DIMACS Influenza Workshop, January 25th, 2006

How big will an epidemic be?Illuminations from a simple model

Collective Dynamics Group, Columbia University

Duncan Watts, Roby Muhamad, Daniel Medina, Peter Dodds

Sociology, Columbia > NSFColumbia Earth Institute > Legg MasonSanta Fe Institute > McDonnell FoundationISERP, Columbia > Office of Naval Research

Main questions:

For novel diseases:

A. Can we predict the size of an epidemic?

B. How important is the reproduction number R0?

Overview:

➊ The reproduction number R0.

➋ Some important aspects of real epidemics:

☛ peculiar distributions of epidemic size

☛ resurgence

➌ Usual simple models do not fare well regarding ➋

➍ A simple model incorporating movement does fare well.

➎ Conclusion.

Compartment models:

SIR model of infectious disease dynamics:Three basic states:

S = SusceptibleI = Infective/InfectiousR = Recovered/Removed/Refractory

I

R

SβI

1 − ρ

ρ

1 − βI

r1 − r

R0 = expected number of infections due to one infective.

R0 has become a summary statistic for epidemics.

Epidemic threshold:

If R0 > 1, ‘epidemic’ occurs.

0 1 2 3 40

0.2

0.4

0.6

0.8

1

R0

Frac

tion

infe

cted

➤ Fine idea from a simple model.

R0 and variation in epidemic sizes:

R0 approximately same for all of the following:

1918-19 “Spanish Flu” ∼ 500,000 deaths in US

1957-58 “Asian Flu” ∼ 70,000 deaths in US

1968-69 “Hong Kong Flu” ∼ 34,000 deaths in US

2003 “SARS Epidemic” ∼ 800 deaths world-wide

Size distributions:

Size distributions are important elsewhere:

➤ earthquakes (Gutenberg-Richter law)

➤ wealth distributions

➤ city sizes, forest fires, war fatalities

➤ ‘popularity’ (books, music, websites, ideas)

(power laws distributions are common but not obligatory)

Size distributions:

What about epidemics?

➤ Simply hasn’t attracted much attention.

➤ Data not as clean as for other phenomena.

Iceland data—monthly counts, 1888–1990:

Caseload recorded monthly for range of diseases in Iceland.

1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 19900

0.01

0.02

0.03

Date

Freq

uenc

y

Iceland: measlesnormalized count

Treat outbreaks separated in time as ‘novel’ diseases.

Iceland data—monthly counts, 1888–1990:

Epidemic size distributions for

Measles, Rubella, Whooping Cough.

0 0.025 0.05 0.075 0.10

1

2

3

4

5

75

N(S

)

S

A

0 0.02 0.04 0.060

1

2

3

4

5

105

S

B

0 0.025 0.05 0.0750

1

2

3

4

5

75

S

C

Spike near S = 0, relatively flat otherwise.

Power law distributions:

X = event size.

Probability distribution function’s tail for large x:

Pr(X = x) ∝ x−θ−1

Complementary cumulative distribution function:

Pr(X > x) =∫ ∞x

Pr(X = x′) dx′ ∝ x−θ

Expect 1 ≤ θ < 2 (finite mean, infinite variance)

Power law distributions:

X = event size.

Probability distribution function’s tail for large x:

Pr(X = x) ∝ x−θ−1

Complementary cumulative distribution function:

Pr(X > x) =∫ ∞x

Pr(X = x′) dx′ ∝ x−θ

Expect 1 � � < 2 (�nite mean, in�nite variance)

Power law distributions:

X = event size.

Probability distribution function’s tail for large x:

Pr(X = x) ∝ x−θ−1

Complementary cumulative distribution function:

Pr(X > x) =∫ ∞x

Pr(X = x′) dx′ ∝ x−θ

Expect 1 ≤ θ < 2 (finite mean, infinite variance)

Iceland data—monthly counts, 1888–1990:

Measles:

0 0.025 0.05 0.075 0.10

1

2

3

4

5

75

N (

ψ)

ψ

A

10−5

10−4

10−3

10−2

10−1

100

101

102

ψN

>(ψ)

Complementary cdf: Pr(X > x) ∝ x−θ

θ = 0.40 and 0.13, both outside of range 1 ≤ θ < 2.

⇒ Distribution seems ‘flat.’

Resurgence—example of SARS:

D

Date of onset

# N

ew c

ases

Nov 16, ’02 Dec 16, ’02 Jan 15, ’03 Feb 14, ’03 Mar 16, ’03 Apr 15, ’03 May 15, ’03 Jun 14, ’03

160

120

80

40

0

Epidemic slows... Then an infective moves to a new context.

Epidemic discovers new ‘pools’ of susceptibles: Resurgence.

➤ Importance of rare, stochastic events.

So...

Can existing simple models produce

broad epidemic distributions + resurgence?

Size distributions:

No: Simple models typically

produce bimodal or

unimodal size distributions.

Including network models:

random, small-world,

scale-free, . . .0 0.25 0.5 0.75 1

0

500

1000

1500

2000A

ψN

(ψ)

R0=3

Burning through the population:

Forest fire models: (Rhodes & Anderson, 1996).

The physicist’s approach:

“if it works for magnets, it’ll work for people...”

Bit of a stretch:

1. epidemics ≡ forest fires

spreading on 3-d and 5-d lattices.

2. Claim Iceland and Faroe Islands exhibit power law

distributions for outbreaks.

3. Original model not completely understood.

Sophisticated metapopulation models:

Community based mixing: Longini (two scales).

Eubank et al.’s EpiSims/TRANSIMS—city simulations.

Spreading through countries—Airlines: Germann et al.,

Vital work but perhaps hard to generalize from...

⇒ Create a simple model involving muliscale travel.

A toy agent-based model:

b=2

i j

x ij =2l=3

n=8

Locally: standard SIR model with random mixing

β = infection probability discrete time simulationγ = recovery probability

P = probability of travelMovement distance: Pr(d) ∝ exp(−d/ξ)

Model output: varying P0 (left) and ξ (right):

P0 = Expected number of infected individuals leaving initiallyinfected context.

Need P0 > 1 for disease to spread (independent of R0).

Restricting frequency of traveland/or range limits epidemic size.

Model output: size distributions:

0 0.25 0.5 0.75 10

100

200

300

400

1942

ψ

N(ψ

)

R0=3

0 0.25 0.5 0.75 10

100

200

300

400

683

ψ

N(ψ

)

R0=12

Flat distributions are possible for certain ξ and P0.

➤ Different R0’s may produce similar distributions.

➤ Same epidemic sizes may arise from different R0’s.

Model output—resurgence:

Standard

model0 500 1000 1500

0

2000

4000

6000

t#

New

cas

es D R0=3

Standard

model with

transport: 0 500 1000 15000

100

200

t

# N

ew c

ases E R

0=3

0 500 1000 15000

200

400

t

# N

ew c

ases G R

0=3

Conclusions-I:

Results suggest that for novel diseases:

A. Epidemic size unpredictable.

B. The reproduction number R0 is not overly useful.

R0 summarises one epidemic after the fact

and enfolds transport, everything.

R0, however measured, is not informative

about (A) how likely the observed epidemic size was

or (B) how likely future epidemics will be.

Conclusions—II:

Disease spread highly sensitive to population structure.

Rare events may matter enormously

(e.g., an infected individual taking an international flight).

More support for controlling population movement

(e.g., travel advisories, quarantine).

Reference:

“Multiscale, resurgent epidemics in a hierarchical

metapopulation model.”

D.J. Watts, R. Muhamad, D. Medina, and P.S. Dodds

Proc. Natl. Acad. Sci., Vol. 102(32), pp. 11157-11162.