Are global epidemicsAre global epidemicspredictable ?predictable ?

V. ColizzaV. Colizza School of Informatics, Indiana School of Informatics, Indiana University, USAUniversity, USA

M. BarthélemyM. Barthélemy School of Informatics, Indiana School of Informatics, Indiana University, USA University, USA

A. BarratA. Barrat Universite Paris-Sud, FranceUniversite Paris-Sud, France

A. VespignaniA. Vespignani School of Informatics, Indiana School of Informatics, Indiana University, USA University, USA “Networks and Complex Systems” talk series

Epidemic spread: Epidemic spread: 1414thth century century

Dec. 1347

Dec. 1347June 1348

Dec. 1348

June 1349

Dec. 1349

June 1350

Dec. 1350

Dec. 1347

Dec. 1350

Black DeathBlack Death

Nov. 2002

Mar. 2003

Epidemic spread: Epidemic spread: nowadaysnowadays

SARSSARS

Epidemic spread: Epidemic spread: nowadaysnowadays

Modeling of global epidemicsModeling of global epidemics

Ravchev, Longini. Mathematical Biosciences (1985)

multi-level description :

intra-city epidemics

inter-city travel

World-wide airport networkWorld-wide airport network

complete 2002 IATA database

V = 3880 airports E = 18810 weighted edges wij #seats / year

Nj urban area population (UN census, …)

V = 3100 airportsE = 17182 weighted edges

>99% of total traffic

Barrat, Barthélemy, Pastor-Satorras, Vespignani. PNAS (2004)

World-wide airport networkWorld-wide airport network<k> = 9.75 kmax = 318

<w> = 74584.4 wmin = 4wmax = 6.167e+06

Frankfurt

Sapporo - Tokyo

Broad distributions strong heterogeneities3 different levels:

degree

weight

population

World-wide airport networkWorld-wide airport networksummarysummary

Epidemics: Stochastic ModelEpidemics: Stochastic Model

compartmental model + air transportation data

NN11

NN22

NN00

NN55

NN44

NN33

ww5454

ww4545SIR model

Susceptible

Infected

Recovered

Stochastic ModelStochastic ModelTravel termTravel term

j

l wjl

tN

wp

j

jljl Travel probability

from j to l

)( jjl X # passengers in class X from j to l multinomial distr.

jl

Stochastic ModelStochastic ModelTravel termTravel term

j

l wjl

%70)1( jlnoisejl ww

l

ljljljj XXX )()(})({

Transport operator:Transport operator:

other source of noise:

two-legs travel: XXX jjj)2()1(

outgoing ingoing

Stochastic ModelStochastic ModelIntra-cityIntra-city

S

I

R

Independent Gaussian noises

Homogeneous assumption

rate of transmission -1 average infectious period

;XKX

compartmental model + air transportation data

SIR model

XXKX

Intra-cities Inter-cities

Epidemics: Stochastic ModelEpidemics: Stochastic Modelsummarysummary

Does it work ?

Case study: SARSCase study: SARS

SusceptibleSusceptible

LatentLatent

InfectedInfected

HospitalizedRHospitalizedRHospitalizedD

HospitalizedD

RecoveredRecoveredDeadDead

d (1-d)

D R

InfectedInfected

HospitalizedHospitalized

Case study: SARSCase study: SARS data: WHO reported cases final report:

• 28 infected countries• 8095 infected cases• 774 deaths

refined compartmentalization parameter estimation:

• literature• best fit

initial condition:• t=0 Feb. 21st

• seed: Hong Kong• I0=1, L0 estimated, S0=N

Case study: SARS Case study: SARS results results

statistical propertiesstatistical propertiesepidemic pattern ?epidemic pattern ?

strong heterogeneity in no. infected cases: 0-103

large fluctuations

Full scale computational study of global epidemics:

statistical properties epidemic pattern effect of complexity of transportation network forecast reliability

SIR model

Results: Geographic spreadResults: Geographic spread

Epidemics starting in Hong Kong

Results: Geographic spreadResults: Geographic spread

Epidemics starting in Hong Kong

Gastner, Newman. PNAS (2004)

Results: Geographic spreadResults: Geographic spread

Epidemics starting in Hong Kong

t=24

day

s

t=48

day

s

t=56

day

s

t=66

day

s

t=16

0 d

ays

maps heterogeneity epidemic spread

appropriate measure

role of specific structural properties: topology, traffic, population

comparison with null hypothesis

11stst PART: Heterogeneity PART: Heterogeneity

Epidemic heterogeneityEpidemic heterogeneityand Network structureand Network structure

HOM HOM P(k)

<k>

P(N)

<N>

P(w)

<w>

WAN WAN WAN WAN

P(w)

<w>

P(k)

k

P(N)

<N>

HETHETkk

P(k) P(w)

w

P(N)

N

HETHETw w

<k>

Epidemic heterogeneityEpidemic heterogeneity

j

jjVtH ln

ln

1)(

j

jj N

tIti

)()(

ll

jj ti

tit

)(

)()(

Entropy:Entropy:

prevalence in city j at time t

normalized prevalence

H [0,1]H=0 most het.H=1 most hom.

Results: Epidemic heterogeneityResults: Epidemic heterogeneity

global properties

average over initial seed

central zone: H>0.9

HETk WAN importance of P(k)

Results: Epidemic heterogeneityResults: Epidemic heterogeneity

epidemics starting from a given city

average entropy profile + maximal dispersion

noise: small effect

Results: Epidemic heterogeneityResults: Epidemic heterogeneity

epidemics starting from a given city

percentage of infected cities

22ndnd PART: Predictability PART: Predictability

t=24

day

s

t=48

day

s

t=56

day

s

t=66

day

s

t=16

0 d

ays

time

One outbreak realization:

Another outbreak realization ?

t=24

day

s

t=48

day

s

t=56

day

s

t=66

day

s

t=16

0 d

ays

? ? ? ? ?

epidemic forecast containment strategies

PredictabilityPredictability

normalized probability

Similarity between 2 outbreak realizations:

Hellinger affinity Overlap function

),(t

ll

jj tI

tIt

)(

)()(

j

IIj

Ijt )(

PredictabilityPredictability

j

IIj

Ijt )(

0)(

1)(

t

t 2 identical outbreaks

2 distinct outbreaks

time t

time t

time t

time t

]1,0[)( t

Results: PredictabilityResults: Predictability

left: seed = airport hubs right: seed = poorly connected airports

HOM & HETw high overlap

HETk low overlap

WAN increased overlap !!

Results: PredictabilityResults: Predictability

j

l wjl

j

l wjl

HOM: few channels high overlap

kki

HETk: broad P(k) lots of channels! low overlap

WAN: broad P(k),P(w) lots of channels! emergence of preferred channels increased overlap !!!

+ degree heterog.

+ weight heterog.

ConclusionsConclusions air transportation network properties

global pattern of emerging disease spatio-temporal heterogeneity of epidemic pattern

quantitative measurement of the predictability of epidemic pattern

epidemic forecast, risk analysis of containment strategies

Ref.: http://arxiv.org/ qbio/0507029