsofageduringanepidemic,Sydney,2009.CommunicableDiseasesIntelligence34(2),116-121.

JOHN,A.M.(1990).Transmissionandcontrolofchildhoodinfectiousdiseases:Doesdemographymatter.PopulationStudies44(2),195-215.[doi:10.1080/0032472031000144556]

KIRK,D.(1996).Demographictransitiontheory.PopulationStudies:AJournalofDemography50(3),361-387.[doi:10.1080/0032472031000149536]

KORB,K.B.,GEARD,N.&DORIN,A.(2013).ABayesianapproachtothevalidationofagent-basedmodels.In:Ontology,Epistemology,andTeleologyofModelingandSimulation(TOLK,A.,ed.).Germany:Springer-Verlag,pp.255-269.[doi:10.1007/978-3-642-31140-6_14]

LONGINI,I.M.,NIZAM,A.,XU,S.,UNGCHUSAK,K.,HANSHAOWORAKUL,W.,CUMMINGS,D.A.T.&HALLORAN,M.E.(2005).Containingpandemicinfluenzaatthesource.Science309(5737),1083-7.[doi:10.1126/science.1115717]

LOPEZ,A.D.,SALOMON,J.,AHMAD,O.,MURRAY,C.J.L.&MAFAT,D.(2000).Lifetablesfor191countries:data,methodsandresults.GPEDiscussionPaper9,WorldHealthOrganisation,Geneva.

MANNION,O.,LAY-YEE,R.,WRAPSON,W.,DAVIS,P.&PEARSON,J.(2012).JAMSIM:amicrosimulationmodellingpolicytool.JournalofArtificialSocietiesandSocialSimulation15(1),8.http://jasss.soc.surrey.ac.uk/15/1/8.html

MOSSONG,J.,HENS,N.,JIT,M.,BEUTELS,P.,AURANEN,K.,MIKOLAJCZYK,R.,MASSARI,M.,SALMASO,S.,TOMBA,G.S.,WALLINGA,J.,HEIJNE,J.,SADKOWSKA-TODYS,M.,ROSINSKA,M.&EDMUNDS,W.J.(2008).Socialcontactsandmixingpatternsrelevanttothespreadofinfectiousdiseases.PLoSMedicine5(3),e74.[doi:10.1371/journal.pmed.0050074]

MURPHY,M.J.(2011).Long-termeffectsofthedemographictransitiononfamilyandkinshipnetworksinbritain.PopulationandDevelopmentReview37(S1),55-80.[doi:10.1111/j.1728-4457.2011.00378.x]

ORCUTT,G.H.(1957).Anewtypeofsocio-economicsystem.TheReviewofEconomicsandStatistics39(2),116-123.[doi:10.2307/1928528]

PARROTT,L.,CHION,C.,MARTINS,C.C.A.,LAMONTAGNE,P.,TURGEON,S.,LANDRY,J.A.,ZHENS,B.,MARCEAU,D.J.,MICHAUD,R.,CANTIN,G.,MéNARD,N.&DIONNE,S.(2011).Adecisionsupportsystemtoassistthesustainablemanagementofnavigationactivitesinthest.lawrenceriverestuary,canada.EnvironmentalModelling&Software26,1403-1418.[doi:10.1016/j.envsoft.2011.08.009]

REPUBLICOFZAMBIACENTRALSTATISTICALOFFICE(2000).HousingandHouseholdCharacteristics:AnalyticalReport.

SILHOL,R.&BOëLLE,P.-Y.(2011).Modellingtheeffectsofpopulationstructureonchildhooddisease:Thecaseofvaricella.PLoSComputationalBiology7(7),e1002105.[doi:10.1371/journal.pcbi.1002105]

SILVERMAN,E.,BIJAK,J.&NOBLE,J.(2011).Feedingthebeast:cancomputationaldemographicmodelsfreeusfromthetyrannyofdata?In:AdvancesinArtificialLife,ECAL2011:ProceedingsoftheEleventhEuropeanConferenceontheSynthesisandSimulationofLivingSystems.MITPress.

SILVERMAN,E.,BIJAK,J.,NOBLE,J.,CAO,V.&HILTON,J.(2012).Semi-artificialmodelsofpopulations:connectingdemographywithagent-basedmodelling.In:4thWorldCongressonSocialSimulation.

SPIELAUER,M.(2010).Whatissocialsciencemicrosimulation?SocialScienceComputerReview29(1),9-20.[doi:10.1177/0894439310370085]

STILLWELL,J.&CLARKE,M.(eds.)(2011).PopulationDynamicsandProjectionMethods,vol.4ofUnderstandingPopulationTrendsandProcesses.Springer.[doi:10.1007/978-90-481-8930-4]

TAPER,M.L.,STAPLES,D.F.&SHEPARD,B.B.(2008).Modelstructureadequacyanalysis:selectingmodelsonthebasisoftheirabilitytoanswerscientificquestions.Synthese163,357-370.[doi:10.1007/s11229-007-9299-x]

WILKINS,R.,WARREN,D.,HAHN,M.&HOUNG,B.(2011).Families,IncomesandJobs,Volume6:AStatisticalReportonWaves1to8oftheHousehold,IncomeandLabourDynamicsinAustraliaSurvey.Melbourne,Australia:MelbourneInstituteofAppliedEconomicandSocialResearch.

WILSON,T.(2011).Areviewofsub-regionalpopulationprojectionmethods.Tech.rep.,QueenslandCentreforPopulationResearch,SchoolofGeography,PlanningandEnvironmentalManagement,TheUniversityofQueensland.

WU,B.M.&BIRKIN,M.H.(2012).Agent-basedextensionstoaspatialmicrosimulationmodelofdemographicchange.In:Agent-BasedModelsofGeographicSystems(HEPPENSTALL,A.J.,CROOKS,A.T.,SEE,L.M.&BATTY,M.,eds.).Springer,pp.347-360.[doi:10.1007/978-90-481-8927-4_16]

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