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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
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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
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Nov. 2002
Mar. 2003
Epidemic spread: Epidemic spread: nowadaysnowadays
SARSSARS
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Epidemic spread: Epidemic spread: nowadaysnowadays
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Modeling of global epidemicsModeling of global epidemics
Ravchev, Longini. Mathematical Biosciences (1985)
multi-level description :
intra-city epidemics
inter-city travel
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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)
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World-wide airport networkWorld-wide airport network<k> = 9.75 kmax = 318
<w> = 74584.4 wmin = 4wmax = 6.167e+06
Frankfurt
Sapporo - Tokyo
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Broad distributions strong heterogeneities3 different levels:
degree
weight
population
World-wide airport networkWorld-wide airport networksummarysummary
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Epidemics: Stochastic ModelEpidemics: Stochastic Model
compartmental model + air transportation data
NN11
NN22
NN00
NN55
NN44
NN33
ww5454
ww4545SIR model
Susceptible
Infected
Recovered
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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
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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
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Stochastic ModelStochastic ModelIntra-cityIntra-city
S
I
R
Independent Gaussian noises
Homogeneous assumption
rate of transmission -1 average infectious period
;XKX
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compartmental model + air transportation data
SIR model
XXKX
Intra-cities Inter-cities
Epidemics: Stochastic ModelEpidemics: Stochastic Modelsummarysummary
Does it work ?
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Case study: SARSCase study: SARS
SusceptibleSusceptible
LatentLatent
InfectedInfected
HospitalizedRHospitalizedRHospitalizedD
HospitalizedD
RecoveredRecoveredDeadDead
d (1-d)
D R
InfectedInfected
HospitalizedHospitalized
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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
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Case study: SARS Case study: SARS results results
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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
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Results: Geographic spreadResults: Geographic spread
Epidemics starting in Hong Kong
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Results: Geographic spreadResults: Geographic spread
Epidemics starting in Hong Kong
Gastner, Newman. PNAS (2004)
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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
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maps heterogeneity epidemic spread
appropriate measure
role of specific structural properties: topology, traffic, population
comparison with null hypothesis
11stst PART: Heterogeneity PART: Heterogeneity
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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>
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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.
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Results: Epidemic heterogeneityResults: Epidemic heterogeneity
global properties
average over initial seed
central zone: H>0.9
HETk WAN importance of P(k)
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Results: Epidemic heterogeneityResults: Epidemic heterogeneity
epidemics starting from a given city
average entropy profile + maximal dispersion
noise: small effect
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Results: Epidemic heterogeneityResults: Epidemic heterogeneity
epidemics starting from a given city
percentage of infected cities
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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
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PredictabilityPredictability
normalized probability
Similarity between 2 outbreak realizations:
Hellinger affinity Overlap function
),(t
ll
jj tI
tIt
)(
)()(
j
IIj
Ijt )(
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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
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Results: PredictabilityResults: Predictability
left: seed = airport hubs right: seed = poorly connected airports
HOM & HETw high overlap
HETk low overlap
WAN increased overlap !!
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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.
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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