Modelling global epidemics: theory and simulationsgalla/... · Modelling global epidemics: theory...
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Modelling global epidemics: theory andsimulations
Marc BarthélemyCEA, IPhT, France
Manchester meeting Modelling complex systems (21-23 june 2010)
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Outline
Introduction
Metapopulation model: applications
SARS Testing strategies
Metapopulation model: theory
Pandemic threshold
Discussion and perspectives
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Epidemiology: an interdisciplinary field
TransportationTransportation systemssystems
Biology,Biology,virologyvirology
UrbanismUrbanism
SocialSocialnetworksnetworks
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Epidemiology and statistical physics
Microscopic level (bacteria, viruses): compartments
Understanding and killing off new viruses
Quest for new vaccines and medicines
Macroscopic level (communities, species)
Integrating biology, movements and interactions
Vaccination campaigns and immunization strategies
statistical physics
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Pandemic spread modeling: past and current
Human movements and disease spread
Black death (14th)Spatial diffusionModel:
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Pandemic spread modeling: past and current Complex movement patterns: different means, different scales (SARS):
Importance of transportation networks (air travel)
Nov. 2002
Mar. 2003
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Modeling in Epidemiology: parameters,realism, simplicity, …
Pandemic modelling(metapopopulation)Social network Intra urban
spread
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Metapopulation model
Baroyan et al, 1969:≈40 russian cities
Rvachev & Longini, 1985: 50 airports worldwide
Grais et al, 1988: 150 airports in the US
Hufnagel et al, 2004: 500 top airports worldwide
Colizza et al, 2006: >99% of the total traffic
l j pop j
pop l
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Stochastic model:
• SIS model:
• SIR model:
• SI model:
λ: proba. per unit time of transmitting the infection µ: proba. per unit time of recovering
One population: Simple models of epidemics
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Stochastic model:
• SIS or SIR model (mean-field):
λ: proba. per unit time of transmitting the infection µ: proba. per unit time of recovering
One population: Simple models of epidemics
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Modeling: SIR with spatial diffusion i(x,t):
where: λ = transmission coefficient (fleas->rats->humans) µ=1/average infectious period S0=population density, D=diffusion coefficient
One population: diffusion effect
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Metapopulation model(mean-field)
Rvachev Longini (1985)
Flahault & Valleron (1985); Hufnagel et al, PNAS 2004, Colizza, Barrat, Barthelemy, VespignaniPNAS 2006, BMB, 2006. Theory: Colizza & Vespignani, Gautreau & al, …
Inner city term(one population homogeneous mixing)
Travel term(network)
Transport operator (mean-field):
Reaction-diffusion modelsFKPP equation
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Stochastic model
compartmental model + air transportation data
Susceptible
Infected
Recovered
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Metapopulation model
!
pPAR,FCO ="PAR,FCONPAR
#t
Travel probabilityfrom PAR to FCO:
ξPAR,FCO
ξPAR,FCO
# passengersfrom PAR to FCO(input data)
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Metapopulation model(mean-field)
Reaction-diffusion models Reaction at each node Diffusion
FKPP equation (continuous limit, d=1)
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Metapopulation model: Applications
Testing against historical examples
SARS (2003)
Testing strategies
Antivirals: cooperative versus egoistic strategies
Travel restrictions
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Application: SARS
pop j
pop i
refined compartmentalization
parameter estimation: clinical data + local fit
geotemporal initial conditions: available empirical data
modeling intervention measures: standard effective modeling
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SARS: predictions
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SARS: predictions (2)
Colizza, Barrat, Barthelemy & Vespignani, bmc med (2007)
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More from SARS - Epidemic pathways
For every infected country:
where is the epidemic coming from ?
- Redo the simulation for many disorder realizations(same initial conditions)
- Monitor the occurrence of the paths(source-infected country)
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Existence of pathways:
confirms the possibility of epidemic forecasting !
Useful information for control strategies
SARS- what did we learn ?
Metapopulation model, no tunable parameter:
good agreement with WHO data !
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Application of the metapopulation model: effect ofantivirals
Threat:Flu
Question: use of antivirals
Best strategy for the countries ?
Model:
Etiology of the disease (compartments)
Metapopulation+Transportation mode (air travel)
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Predictions: pandemic flu
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Effect of antivirals
Comparison of strategies
“Uncooperative”: each country stockpiles AV
“Cooperative”: each country gives 10% (20%) of its own stock
Baseline: reference point (no antivirals)
Travel restrictions
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Travel limitations….
Colizza, Barrat, Barthélemy, Valleron, Vespignani. PLoS Medicine (2007)
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Effect of antivirals: Strategy comparison
Best strategy: Cooperative !
Colizza, Barrat, Barthelemy, Valleron, Vespignani, PLoS Med (2007)
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Metapopulation model: theory
Theoretical questions
Effect of heterogeneity on the predictability
Arrival time ?
Epidemic threshold ? At what conditions can a diseasebecome a pandemic ?
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One outbreak realization:
Another outbreak realization ? Effect of noise ?
? ? ? ? ? ?
Predictability
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Overlap measure
Similarity between 2 outbreak realizations:
Overlap function
time t time t
1)( =! t
time t time t
1)( <! t
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Predictability
no degree fluctuationsno weight fluctuations
+ degreeheterogeneity
+ weightheterogeneity
Colizza, Barrat, Barthelemy & Vespignani, PNAS (2006)
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Effect of heterogeneity:
degree heterogeneity: decreases predictability
Weight heterogeneity: increases predictability !
Good news: Existence of preferred channels !
Epidemic forecast, risk analysis of containment strategies
j
lwjl
Predictability
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Once a city is infected:
- what is the condition for non-extinction in the city ? (R0>1)
- will it spread to other cities ?
- will it invade the whole world ?
- can we express a condition of the form R*>1 ?
Pandemic threshold
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One population: SIR (stochastic version)
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• Epidemic threshold =critical point• Prevalence i =orderparameter
Epidemic Threshold λc (SIR, SIS,…)
i
λλ c
Active phaseAbsorbingphase
Finite prevalenceVirus death
Density of infected
One population: main results
Basic reproductive number:
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Probability of extinction (stochastic version):
One population: main results
Number of infected individuals at t=0
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Assumptions:
- Cities with the same population N
- p: probability per unit time for any individual to jump from one node to one of its neighbor
Many populations connectedthough a network
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Assumptions (cont’d): Uncorrelated, complexnetwork:
- Degree distribution P(k), moments
- Percolation threshold:
- Scale-free networks:
Many populations connectedthough a network
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Building block: two cities
City 0 City 1
Proba p
Proba q
p=0.01, q=0R0=2
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Arrival time probability in city 1 of an infected individual:
Building block: two cities
Cumulative:
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Condition for a network
Condition for a pandemic spread:
Explicitely (SIR):
Barthelemy, Godreche, Luck (2010)
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Consequences (1)
Scale-free network:
Travel restrictions inefficient !
In agreement with a simple mean-field argument[Colizza & Vespignani, PRL (2007)]
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Consequences (2)
d=1: pc=1 finite cluster
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Consequences (3)
d=2 and Bethe lattice: pc<1
Fraction ofinfected
cities
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Conclusions
Metapopulation model
Works for modeling pandemic spread
Mostly numerical results, many theoretical problems(reaction-diffusion on networks)
Existence of a “pandemic threshold” connected topercolation
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Perspectives
Population and travel probabilities (broadly)distributed: effect on the pandemic threshold ?
Challenges of modern epidemiology:
Smaller scales ? (urban area)
Human mobility and city structure: statisticalcharacterization ? Models ?
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Numerical studies on the metapopulation model
- A. Barrat (CPT, Marseille)
- V. Colizza (ISI, Turin)
- A.-J. Valleron (Inserm, Paris)
- A. Vespignani (IU, Bloomington)
Theoretical analysis of the metapopulation model
- C. Godrèche (IPhT, CEA)
- J.-M. Luck (IPhT, CEA)
Collaborators (metapopulation model)
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Thank you.