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The impact of mobility networks on the worldwide spread of epidemics

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## The impact of mobility networks on the worldwide spread of epidemics

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### The impact of mobility networks on the worldwide spread of epidemics

Department of Informatics

Indiana University

Alessandro Vespignani

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]

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)

ORD Chicago

LAX Los Angeles

DFW Dallas

PHX Phoenix

DEN Denver

DTW Detroit

MSP Minneapolis

IAH Houston

SFO San Francisco

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

MSP

DTW

DEN

ORD

SFO

ATL

PHX

LAX

DFW

IAH

Statistical distribution…

- Skewed
- Heterogeneity and high variability
- Very large fluctuations

(variance>>average)

complex realities

Mechanistic meta-population models

City i

City a

City j

Intra-population infection dynamics by stochastic compartmental modeling

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

World-wide airport network

- completeIATA database
- V = 3100 airports
- E = 17182 weighted edges
- wij #seats / (different time scales)
- Nj urban area population

(UN census, …)

>99% of total traffic

Barrat, Barthélemy, Pastor-Satorras,

Vespignani. PNAS (2004)

World-wide airport network complex properties…

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

b

m

S

I

R

I

St+Dt = St - Binom(St , bDt It/N)

It+Dt = It + Binom(St , bDtIt/N) – Binom(It,mDt)

Rt+Dt = Rt + Binom(It , mDt)

Global spread of infective individuals

wjl

- Probability that any individual in the class X travel from j→l
- Proportional to the traffic flow
- Inversely proportional to the population

j

l

Stochastic travel operator

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

- Net balance of individuals in the class Xarriving and leaving the city j

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 , bDtIj,t/N) – Binomj(Ij,t,mDt) + j (I)

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

- 3100 x 3 differential coupled stochastic equations

Stochastic

coupling terms

=

Travel

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)

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%

Republic

of Korea

China

United

Kingdom

Japan

India

Germany

Taiwan

Thailand

Vietnam

Switzerland

Philippines

France

Malaysia

Italy

Singapore

Spain

Indonesia

Australia

Infectious

Sympt. Not Tr.

rbb

b

Infectious

Asympt.

Infectious

Sympt. Tr.

Latent

e (1-pa ) pt

e (1-pa ) (1-pt )

e pa

Infectious

Sympt. Tr.

Infectious

Asympt.

Infectious

Sympt. Not Tr.

m

m

m

Recovered /

Removed

Guessing exercise: similarities with influenza….I Sympt.

infectiousness

I Asympt.

S

L

R

time

(days)

1.9

3

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

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

Apr 2008

Feb 2008

0

Pandemic forecast…Pandemic with R0=1.6 starting from Hanoi (Vietnam) in October 2006

Baseline scenario

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)

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

different scenarios

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

Data integration + algorithms

- Stochastic epidemic models
- Network models
- Data:
- Census
- 3x105 grid population
- IATA
- Mobility (US, Europe (12), Australia, Asia)
- Visualization packages

V. Colizza

A. Barrat

M. Barthelemy

R. Pastor Satorras

Collaborators- A.J. Valleron

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

More Information/paper/data

http://cxnets.googlepages.com

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