How big will an epidemic be? Illuminations from a simple model - - PowerPoint PPT Presentation

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 > NSF Columbia Earth Institute > Legg Mason Santa Fe Institute > McDonnell Foundation ISERP, 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 = Susceptible I = Infective/Infectious R = Recovered/Removed/Refractory

I R S

βI 1 − ρ ρ 1 − βI r 1 − 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.

1 2 3 4 0.2 0.4 0.6 0.8 1

R0 Fraction infected

➤ 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 1990 0.01 0.02 0.03 Date Frequency

Iceland: measles normalized count

Treat outbreaks separated in time as ‘novel’ diseases.

Iceland data—monthly counts, 1888–1990:

Epidemic size distributions for Measles, Rubella, Whooping Cough.

0.025 0.05 0.075 0.1 1 2 3 4 5 75

N(S) S

A

0.02 0.04 0.06 1 2 3 4 5 105

S

B

0.025 0.05 0.075 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 ❊①♣❡❝t ✶ ✔ ✒ ❁ ✷ ✭☞♥✐t❡ ♠❡❛♥✱ ✐♥☞♥✐t❡ ✈❛r✐❛♥❝❡✮

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:

0.025 0.05 0.075 0.1 1 2 3 4 5 75

N (ψ) ψ

A

10

−5

10

−4

10

−3

10

−2

10

−1

10 10

1

10

2

ψ N>(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 # New cases

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

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, . . .

0.25 0.5 0.75 1 500 1000 1500 2000 A

ψ N(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 xij =2 l=3 n=8 Locally: standard SIR model with random mixing β = infection probability discrete time simulation γ = recovery probability P = probability of travel Movement distance: Pr(d) ∝ exp(−d/ξ)

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

P0 = Expected number of infected individuals leaving initially infected context. Need P0 > 1 for disease to spread (independent of R0). Restricting frequency of travel and/or range limits epidemic size.

Model output: size distributions:

0.25 0.5 0.75 1 100 200 300 400 1942

ψ N(ψ)

R0=3 0.25 0.5 0.75 1 100 200 300 400 683

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 model

500 1000 1500 2000 4000 6000

t # New cases

D R0=3

Standard model with transport:

500 1000 1500 100 200

t # New cases

E R0=3 500 1000 1500 200 400

t # New cases

G R0=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

• r (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.