The impact of mobility networks on the worldwide

spread of epidemics

Alessandro Vespignani

Complex Systems GroupDepartment of InformaticsIndiana University

Complex Systems GroupDepartment of InformaticsIndiana University

Weather forecast

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]

# 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...

Parameters

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.

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• ……

Why not forecast on…

Emerging disease spreading evolution

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.

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.

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)

Mobility networks

Airport network

Each edge is characterized by weight wij defined as the number of passengers in the year

LAX

ORD

ATL

DFW

SFO

DEN

PHX

DTW

IAH

MSP

ATL AtlantaORD ChicagoLAX Los AngelesDFW DallasPHX PhoenixDEN DenverDTW DetroitMSP MinneapolisIAH HoustonSFO San Francisco

Statistical distribution…

Skewed Heterogeneity and high

variability Very large fluctuations

(variance>>average)

Computational epidemiology in complex realities

Mechanistic meta-population models

City a

City j

City i

Intra-population infection dynamics by stochastic compartmental

modeling

• Ravchev et al. (in russian) 197740-80 russian cities

• Ravchev, Longini. Mathematical Biosciences (1985)50 urban areas worldwide

Global spread of epidemics on the airport network

Urban areas+

Air traffic flows

•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

>99% of total traffic

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

complete IATA database V = 3100 airports E = 17182 weighted edges wij #seats / (different time scales)

Nj urban area population

(UN census, …)

World-wide airport networkWorld-wide airport network

World-wide airport network complex properties…

Colizza, Barrat, Barthélemy, Vespignani. PNAS (2006)

S I R

time

Homogenous mixing assumption

S

Intra-city infection dynamics

St+t = St - Binom(St , t It/N)

It+t = It + Binom(St , tIt/N) – Binom(It,t)

Rt+t = Rt + Binom(It , t)

S II

R

Global spread of infective individuals

jlwjl

tN

wp

j

jljl

Probability that any individual in the class X travel from j→l

Proportional to the traffic flow Inversely proportional to the population

Stochastic travel operator

Probability that individuals travel from j→l given a population Xj

l

lljjjlj XXX )()(})({

l

X

ljljl

ljl

ljlj

jl

l jlj

jl ppX

XP

)(

1!)!(

!})({

Net balance of individuals in the class X arriving and leaving the city j

Meta-population SIR model

Stochasticcoupling terms

=Travel

3100 x 3 differential coupled stochastic equations

Sj,t+Dt = Sj,t - Binomj(Sj,t , t Ij,t/N) + j (S)

Ij,t+Dt = Ij,t + Binomj(Sj,t , tIj,t/N) – Binomj(Ij,t,t) + j (I)

Rj,t+Dt = Rj,t + Binomj(Ij,t , t) + j (R)

Directions…..

Basic theoretical questions…

Applications… Historical data Scenarios forecast

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)

Historical data :The SARS case…

Statistical Predictions…

Quantitatively speaking

Correct predictions in 210 countries over 220 Quantitatively correct

How is that possible?Stochastic noise + complex network

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

10%

100%

Germany

FranceItaly

Switzerland

United Kingdom

Spain

Australia

China JapanIndia

Indonesia

Malaysia

PhilippinesThailand

Vietnam

Republicof Korea

Taiwan

Singapore

Avian H5N1 Pandemic ???Avian H5N1 Pandemic ???

H5N1

H3N2

reassortmentreassortment

mutationmutation

165 cases88 deaths(Feb 6th, 2006)

Recovered /Removed

InfectiousAsympt.

Latent

Susceptible

InfectiousSympt. Not Tr.

InfectiousSympt. Tr.

InfectiousAsympt. Infectious

Sympt. Tr.

InfectiousSympt. Not Tr.r

pa(1-pa ) pt (1-pa ) (1-pt )

time(days)

S

L R

I Sympt.

I Asympt.

1.9 3

Longini et al. Am. J. Epid. (2004)

infe

cti

ou

sn

ess

Guessing exercise: similarities with influenza….

A convenient quantity

Basic reproductive number

The number of offspring cases generated by an infected individual in a susceptible population

R0

Estimates for R0 = 1.1 - 30 !!(most likely [1.5 - 3.0])

Feb 2007 May 2007 Jul 2007

Dec 2007 Feb 2008 Apr 20080

max

Pandemic with R0=1.6 starting from Hanoi (Vietnam) in October 2006Baseline scenario

Pandemic forecast…

Country level

City level

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)

Travel restrictions….

Antivirals….

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

Use of AV stockpiles in the different scenarios

Cooperative versus uncooperative

Geographical regions…

Beneficial also for the donors

Uncooperative

Cooperative

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 …………

What’s for the future..

Refined census data 2.5 arc/min resolution Global Rural-Urban

Mapping Project (GRUMP)

Voronoi tassellation

Boundary mobility

Boundary mobility

World-wide scale

# ca

ses

Same resolution worldwide…

Data integration + algorithms

Stochastic epidemic models Network models Data:

Census 3x105 grid population IATA Mobility (US, Europe (12), Australia, Asia)

Visualization packages

Collaborators

V. Colizza A. Barrat

M. Barthelemy R. Pastor Satorras

More Information/paper/data

http://cxnets.googlepages.com

• A.J. Valleron

PNAS, 103, 2015-2020 (2006) Plos Medicine, 4, e13 (2007) Nature Physics, 3, 276-282 (2007)