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Original citation:
Pellis, Lorenzo, Ball, Frank, Bansal, Shweta, Eames, Ken, House, Thomas A., Isham,
Valerie and Trapman, Pieter. (2015) Eight challenges for network epidemic models.
Epidemics, 10 .
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ContentslistsavailableatScienceDirect
Epidemics
jo u rn al h om ep age : w w w . e l s e v i e r . c o m / l o c a t e / e p i d e m i c s
Eight
challenges
for
network
epidemic
models
Lorenzo
Pellis
a,∗,
Frank
Ball
b,
Shweta
Bansal
c,d,
Ken
Eames
e,
Thomas
House
a,
Valerie
Isham
f,
Pieter
Trapman
gaWarwickInfectiousDiseaseEpidemiologyResearchCentre(WIDER)andWarwickMathematicsInstitute,UniversityofWarwick,CoventryCV47AL,UK bSchoolofMathematicalSciences,UniversityofNottingham,UniversityPark,NottinghamNG72RD,UK
cDepartmentofBiology,GeorgetownUniversity,Washington,DC20057,USA dFogartyInternationalCenter,NationalInstitutesofHealth,Bethesda,MD,USA
eCentreforMathematicalModellingofInfectiousDiseases,LondonSchoolofHygieneandTropicalMedicine,LondonWC1E7HT,UK fDepartmentofStatisticalScience,UniversityCollegeLondon,LondonWC1E6BT,UK
gDepartmentofMathematics,StockholmUniversity,Stockholm10691,Sweden
a
r
t
i
c
l
e
i
n
f
o
Articlehistory:
Received15February2014 Receivedinrevisedform25July2014 Accepted28July2014
Availableonline4August2014
Keywords:
Infectiousdiseasemodels Transmissiondynamics Contactnetworks Randomgraphs Dynamicnetworks Controlmeasures
a
b
s
t
r
a
c
t
Networksofferafertileframeworkforstudyingthespreadofinfectioninhumanandanimalpopulations. However,owingtotheinherenthigh-dimensionalityofnetworksthemselves,modellingtransmission throughnetworksismathematicallyandcomputationallychallenging.Eventhesimplestnetwork epi-demicmodelspresentunansweredquestions.Attemptstoimprovethepracticalusefulnessofnetwork modelsbyincludingrealisticfeaturesofcontactnetworksandofhost–pathogenbiology(e.g.waning immunity)havemadesomeprogress,butrobustanalyticalresultsremainscarce.Amoregeneraltheory isneededtounderstandtheimpactofnetworkstructureonthedynamicsandcontrolofinfection.Here weidentifyasetofchallengesthatprovidescopeforactiveresearchinthefieldofnetworkepidemic models.
©2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense
(http://creativecommons.org/licenses/by/3.0/).
Introduction
Networks(orgraphs)areextremelyflexibletoolsfor represent-ingcomplexsystemsofinteractingcomponents(Boccalettietal., 2006;Durrett,2007;Newman,2010).Eachcomponentis repre-sentedbyanode(orvertex)andeachlink(oredge)betweennodes describessomesortofinteractionbetweenthem.Here,wefocuson thespecificapplicationofnetworksinthefieldofinfectiousdisease modelling(Andersson,1999;Danonetal.,2011).
Becauseoftheirflexibility,networkshavebeenusedtomodel infectionspreadindifferentforms.Nodescandescribesingle indi-viduals,groups of individuals (e.g.households, farms,cities) or locationstowhichindividualsareconnected(e.g.seeRileyetal.,in thisissue).Linkscanrepresentinfectiousattemptsortransmission events(inwhichcasethenetworkisdirected)orsimply acquain-tancesbetweenthem(socialorsexualrelationshipsthroughwhich theinfectioncanspread,usuallyinbothdirections),movementsof animalsbetweenfarms(directorviaintermediatemarkets),flight routes,etc.
∗Correspondingauthor.Tel.:+442476524402.
E-mailaddress:[email protected](L.Pellis).
Thisapparentsimpleandintuitiverepresentationofa popula-tionofinteractingcomponentshasthedrawbackthatitmightbe difficulttoworkwith.Eveninthecaseofasimpleundirected net-workwithnnodes,westillneedn(n−1)/2binarydigitstofully describethepresenceorabsenceofeachpossibleedge.Thus, par-ticularlyforlargenetworks,thegeneralapproachistosummarise mostofthenetworkinformationinasmallsetofstatisticsandthen studytheirimpactoninfectionspread.Amongthemyriadnetwork properties(Boccalettietal.,2006;Newman,2010),inthispaper weconsidersomeofthosethatappearbothepidemiologically rel-evantandamenabletoanalysis,suchas:degreedistribution,the distributionofthenumberoflinksfromeachnode;assortativity, thepropensityofepidemiologicallysimilarnodestobeconnected toeachother,animportantexampleofwhichisthedegree correla-tionbetweenneighbouringnodes;clustering,thepropensityoftwo nodeswithacommonneighbourtobeneighboursofeachother(i.e. thefractionoftripletsthatformtriangles);modularity,the parti-tioningofthenetworkintointernallywell-connectedgroups;and
betweennesscentralityofanode,i.e.thenumberofshortestpaths betweenallpairsofnodesthatpassthroughthatnode.
Here,wehaveinmindnodesasindividualsandlinksas acquain-tancesbetweenthem,andthereforeprimarilyconsiderinfection
spread on undirected networks. Furthermore, we mostly have
http://dx.doi.org/10.1016/j.epidem.2014.07.003
in mind permanently immunising infections (i.e. SIR epidemic models).Althoughmostchallengesapplyalsointheabsenceof per-manentimmunity(i.e.SISandSIRSmodels),thisanalyticallymuch hardercaseisthefocusofSection‘Incorporatingwaningimmunity innetworkepidemicmodels’.InSection‘Understandingtheeffect ofheterogeneityonparameterestimationandepidemicoutcome’, weconsidertheso-calledconfigurationmodel(Danonetal.,2011;
Durrett,2007,Chapter3):besidetheErdös-Rényirandomgraph (Durrett,2007,Chapter2), thisisthemostanalyticallytractable networkbecauseofitslocallytree-likestructure,butitlacksmany featuresofreal-worldnetworksthatcandramaticallyimpact
trans-missiondynamics. We then discusscomplex networks (i.e.not
locally tree-like),first unweightedand static(Section ‘Develop-inganalyticalmethodstogenerateandstudyepidemicsonstatic
unweightedcomplexnetworks’)andthenweightedanddynamic
(Section‘Developinganalyticalmethodstomodelweightedand
dynamicnetworksandepidemicsthereon’).Approximatemethods
arediscussedinSection‘Developingandvalidatingapproximation schemesforepidemicsonnetworks’.Finally,inSections‘Clarifying theimpactofnetworkpropertiesonepidemicoutcome’, ‘Strength-eningthelinkbetweennetworkmodellingandepidemiologically
relevant data’ and ‘Designing network-based interventions’ we
discusstheimpactofnetworkstructureoninfectionspread,the relationshipbetweennetworkmodelsanddata,andinterventions, respectively.
1. Understandingtheeffectofheterogeneityonparameter estimationandepidemicoutcome
In homogeneously mixing populations, the relationships
betweenkeyepidemiologicalquantitiesaregenerallywell under-stood.Forexample,itiswellknownthatforSIRepidemicsinthe largepopulationlimit(startingwithanegligiblefractionofthe pop-ulationinfected),R0andthefinalsizeofalargeoutbreak,zsay,are stronglylinkedbythesimplerelationship1−z=e−R0z(Diekmann
etal.,2013).
However,evenforanSIRepidemiconaconfiguration-type net-work,thissimplerelationshipislost:R0andfinalsizeofalarge outbreakbothdependonthedegreedistribution,buttheformeris affectedbythedegreevariance,whichismuchmoresensitiveto changesinprobabilitiesofhigh-degreethanlow-degreevertices, whilethelatterishighlydependentontheexactprobabilitiesof low-degreevertices,buthardlydependsonhigh-degreeones. Sim-ilarconsiderationsapplywhenindividualsvaryinsusceptibility and/orinfectivity,withtheadditionalproblemthatattainabledata areunlikelytoprovidemuchinformationofthistype.
Itthereforeremainsanimportantproblemtounderstandhow, notonlyR0,probabilityofalargeoutbreakanditsfinalsize,butalso durationoftheepidemicandpeakincidence,relatetoeachother andhowthedependenciesareaffectedbypotentiallyunobserved heterogeneityinsusceptibility/infectivityanddegree.
Furthermore,duringanoutbreak,earlypredictionsforpublic healthpurposesaretypicallyneeded.Therefore,itisimportantto quantifyhowsuchheterogeneitiesaffectearlyparameterestimates (e.g.ofR0)andtherepercussionsofpotentialestimationbiaseson epidemicpredictions.
2. Developinganalyticalmethodstogenerateandstudy epidemicsonstaticunweightedcomplexnetworks
Althoughconvenientforitsanalyticaltractability,the configu-rationmodelfailstocapturesomeimportantpropertiesofrealistic
contact networks. The POLYMOD study (Mossong et al., 2008)
revealedstrongassortativitybyage(peoplemakemorecontacts of similaragetotheirown than ofothers) withtheadditional
trans-generational contact between children and adults, while
Readetal.(2008)highlightedsignificantclusteringinan
empir-ically measuredsocial network. Metapopulation and multitype
epidemicmodels(seeBalletal.,inthisissue)are epidemiologi-callyimportantexamplesofmodularnetworks.Spatial(seeRiley et al.,in this issue) andhighly heterogeneousnetworks ofsize
n,unliketheconfigurationmodel,exhibit pathlengthsoforder otherthanlog(n).Finally,higher-ordercorrelationssuchas four-motifstructureorcorrelationsatthetriplelevelarelikelytooccur in any network generated by complex social processes (Miller, 2009).
Anumberofmodelsforconstructingrandomnetworkshave
beendevelopedtoincorporaterealisticgraphproperties.Generally,
astherandomgraphmodelunderconsiderationbecomesmore
complex,rigorousresultsaboutthepropertiesoftheresulting net-work,and ofepidemics runningonit, becomelessgeneral.For
example,thepreferentialattachmentnetwork modelallowsfor
rigorousanalysisofmostnetworkpropertiesandalsoasymptotic epidemicthresholdbehaviour(Durrett,2007,Chapter4).For ran-domgeometricgraphsnetworkpropertiesareknownbutanalysis ofepidemicdynamicshassofarrequiredMonteCarlosimulation (Ishametal.,2011).Forexponentialrandomgraphs(Danonetal.,
2011)and related modelsthat seektogeneratenetworks with
specifiedpropertiesinthemostrandomwaypossible,thereare essentiallynoexactresults.
Rigorousanalysisis,however,possibleforSIRepidemicsdefined
onsomerandomnetworkmodelswithclustering.Theseinclude
models incorporating small cliques of individuals, e.g. random intersectiongraphs,triangle-orhousehold-basedmodels(seeBall etal.,2013,andreferencestherein).However,analyticaltractability stemsfromthefactthatallsuchmodelshaveatree-likestructure atsomelevel(e.g.atreeoffullyconnectedcliques).
Althoughthesemodelsenableanalysisoftheeffectof cluster-ingandsometimesalsodegreecorrelationonepidemicproperties, itmustberecognisedthatthenetworkstheyproducearerather special andnot easilygeneralisable.Also, epidemicsondistinct
network modelshavingcommondegreedistribution,clustering
coefficientanddegreecorrelationmayhavedifferentproperties (Balletal.,2013).Therefore,majorchallengesinvolveidentifying which,ifany,ofthecurrentmodelsreflectsrealitywellenoughfor thequestionathandanddevelopingothernetworkmodelsthatare bothsufficientlyrealisticandamenabletorigorousmathematical analysis.
3. Developinganalyticalmethodstomodelweightedand dynamicnetworksandepidemicsthereon
Linkswithinreal-worldsocial networksarenot allidentical: someinteractionscarryagreaterriskofdiseasetransmissionthan others.Toaccountforthisadditionalheterogeneity,wecan
con-sider weightednetworks, in which a link’s weight(which may
vary over time) can be thought of as its relative transmission
potential.Some modelshave attempted toinclude information
inwhich unobstructedsensorswerewithina givenfunctioning distance.
On the other hand, social contacts are neither continuous
norpermanent.Variousforms ofnetwork dynamicsareknown
toberelevanttoinfectious diseaseepidemiology(Bansalet al., 2010): extrinsic processes (e.g. births, deaths, school terms, changesinsocialrelationships,migration,hostmobility,seasonal orlong-term sociallyor economically-driven changes); individ-uals’spontaneouschanges(avoidancebehaviour)orpublichealth interventions (vaccination, school closure); and the spread of theinfection itself (recoveredindividuals become irrelevant in futurechainsoftransmission,infectedindividualsmayaltertheir behaviour).
Thesechangescanalterlocalnetworktopology(intheformof added/removednodesandedges,orasalterededgeweights)and evenaffectglobalnetworkstructureandproperties.Inresponseto eachoftheprocesseshighlightedabove,respectively:
a.Models have successfully included varying contact durations (KretzschmarandMorris,1996),formationanddissolution of contacts(EamesandKeeling,2002),contactexchange(Volzand Meyers,2007).However,theinclusionofdemographicprocesses inatractableandrealisticmannerremainselusive(withafew recentexceptions;seee.g.Kamp,2010).
b.Modelshaveincludedinfection-avoidanceusingnetwork mod-elswithadaptivecontactexchange(e.g.susceptiblesreplacing
infected neighbours withother randomly chosen susceptible
ones;Grossetal.,2006)orwithserosortingmodelsforHIVwhere individualschoosesexualpartnersmatchingtheirinfections sta-tus(Volzetal.,2010).Thesemodelsshowasignificantimpact onepidemiologicaloutcomesofthisbehaviour;however,itis
unclearwhetherdatasupportsuchmodellingassumptionsas
realisticbehaviouralresponsestoongoingepidemics(Funketal., in this issue). Publichealth interventionsarediscussed more broadlyinSection‘Designingnetwork-basedinterventions’. c.Finally, for respiratory diseases suchas influenza,illness has
beenfoundtoreducecontactandgenerateashiftinage-specific mixing(vanKerckhoveetal.,2013).However,amorecomplete understandingoftheimpactofdiseaseoncontactstructureis necessaryforabroadclassofpathogens.
Theserecentdevelopmentsarepromising,butwestilllacka mathematicalframeworkthattractablyhandlesabroadrangeof realisticdynamicnetworks.
4. Incorporatingwaningimmunityinnetworkepidemic models
Mostofthe theoryofepidemics onstaticrandom networks
concerns the SIR model because the assumption of
perma-nentimmunitysignificantlyincreasesanalyticaltractability.Many
quantitiesdo not dependon when eventshappen but onlyon
whethertheyhappenor not:therefore, thereal-time dynamics
canoftenbeignoredand propertiessuchasR0,theprobability ofalargeoutbreakanditsfinalsizecanbecomputedusing the-oryfrombranchingprocesses(Jagers,1975)orpercolationtheory (Grimmett,1999).Whenimmunityislackingorwaningatthesame timescaleastheinfectiondynamics(e.g.SIS,SIRSmodels), rigor-ousanalysisbecomemuchharder:thetimeatwhicheventsoccur cannotbeignored,anddependenciesappearnotonlybetweenthe statesofneighboursbutalsobetweenthoseofdistantindividuals.
Modelswithoutpermanent immunityareseldom studiedin
a rigorous way, with thenotable exception of the Markov SIS
epidemic(i.e.withconstantinfectionandrecoveryrates), exten-sivelyconsideredinthephysicsliteratureasthecontactprocess
(Liggett,1999).However,eveninthesimplecaseoftheMarkovSIRS epidemictherearenorigorousresultsaboutthesurvival probabil-ityonaninfinitegraphandwhetheritincreasesastheinfectionrate increases(e.g.highratesmightnotgiveenoughtimeforrecovered individualstoregain susceptibilitybeforeinfectiongoesextinct locally;vandenBergetal.,1998).Furthermore,itisnotknown whetheranepidemicthatsurvivesforalongtimereaches ende-micityinallpartsofthenetworkorwhetherdifferentpartsofthe networkexperiencerecurrentwavesofinfection.Thisproblemis closelyrelatedtoweakandstrongsurvivalinthecontactprocess (Liggett,1999).
5. Developingandvalidatingapproximationschemesfor epidemicsonnetworks
Approximateresultsareavailablethroughagreatmany meth-ods.Theseareusedtodescribethelimitingdynamicsofstochastic epidemicsonnetworksintermsofsetsofdifferentialequations (e.g. pairapproximations, triple-based models, effective-degree approaches). For some locally tree-like networks a differential equationmodelisasymptoticallyexact,butforclusterednetworks
the situation is much more complex. Typically, the heuristic
argumentsusedtomotivateapproximationsrelyonanimplicit
assumptionsuchasthatthenetworkinquestionisselected
uni-formly at random from the set of all graphs having specified
properties.Forexample,forclusterednetworks,approximations usuallyassumethatallencounteredtripletsformclosedtriangles independentlywithconstantprobability(Danonetal.,2011)and hencearenotdesignedfornetworkswhere,say,trianglesallcluster incliques(e.g.households,seeSection‘Developinganalytical meth-odstogenerateandstudyepidemicsonstaticunweightedcomplex networks’).Asyet,however,thereisnocompletetheoretical
under-standingofwhenagivenapproximationwillwork,andamajor
challengeistoputsuchapproachesona rigorousmathematical footing,forexamplebyfindinganasymptoticregimeunderwhich theapproximationbecomesexactasthepopulationsizetendsto infinity.
6. Clarifyingtheimpactofnetworkpropertiesonepidemic outcome
Acommonlystatedchallengeforcomplexnetworkmodelsis
tounderstandhownetworkcharacteristicsaffectepidemiological quantitiesofinterest.Theproblemsaresimilartothosehighlighted forsimplenetworksinSection‘Understandingtheeffectof het-erogeneityonparameterestimationandepidemicoutcome’,with additionalcomplicationsduetotheshortageofanalyticalresults. EvensimplequestionslikethedependenceofR0andthe probabil-ityandsizeofalargeoutbreakonclusteringanddegreecorrelation (Section‘Developinganalyticalmethodstogenerateandstudy
epi-demicsonstatic unweightedcomplex networks’) need care,as
structurallydifferentnetworkscanexhibitthesameclusteringand correlation(Balletal.,2013),andanswerswilldependonother aspectsofnetworktopology.
Forweightednetworks,modelscouldbeusedtoconsiderthe impactof:thedistributionoflinkweights;theroleofcorrelationof weights(doesitmatterwhetherweightsaredistributedrandomly, orwhetherweightsarecorrelatedattheindividuallevelor‘locally’ withinthenetwork?);therelationshipbetweenweightanddegree (dopeoplewithmorecontactshavecontactsoflowerweight?);the relevanceoflow-weightlinks(cansuchlinksbeignored,ordothey drivetheemergenceofinfectionfromdenselocalcliques?).
exchange(Volz and Meyers,2007)for respiratory diseases,can influencediseasedynamics.Amorecompleteunderstandingofthe epidemiological significanceof dynamic contactpatterns across allclasses of pathogens in influencing both diseasespread and theefficacy of various intervention strategies (Section ‘Design-ingnetwork-basedinterventions’)isneeded.Thiswillinevitably depend onpathogen-specific characteristics, disease timescales andthequestionsathand.
Answerstothesequestionsarevitalforpublichealthmodelling byprovidingguidanceonthelevelsofheterogeneityanddetailthat arerequired(e.g.caninteractionsthatunderliedisease transmis-sionbeadequatelycapturedbyastaticnetworkmodelorshould complexdynamicsbemodelledexplicitly?).
7. Strengtheningthelinkbetweennetworkmodellingand epidemiologicallyrelevantdata
Thechallengesdescribedabovearerathertheoreticalinnature, butarestronglymotivatedbytheneedtocapturethose charac-teristics ofsocial behaviourthat are deemedtoaffectinfection spread.Asmoredatabecomeavailable,modellersneedtoimprove theiranalytical and computationaltoolkit.Agent-based simula-tionsareundoubtedlyuseful,butoftenfacesignificantalgorithmic andcomputationalproblemswithrespecttonetwork representa-tion,measurementoftopologicalfeatures,anddynamicalmodels forpathogenspread,aswellasalackofgeneralityandunproven robustnesstouncertaintiesinmodelstructureandparameter val-ues.Advancesindata-drivenanalyticmodellingwouldsolvesome ofthesechallengeswhileenhancingunderstandingofthe determi-nantsofmodelbehaviour.Atthesametime,modellingshouldplay animportantroleinguidingfuturedatacollection,inparticularby highlightingthosedatatowhichepidemicoutcomesaremost sen-sitive,thusclosingavirtuousfeedbackloopbetweentheoretical understandingandreal-worldobservations.
Thepastdecadehasseensuchafeedbackloopmoreheavily
tiltedtowardstheanalyticalmodellingside.Althoughfurtherwork inthatdirectionisneeded,particularlyexcitingistheemerging worldof‘bigdata’,intheformofgeneticinformation(Cottametal., 2008;Ypmaet al.,2012), contactdiaries (Mossong etal.,2008; vanKerckhoveetal.,2013)andelectronicsensors(Salathéetal., 2010;Stehléetal.,2011),andtheincreasedpowerofmodern sta-tisticalmethodstodealwiththesedata(Cauchemezetal.,2011). Theseraisethepossibilityofmoredirectobservationofepidemic networksthanhaspreviouslybeenpossible,andarediscussedin Frostetal.(inthisissue),Eamesetal.(inthisissue),Lessleretal. (inthisissue)andDeAngelisetal.(inthisissue).
However, connecting observations to model structure and
parametersisfarfromtrivial.Forexample,littleworkhasbeen doneto relatequantifiable measuresof link weight directlyto theriskof transmissionacrossthelink.Studiesarerequiredto collectbothsocialcontactandepidemiologicaldatato systemat-icallyassessawiderangeof‘weight’measuresandtodetermine
the relevant mapping between weight and risk. Those studies
thathavebeencarriedouthaveindicatedthatriskof transmis-sionvariesby(amongotherfactors)typeofsexualcontact(Boily etal.,2009),andbysocialsetting(Cauchemezetal.,2011;teBeest etal.,2013).Itremainsunclearhowgenerallysuchresultscanbe applied,andwhatroleisplayedbythevariouspropertiesofthe individualsandtheirrelationship:forexample,isalinkbetween twoschoolfriendsofhighweightbecausetheyareatschool,or becausetheyareofparticular(andsimilar)ages,orbecausethey shareothersocialactivities?Theappropriatemeasureswill dif-ferfordifferentpathogens–consider,forexample,influenzaand HIV–sostudiesshouldincludepathogenswithdifferentmodesof transmission.
8. Designingnetwork-basedinterventions
Public health interventions can aim to reduce transmission
along network edges without fundamentally altering the
net-worktopology(facemasks,handwashing)orcanhavelocaland
population-scaleeffectsonthetopologyofthecontactnetwork (e.g.schoolclosures,socialdistancingorvaccination,whichreduce contactsbyremovingnetworkedges).
Understandingnetworkstructureisvital,asnetworkfeatures canalsobeexploitedtodesignoptimalstrategies.Twosuch
strate-giesinclude targeting high-degree nodes tomake the network
sparserandtargetingcentralnodestofragmentthepopulationinto hard-to-reachsubgroups.Whiletheoreticallysoundideassuchas theseareingeneraldifficulttoimplementinpracticewhenlacking knowledgeofthecompletenetwork,afewrecentapproacheshave beenproposedtomakethesestrategiesfeasible:fortheformer, identifyinghigh-degreenodes(e.g.acquaintanceimmunization)or identifyingindividualtraitsthatserveasproxiesforhigh connec-tivity(e.g.ageandoccupationinhumanpopulations:Bansaletal., 2006;socialroleinwildlifepopulations:OtterstatterandThomson, 2007; oractivityin livestockpopulations: Shirleyand Rushton, 2005);forthelatter,identifyingsocialrolesoroccupationsthat cor-relatewithhighbetweenness(e.g.sexworkers:Mishraetal.,2012) oremployinglocalalgorithmsthatidentifyhighlycentral individ-ualswithoutrequiringknowledgeoftheentirenetwork(e.g.the communitybridgefinderalgorithm:SalathéandJones,2010). Fur-thersuchworkisrequiredforefficientandfeasiblenetwork-based interventionstrategiesintheabsenceofcompletenetworkdata, andforabetterunderstandingoftherelationshipbetweenpartial networkdataandinterventionefficacy.
Contacttracing(i.e.real-timetrackingofinfectedindividuals andtheirexposedcontacts)isatypicalnetwork-basedintervention (andisthestandardofcareinsomelocations,e.g.syphilisinthe UnitedStates).Byautomaticallyidentifyinghigh-riskindividuals, itcanbehighlyeffectiveasapreventativeorcontrolstrategy,and isparticularlyusefulforasymptomaticinfections.Previouswork indicatesthatcontacttracingeffectivenessincreaseswith cluster-ing(EamesandKeeling,2003),butquestionsremainabouttracing of‘high-risk’individuals,theoptimaltimingofcontacttracing,the interactionsbetweentimescalesoftracingandthenaturalhistory ofinfection,aswellasinteractionswithotherinterventions.
Anadditional challengeliesin themodelling ofbehavioural
responses to interventions as they pertain to changes in
net-workstructure.Examplesincludechangingofage-specificmixing patternsduringschoolclosurestocontrolrespiratorydisease out-breaks (Cauchemez et al., 2008) or rewiring of links during a movementstandstillimplementedtocontrollivestockdisease out-breaks(Robinsonetal.,2007).Thischallengeisdiscussedfurther inFunketal.(inthisissue).
Conclusions
Modellingtransmissionwithinnetworksisabroadand
chal-lenging field. As we have outlined above, it offers a range of
problemsincludingfundamentaltheoreticalwork,understanding andcapturingobservednetworkdata,andguidingnetwork-based public-healthinterventions.Whilethelistofpotentialchallenges is practicallyendless,herewehave attemptedtoidentifya set ofproblems,coveringarangeoffacets,thatmeritstudy.Within thisissuecanbefoundreferencetorelatedchallengesincluding networksinphylodynamics(Frostetal.,inthisissue), measure-mentofnetworkdata(Eamesetal.,inthisissue),andtheplace ofnetworkmodelsinrelationtoothermodellingstructures(Riley etal.,inthis issue;Balletal.,in thisissue).While wecertainly
urgent–questionsinnetworkmodelling,wehopethatthispaper willplayaroleinspurringadvancesinthisimportantand fascinat-ingfield.
Acknowledgments
TheauthorswanttoacknowledgetheIsaacNewtonInstitutefor MathematicalSciences,Cambridge,UK,forhostingtheInfectious DiseaseDynamicsprogrammethatresultedinthiscollaboration, anditsfollow-upprogrammewherethecollaborationscontinued. THandLParesupportedbytheEngineeringandPhysicalSciences ResearchCouncil;PTbyVetenskapsrådet(SwedishResearch Coun-cil)projectnr.2010-5873;KEbyaCareerDevelopmentFellowship awardfromtheNationalInstituteforHealthResearch(Grant
NIHR-CDF-2001-04-019);andSBbytheRAPIDDProgramoftheScience
&TechnologyDirectorate,DepartmentofHomelandSecurity,and theFogartyInternationalCenter,NationalInstitutesofHealth.We wouldalsoliketothanktheanonymousrefereeforhelpful
com-mentsandimprovements.
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