The impact of mobility networks on the worldwide spread of epidemics

Alessandro Vespignani

Complex Systems Group Department of Informatics Indiana University

Weather forecast

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] 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

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 Ability to explain (caveats) trends at a population level Realistic 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 SFO LAX MSP DEN PHX DFW IAH ORD DTW ATL ATL Atlanta ORD Chicago LAX Los Angeles DFW Dallas PHX Phoenix DEN Denver DTW Detroit MSP Minneapolis IAH Houston SFO San Francisco

Statistical distribution…

 Skewed  Heterogeneity and high variability  Very large fluctuations (variance>>average)

Computational epidemiology in 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)

40-80 russian cities 1977

•

R. Grais et al 150 urban areasin the US

•

Ravchev, Longini. Mathematical Biosciences (1985)

T. Hufnagel et al. PNAS (2004)

50 urban areas worldwide 500 top airports

Colizza, Barrat, Barthelemy, A.V. PNAS 103 (2006)

3100 urban areas+airports, 220 countries, 99% traffic

World-wide airport network



complete IATA database



V = 3100

airports

 

E = 17182

weighted edges

w ij

#seats / (different time scales)



N j

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)

S

b

I

m

Homogenous mixing assumption R S

time

Intra-city infection dynamics

b

S I

m

R I

S t+

D

t = S t - Binom(S t ,

bD

t I t /N) I t+

D

t = I t + Binom(S t ,

bD

t I t /N) – Binom(I t ,

mD

t) R t+

D

t = R t + Binom(I t ,

mD

t)

Global spread of infective individuals

w jl l j

 Probability that any individual in the class X travel from

j

→

l

  Proportional to the traffic flow Inversely proportional to the population

p jl



w jl

D

t N j

Stochastic travel operator

P

({ x

l

})  (

X j

 

l X

x

jl j

!

)!



l

x

jl

!



l p

x

jl jl

  1  

l p jl

  (

X j

 

l

x

jl

)  Probability that x population

X j

individuals travel from

j→l

given a 

j

({

X

}) 

l

  x

jl

(

X j

)  x

lj

(

X l

)   Net balance of individuals in the class

X

leaving the city j arriving and

Meta-population SIR model

S j,t+Dt = S j,t - Binom j ( S j,t ,

bD

t I j,t /N ) +



j (S) I j,t+Dt = I j,t + Binom j ( S j,t ,

bD

t I j,t /N ) – Binom j ( I j,t ,

mD

t) +



j (I) R j,t+Dt = R j,t + Binom j (I j,t ,

mD

t ) +



j (R)

Stochastic coupling terms = Travel 

3100 x 3 differential coupled stochastic equations

Directions…..



Applications…



Historical data



Scenarios forecast

 Basic

theoretical

questions…

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

100% 10% United Kingdom Germany Spain France Switzerland Italy China India Thailand Vietnam Singapore Malaysia Republic of Korea Japan Taiwan Philippines Indonesia Australia

Avian H5N1 Pandemic ???

H3N2 H5N1 165 cases 88 deaths

(Feb 6 th , 2006)

mutation reassortment

Guessing exercise: similarities with influenza….

S L 1.9

I Sympt.

I Asympt.

3 R time (days) Infectious Asympt.

r

b b

Susceptible

b

Latent

e

(1-p a ) p t

Infectious Sympt. Tr.

m e

p a

Infectious Asympt.

m e

Infectious Sympt. Not Tr.

Infectious Sympt. Tr.

(1-p a ) (1-p t )

Infectious Sympt. Not Tr.

m

Recovered / Removed Longini et al. Am. J. Epid. (2004)

A convenient quantity

 Basic reproductive number

R 0

 The number of offspring cases generated by an infected individual in a susceptible population Estimates for R 0 = 1.1 - 30 !!

(most likely [1.5 - 3.0])

Pandemic forecast… Feb 2007 May 2007 Jul 2007 Dec 2007 Feb 2008 Apr 2008 Pandemic with

R 0 =1.6

Baseline scenario starting from Hanoi (Vietnam) in October 2006 0

r

max

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…

Uncooperative

Beneficial also for the donors

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

Same resolution worldwide…

Data integration + algorithms

 Stochastic epidemic models  Network models  Data:   Census 3x10 5 grid population   IATA Mobility (US, Europe (12), Australia, Asia)  Visualization packages

Collaborators

 

V. Colizza



M. Barthelemy A. Barrat

 •

A.J. Valleron R. Pastor Satorras

 PNAS,

103

, 2015-2020 (2006)  Plos

Medicine

,

4

, e13 (2007)  Nature Physics,

3,

276-282 (2007)

More Information/paper/data

http://cxnets.googlepages.com