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.