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The impact of mobility networks on the worldwide spread of epidemics
Alessandro Vespignani Complex Systems Group Department of Informatics Indiana University
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Weather forecast Parameters
# u is the zonal velocity (velocity in the east/west direction tangent to the sphere). # v is the meridional velocity (velocity in the north/south direction tangent to the sphere). # ω is the vertical velocity # T is the temperature # φ is the geopotential # f is the term corresponding to the Coriolis force, and is equal to 2Ωsin(φ), where Ω is the angular rotation rate of the Earth (2π / 24 radians/hour), and φ is the latitude. # R is the gas constant # p is the pressure # cp is the specific heat # J is the heat flow per unit time per unit mass # π is the exner function # θ is the potential temperature . Numerical weather prediction uses mathematical models of the atmosphere to predict the weather. Manipulating the huge datasets with the most powerful supercomputers in the world. The primitive equations can be simplified into the following equations: # Temperature: ∂T/∂t = u (∂Tx/∂X) + v (∂Ty/∂Y) + w (∂Tz/∂Z) # Wind in E-W direction: ∂u/∂t = ηv - ∂Φ/∂x – Cp θ (∂π/∂x) – z (∂u/∂σ) – [∂(u2 + y) / 2] / ∂x # Wind in N-S direction: ∂v/∂t = -η(u/v) - ∂Φ/∂y – Cp θ (∂π/∂y) – z (∂v/∂σ) – [∂(u2 + y) / 2] / ∂y # Precipitable water: ∂W/∂t = u (∂Wx/∂X) + v (∂Wy/∂Y) + z (∂Wz/∂Z) # Pressure Thickness: ∂(∂p/∂σ)/∂t = u [(∂p/∂σ)x /∂X] + v [(∂p/∂σ)y /∂Y] + z [(∂p/∂σ)z /∂Z]
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Super-computer simulations
Fracture in 1.6 millions atoms material 6.8 billion finite elements plasma Ab initio simulations thousand of atoms pico-second scale ……
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Why not forecast on… Emerging disease spreading evolution
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Wide spectrum of complications and complex features to include…
Simple Realistic Ability to explain (caveats) trends at a population level Model realism looses in transparency. Validation is harder.
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Collective human behavior….
Social phenomena involves large numbers of heterogeneous individuals over multiple time and size scales huge richness of cognitive/social science In other words The complete temperature analysis of the sea surface, and satellite images of atmospheric turbulence are easier to get than the large scale knowledge of commuting patterns or the quantitative measure of the propensity of a certain social behavior.
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Unprecedented amount of data…..
Transportation infrastructures Behavioral Networks Census data Commuting/traveling patterns Different scales: International Intra-nation (county/city/municipality) Intra-city (workplace/daily commuters/individuals behavior)
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Mobility networks
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Airport network Each edge is characterized by weight wij defined as the number of passengers in the year MSP DTW DEN ATL Atlanta ORD Chicago LAX Los Angeles DFW Dallas PHX Phoenix DEN Denver DTW Detroit MSP Minneapolis IAH Houston SFO San Francisco ORD SFO LAX PHX ATL DFW IAH
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Statistical distribution…
Skewed Heterogeneity and high variability Very large fluctuations (variance>>average)
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Computational epidemiology in
complex realities
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Mechanistic meta-population models
City i City a City j Intra-population infection dynamics by stochastic compartmental modeling
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Global spread of epidemics on the airport network
Urban areas + Air traffic flows Ravchev et al. (in russian) 1977 40-80 russian cities Ravchev, Longini. Mathematical Biosciences (1985) 50 urban areas worldwide R. Grais et al 150 urban areasin the US T. Hufnagel et al. PNAS (2004) 500 top airports Colizza, Barrat, Barthelemy, A.V. PNAS 103 (2006) 3100 urban areas+airports, 220 countries, 99% traffic
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World-wide airport network
complete IATA database V = 3100 airports E = weighted edges wij #seats / (different time scales) Nj urban area population (UN census, …) >99% of total traffic Barrat, Barthélemy, Pastor-Satorras, Vespignani. PNAS (2004)
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World-wide airport network complex properties…
Colizza, Barrat, Barthélemy, Vespignani. PNAS (2006)
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b m S I R S Homogenous mixing assumption time
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Intra-city infection dynamics
b m S I R I St+Dt = St - Binom(St , bDt It/N) It+Dt = It + Binom(St , bDt It/N) – Binom(It,mDt) Rt+Dt = Rt + Binom(It , mDt)
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Global spread of infective individuals
wjl j l Probability that any individual in the class X travel from j→l Proportional to the traffic flow Inversely proportional to the population
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Stochastic travel operator
Probability that x individuals travel from j→l given a population Xj Net balance of individuals in the class X arriving and leaving the city j
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Meta-population SIR model
Sj,t+Dt = Sj,t - Binomj(Sj,t , bDt Ij,t/N) + j (S) Ij,t+Dt = Ij,t + Binomj(Sj,t , bDt Ij,t/N) – Binomj(Ij,t,mDt) + j (I) Rj,t+Dt = Rj,t + Binomj(Ij,t , mDt) + j (R) Stochastic coupling terms = Travel 3100 x 3 differential coupled stochastic equations
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Directions….. Applications… Basic theoretical questions…
Historical data Scenarios forecast Basic theoretical questions…
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Prediction and predictability
Q1: Do we have consistent scenario with respect to different stochastic realizations? Q2: What are the network/disease features determining the predictability of epidemic outbreaks Q3:Is it possible to have epidemic forecasts? Colizza Barrat, Barthélemy, Vespignani. PNAS 103, 2015 (2006); Bulletin Math. Bio. (2006)
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Historical data : The SARS case…
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Statistical Predictions…
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Quantitatively speaking
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Correct predictions in 210 countries over 220 Quantitatively correct
How is that possible? Stochastic noise + complex network
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Taking advantage of complexity…
Two competing effects Paths degeneracy (connectivity heterogeneity) Traffic selection (heterogeneous accumulation of traffic on specific paths) Definition of epidemic pathways as a backbone of dominant connections for spreading
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100% 10% Republic of Korea United Kingdom China Japan Germany India
Taiwan Thailand Switzerland Vietnam Philippines France Italy Singapore Malaysia Spain Indonesia Australia
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Avian H5N1 Pandemic ??? reassortment mutation H3N2 H5N1 165 cases
88 deaths (Feb 6th, 2006) mutation
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Guessing exercise: similarities with influenza….
Recovered / Removed Infectious Asympt. Latent Susceptible Sympt. Not Tr. Sympt. Tr. rb b b e pa e (1-pa ) pt e (1-pa ) (1-pt ) m infectiousness I Sympt. I Asympt. S L R time (days) 1.9 3 Longini et al. Am. J. Epid. (2004)
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A convenient quantity R0 Estimates for R0 = 1.1 - 30 !!
Basic reproductive number The number of offspring cases generated by an infected individual in a susceptible population R0 Estimates for R0 = !! (most likely [ ])
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Pandemic forecast… Feb 2007 May 2007 Jul 2007 Dec 2007 Feb 2008 Apr 2008 rmax Pandemic with R0=1.6 starting from Hanoi (Vietnam) in October 2006 Baseline scenario
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Country level City level
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Containment strategies….
Travel restrictions Partial Full (country quarantine???) Antiviral Amantadine and Rimantadine (inhibit matrix proteins) Zanamivir and Oseltamivir (neuraminidase inhibitor) Vaccination Pre-vaccination to the present H5N1 Vaccine specific to the pandemic virus (6-9 months for preparation and large scale deployment)
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Travel restrictions….
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Antivirals….
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Stockpiles management
Scenario 2 Stockpiles sufficient for 10% of the population in a limited number of countries + WHO emergency supply deployment in just two countries uncooperative strategy Scenario 3 Global stockpiles management with the same amount of AV doses. Cooperative Strategy
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Use of AV stockpiles in the
different scenarios
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Cooperative versus uncooperative
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Geographical regions…
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Beneficial also for the donors
Uncooperative Beneficial also for the donors Cooperative
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What we learn… Complex global world calls for a non-local perspective
Preparedness is not just a local issue Real sharing of resources discussed by policy makers …………
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What’s for the future.. Refined census data Voronoi tassellation
2.5 arc/min resolution Global Rural-Urban Mapping Project (GRUMP) Voronoi tassellation Boundary mobility
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Boundary mobility
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World-wide scale
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# cases
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Same resolution worldwide…
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Data integration + algorithms
Stochastic epidemic models Network models Data: Census 3x105 grid population IATA Mobility (US, Europe (12), Australia, Asia) Visualization packages
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Collaborators http://cxnets.googlepages.com M. Barthelemy V. Colizza
A. Barrat M. Barthelemy R. Pastor Satorras A.J. Valleron PNAS, 103, (2006) Plos Medicine, 4, e13 (2007) Nature Physics, 3, (2007) More Information/paper/data
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