The Hidden Geometry of Complex, Network-Driven Contagion Phe(2)

tIn¼aInSn=Nn bInn¼1,…,M

(1)

whereNnisthepopulationsizeofpopulationn,Misthenumberofpopulations,andSn,In,Rn¼Nn Sn Inareabsolutenumbersofsus-ceptible,infected,andrecoveredindividuals,re-spectively.Parameterbisthemeanrecoveryrateofindividuals(forinfluenza-likediseasesb–1=3to5days),andR0=a/bisthebasicrepro-ductionratio(forwhichweassumetypicalvaluesintherange1.4to2.9).(ThefocusonSIRkineticsisnotessential,asthefollowingresultsarealsovalidforothertypesoflocaldynamics.)Eachlo-calpopulationrepresentsanodenintheglobalmobilitynetwork(GMN),depictedinFig.1A.Inadditiontothelocaldynamics,individualstravelbetweennodesaccordingtotherateequation

tUn¼∑wnmUm wmnUn

m≠n

ð2Þ

whereUnisaplaceholderfortheclassesSn,In,andRn.Thequantitieswnm=Fnm/Nmrepresenttheper-capitatrafficfluxfromlocationmton.

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WeightedlinksFnmquantifydirectairtraffic(passengersperday)fromnodemtonoden.TheGMNisconstructedfromtheworldwideairtrafficbetween4069airportswith25,453directconnections.Detailsonthedataandnetworkcon-structionareprovidedinthesupplementarymate-rials(e.g.,fig.S1andtableS1)(5,13,20,29).ThetotalnetworktrafficisapproximatelyF¼8:91Â106passengersperday.Assumingthatthetotaltrafficinandoutofanodeisproportionaltoitspopulationsize,Eqs.1and2canberewrittenas

tjn¼asnjnsðjn=eÞ bjnþg∑Pmnðjm jnÞ tsn¼ asnjnsðjn=eÞþg∑Pmnðsm snÞ

m≠n

m≠n

emanatingfromnoden,i.e.,Pmn=Fmn/Fn,whereFn¼∑Fmn.Theadditionalsigmoidfunc-tionsðxÞ¼x=ð1þxhÞwithgainparameterh>>0accountsforthelocalinvasionthresholdeandfluctuationeffectsforjn<e(30–32).Typicalparameterchoicesforeandhareh¼4,8,∞and log10e¼4,…,6.Ourresultsarerobustwithre-specttochangesintheseparameters(e.g.,figs.S5andS13).

Figure1BshowsatemporalsnapshotofthedynamicalsystemdefinedbyEq.3forahy-potheticalpandemicwithinitialoutbreakloca-tion(OL)inHongKong(HKG)(seealsoFig.2Bandfig.S2fortemporalsequencesofthedy-namicalsystemforvariousotherOLs).General-ly,themetapopulationmodelaboveandrelatedmodelsusedinthepastgeneratesolutionsthatarecharacterizedbysimilarqualitativefeatures.First,onlyduringtheearlystageoftheprocessdoestheprevalencejn(t)(i.e.,thefractionofinfectedindividuals)correlatesignificantlywithgeographicdistancefromtheOL.Second,atin-mh

termediateandlaterstages,themultiscalestructureoftheGMNinducesaspatialdecoherenceofthespreadingpattern.Third,despitetheglobalconnectivity,thespatiotemporalpatternsdonotconvergetothesamepattern,i.e.,spatiotemporaldifferencesarenotatransienteffect(figs.S3toS6andmoviesS1toS3).Thistypeofcomplexitysharplycontraststhegenericbehaviorofordinaryreaction-diffusionsystems,whichtypicallyex-hibitspatiallycoherentwavefronts.

MostProbablePathsandEffectiveDistanceThekeyideawepursuehereisthat,despitethestructuralcomplexityoftheunderlyingnetwork,theredundancyofconnections,andthemultiplic-ityofpathsacontagionphenomenoncantake,thedynamicprocessisdominatedbyasetofmostprobabletrajectoriesthatcanbederivedfromtheconnectivitymatrixP.Thishypothesisisanalogoustothedominanceofthesmallestresistorinastrong-lyheterogeneouselectricalnetworkwithparallelconductinglines.Giventheflux-fraction0≤Pmn≤1,i.e.,thefractionoftravelersthatleavenoden

and

(3)

withsn=Sn/Nn,jn=In/Nn,andrn=1–sn–jn.Adetailedderivationisprovidedinthesupplemen-tarytext.Themobilityparametergistheaveragemobilityrate,i.e.,g¼F=W,whereW¼∑nNnisthetotalpopulationinthesystem.Thisyieldsnu-mericalvaluesintherangeg=0.0013–0.0178day–1.ThematrixPwith0≤Pmn≤1quantifiesthefractionofthepassengerfluxwithdestinationm

AB

C

ETa [days]

T [days]

a

g

3

g

3

D [10 km]

g

D [10 km]

plexityinglobal,network-drivencontagionphenomena.(A)Theglobalmobilitynetwork(GMN).Graylinesrepresentpassengerflowsalongdirectconnectionsbetween4069airportsworldwide.Geographicregionsaredistinguishedbycolor[classifiedaccordingtonetworkmodularitymaximization(39)].(B)Temporalsnapshotofasimulatedglobalpandemicwithinitialoutbreaklocation(OL)inHongKong(HKG).ThesimulationisbasedonthemetapopulationmodeldefinedbyEq.3withparametersR0=1.5,b=0.285day–1,g=2.8×10–3day–1,e=10–6.Redsymbolsdepictlocationswithepidemicarrivaltimesinthetimewindow105days≤Ta≤110days.Becauseofthemultiscalestructureoftheunderlyingnetwork,thespatialdistributionofdiseaseprevalence(i.e.,thefractionofinfectedindividuals)lacksgeometriccoherence.Noclearwave-frontisvisible,andbasedonthisdynamicstate,theOLcannotbeeasilydeduced.(C)Forthesamesimulationasin(B),thepaneldepictsarrivaltimesTaasafunctionofgeographicdistanceDgfromtheOL[nodesarecoloredaccordingtogeographicregionasin(A)]foreachofthe4069nodesinthenetwork.Ona

globalscale,TaweaklycorrelateswithgeographicdistanceDg(R2=0.34).Alinearfityieldsanaverageglobalspreadingspeedofvg=331km/day(seealsofig.S7).UsingDgandvgtoestimatearrivaltimesforspecificlocations,however,doesnotworkwellowingtothestrongvariabilityofthearrivaltimesforagivengeographicdistance.Theredhorizontalbarcorrespondstothearrivaltimewindowshownin(B).(D)Arrivaltimesversusgeographicdistancefromthesource(Mexico)forthe2009H1N1pandemic.Symbolsrepresent140affectedcountries,andsymbolsizequantifiestotaltrafficpercountry.Arrivaltimesaredefinedasthedateofthefirstconfirmedcaseinagivencountryaftertheinitialoutbreakon17March2009.Asinthesimulatedscenario,arrivaltimeandgeographicdistanceareonlyweaklycorrelated(R2=0.0394).(E)Inanalogyto(D),thepaneldepictsthearrivaltimesversusgeographicdistancefromthesource(China)ofthe2003SARSepidemicfor29affectedcountriesworldwide.ArrivaltimesaretakenfromWHOpublisheddata(2).Asin(C)and(D),arrivaltimecorrelatesweaklywithgeographicdistance.SCIENCE

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arriveatnodem,wedefinetheeffectivedistancednmfromanodentoaconnectednodemas

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