The Hidden Geometry of Complex, Network-Driven Contagion Phe(3)

Figure2CpresentsthecorrelationofarrivaltimesTawitheffectivedistancesDeffforthe

T=41 d.T=51 d.T=62 d.T=72 d.

C

ET [days]

Ta [days]

a

DeffFig.2.Understandingglobalcontagionphenomenausingeffective

distance.(A)Thestructureoftheshortestpathtree(ingray)fromHongKong(centralnode).RadialdistancerepresentseffectivedistanceDeffasdefinedbyEqs.4and5.NodesarecoloredaccordingtothesameschemeasinFig.1A.(B)Thesequence(fromlefttoright)ofpanelsdepictsthetimecourseofasimulatedmodeldiseasewithinitialoutbreakinHongKong(HKG),forthesameparam-etersetasusedinFig.1B.Prevalenceisreflectedbytherednessofthesymbols.Eachpanelcomparesthestateofthesystemintheconventionalgeographicrepresentation(bottom)withtheeffectivedistancerepresentation(top)

SCIENCE

neouswavethatpropagatesoutwardsatconstanteffectivespeedintheeffectivedistancerepresentation.(C)EpidemicarrivaltimeTaversuseffectivedistanceDeffforthesamesimulatedepidemicasin(B).Incontrasttogeographicdistance(Fig.1C),effectivedistancecorrelatesstronglywitharrivaltime(R2=0.973),i.e.,effectivedistanceisanexcellentpredictorofarrivaltimes.(DandE)Linearrelationshipbetweeneffectivedistanceandarrivaltimeforthe2009H1N1pandemic(D)andthe2003SARSepidemic(E).ThearrivaltimedataarethesameasinFig.1,DandE.Theeffectivedistancewascomputedfromtheproj-ectedglobalmobilitynetworkbetweencountries.Asinthemodelsystem,weobserveastrongcorrelationbetweenarrivaltimeandeffectivedistance.VOL342

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paredtoFig.1C,thisdemonstratesthateffectivedistancegeneratesamuchhighercorrelationthangeographicdis-2

tance(R2eff¼0:97comparedtoRgeo¼0:34;seetablesS2andS3andfig.S12formoreexamples).Furthermore,therelationshipofTaandDeffislinear,whichmeansthattheeffectivespeedveff=Deff/Taofthewavefrontisawell-definedcon-stant.Tocomparetheregressionquality,wecom-putedthedistributionofrelativeresidualsr=dTa/Ta,usingeffectiveorgeographicdistanceasaregressor.Theratioofresidualvariancesimpliesamorethan50-foldhigherpredictionquality(tableS3andfig.S13).

Althoughwehavedemonstratedtheclearlinearfunctionalrelationshipforsimulated,hy-potheticalscenariosofglobaldiseasespread,itiscrucialtotestthevalidityandusefulnessoftheeffectivedistanceapproachonempiricaldata.Figure2,DandE,depictarrivaltimeversusef-fectivedistanceonthebasisofdataforthe2009H1N1pandemicandtheglobal2003SARSepi-demic,respectively(figs.S14toS16andtableS4).ArrivaltimesarethesameasinFig.1,DandE,butshownacrosseffectiveratherthangeographicdistances.Astheempiricaldataareavailableonacountryresolution,wedeterminedthetrafficbe-tweencountriesbyaggregationtospecifyacoarse-grainednetwork(GMNc)(189nodes,5004links)andeffectivedistancesfromtheoriginlocationineachcase(seesupplementarytextfordetails).

BoththeH1N1andSARSdataexhibitaclearlinearrelationshipbetweenarrivaltimeandef-fectivedistancefromthesource,eventhoughadditionalfactorscomplicatethespreadingofrealdiseases.Fluctuations,effectsduetocoarsegraining,anderrorsinarrival-timemeasurementscanaddnoisetothesystem,whichincreasesthescatterinthelinearrelationship.Toaddressthegeneralvalidityoftheobservedeffects,wealsoanalyzeddatageneratedbytheglobalepidemicandmobilitymodel(GLEAM)(),asophisticatedepidemicsimulationframe-work(21).GLEAMincorporatesairtransporta-tionandlocalcommutertrafficonaglobalscale,isfullystochastic,andpermitsthesimulationofinfectiousstate–dependentmobilitybehavior,clin-icalstates,antiviralstatement,andmore.There-sultsofthisanalysisareshowninfigs.S17toS19andareconsistentwithourclaims.

RelativeArrivalTimesAreIndependentofEpidemicParameters

Ourresultsrevealanimportant,approximaterelationshipbetweenthesystemparameters,whichcanbesummarizedasfollows:

Ta¼DeffðPÞ=veffða,R0,g,eÞ

︸eff:distance︸eff:speed

ð6Þ

fectivedistancesDeffandeffectivespreading

speedveff,andthateachfactordependsondif-ferentparametersofthedynamicalsystem.Theepidemiologicalparametersdeterminetheeffec-tivespeed,whereaseffectivedistancedependsonlyonthetopologicalfeaturesofthestaticunderlyingnetwork,i.e.,thematrixP.Whenconfrontedwiththeoutbreakofanemergentin-fectiousdisease,oneofthekeyproblemsisthatthedisease-specificparametersaretypicallyun-knowninthebeginning,andsimulationsbasedonplausibleparameterrangestypicallyexhibitsubstantialvariabilityinpredictedoutcomes.However,Eq.6allowsustocomputerelativearrivaltimeswithoutknowledgeofthesepa-rameters.If,forexample,theoutbreaknodeislabeledk,whilenandmarearbitrarynodes,thenTa(n|k)/Ta(m|k)=Deff(n|k)/Deff(m|k).Equa-tion6statesthattheeffectivespeedveffisaglobalproperty,independentofthemobilitynet-workandtheoutbreaklocation.Thus,irrespec-tiveofmobilityandOL,onecaninvestigatehowtheeffectivespeeddependsonrateparam-etersofthesystem.

OriginofOutbreakReconstructionBasedonEffectiveDistance

Theconceptofeffectivedistanceisparticularlyvaluableforsolvingtheaforementionedin-verseproblem:Givenaspatiallydistributedprevalencepatternthatwasgeneratedby

an

Thisequationstatesthatarrivaltimescanbecomputedwithhighfidelitybasedontheef-

A

C

SVOFig.3.Qualitativeoutbreakreconstructionbasedoneffectivedistance.(A)Spatialdistributionofprevalencejn(t)attimeT=81daysforOLChicago(parametersb=0.28day–1,R0=1.9,g=2.8×10–3day–1,ande=10–6).Afterthistime,itisdifficult,ifnotimpossible,todeterminethecorrectOLfromsnapshotsofthedynamics.(B)CandidateOLschosenfromdifferentgeographicregions.(C)Panelsdepictthestateofthesystemshownin(A)fromtheperspectiveofeachcandidateOL,usingeachOL’sshortestpathtreerepresen-tation.OnlytheactualOL(ORD,circledinblue)producesacircularwavefront.EvenforcomparableNorthAmericanairports[Atlanta(ATL),Toronto(YYZ),andMexicoCity(MEX)],thewavefrontsarenotnearlyasconcentric.EffectivedistancesthuspermittheextractionofthecorrectOL,basedoninformationonthemobilitynetworkandasinglesnapshotofthedynamics.SCIENCE

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