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Biological ContagionPrinciples of Complex Systems
CSYS/MATH 300, Fall, 2011
Prof. Peter Dodds
Department of Mathematics & Statistics | Center for Complex Systems |Vermont Advanced Computing Center | University of Vermont
Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.
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Outline
Introduction
Simple disease spreading modelsBackgroundPredictionMore modelsToy metapopulation modelsModel outputConclusionsPredicting social catastrophe
References
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Contagion
A confusion of contagions:I Was Harry Potter some kind of virus?I What about the Da Vinci Code?I Did Sudoku spread like a disease?I Language?I Religion?I Democracy...?
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Contagion
NaturomorphismsI The feeling was contagious.I The news spread like wildfire.I Freedom is the most contagious virus known to
man.Hubert H. Humphrey, Johnsons vice president
I Nothing is so contagious as enthusiasm.Samuel Taylor Coleridge
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Social contagion
Optimism according to Ambrose Bierce: ()The doctrine that everything is beautiful, including what isugly, everything good, especially the bad, and everythingright that is wrong. ... It is hereditary, but fortunately notcontagious.
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Social contagion
Eric Hoffer, 19021983There is a grandeur in the uniformity of the mass. Whena fashion, a dance, a song, a slogan or a joke sweepslike wildfire from one end of the continent to the other,and a hundred million people roar with laughter, swaytheir bodies in unison, hum one song or break forth inanger and denunciation, there is the overpoweringfeeling that in this country we have come nearer thebrotherhood of man than ever before.
I Hoffer () was an interesting fellow...
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The spread of fanaticism
Hoffers acclaimed work: The True Believer:Thoughts On The Nature Of Mass Movements (1951) [3]
Quotes-aplenty:I We can be absolutely certain only about things we
do not understand.I Mass movements can rise and spread without belief
in a God, but never without belief in a devil.I Where freedom is real, equality is the passion of the
masses. Where equality is real, freedom is thepassion of a small minority.
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Imitation
despair.com
When people are freeto do as they please,they usually imitateeach other.
Eric HofferThe Passionate Stateof Mind [4]
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The collective...
despair.com
Never Underestimatethe Power of StupidPeople in LargeGroups.
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Contagion
DefinitionsI (1) The spreading of a quality or quantity between
individuals in a population.I (2) A disease itself:
the plague, a blight, the dreaded lurgi, ...I from Latin: con = together with + tangere to touch.I Contagion has unpleasant overtones...I Just Spreading might be a more neutral wordI But contagion is kind of exciting...
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Examples of non-disease spreading:
Interesting infections:I Spreading of buildings in the US... ()
I Viral get-out-the-vote video. ()
Lavf50.5.0
Convertified by iSquint - http://www.isquint.org
walmartspread.mp4Media File (video/mp4)
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Contagions
Two main classes of contagion1. Infectious diseases:
tuberculosis, HIV, ebola, SARS, influenza, ...
2. Social contagion:fashion, word usage, rumors, riots, religion, ...
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Mathematical Epidemiology
The standard SIR model [8]
I = basic model of disease contagionI Three states:
1. S = Susceptible2. I = Infective/Infectious3. R = Recovered or Removed or Refractory
I S(t) + I(t) + R(t) = 1I Presumes random interactions (mass-action
principle)I Interactions are independent (no memory)I Discrete and continuous time versions
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Mathematical Epidemiology
Discrete time automata example:
I
R
SI
1
1 I
r1 r
Transition Probabilities:
for being infected givencontact with infectedr for recovery for loss of immunity
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Mathematical Epidemiology
Original models attributed toI 1920s: Reed and FrostI 1920s/1930s: Kermack and McKendrick [5, 7, 6]
I Coupled differential equations with a mass-actionprinciple
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Independent Interaction models
Differential equations for continuous modelddt
S = IS + R
ddt
I = IS rI
ddt
R = rI R
, r , and are now rates.
Reproduction Number R0:I R0 = expected number of infected individuals
resulting from a single initial infectiveI Epidemic threshold: If R0 > 1, epidemic occurs.
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Reproduction Number R0
Discrete version:I Set up: One Infective in a randomly mixing
population of SusceptiblesI At time t = 0, single infective random bumps into a
SusceptibleI Probability of transmission = I At time t = 1, single Infective remains infected with
probability 1 rI At time t = k , single Infective remains infected with
probability (1 r)k
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Reproduction Number R0
Discrete version:I Expected number infected by original Infective:
R0 = + (1 r) + (1 r)2 + (1 r)3 + . . .
= (
1 + (1 r) + (1 r)2 + (1 r)3 + . . .)
= 1
1 (1 r)= /r
For S0 initial infectives (1 S0 = R0 immune):
R0 = S0/r
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Independent Interaction models
For the continuous versionI Second equation:
ddt
I = SI rI
ddt
I = (S r)I
I Number of infectives grows initially if
S(0) r > 0 S(0) > r S(0)/r > 1
I Same story as for discrete model.
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Independent Interaction models
Example of epidemic threshold:
0 1 2 3 40
0.2
0.4
0.6
0.8
1
R0
Frac
tion
infe
cted
I Continuous phase transition.I Fine idea from a simple model.
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Independent Interaction models
Many variants of the SIR model:I SIS: susceptible-infective-susceptibleI SIRS: susceptible-infective-recovered-susceptibleI compartment models (age or gender partitions)I more categories such as exposed (SEIRS)I recruitment (migration, birth)
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Disease spreading models
For novel diseases:1. Can we predict the size of an epidemic?2. How important is the reproduction number R0?
R0 approximately same for all of the following:I 1918-19 Spanish Flu 500,000 deaths in USI 1957-58 Asian Flu 70,000 deaths in USI 1968-69 Hong Kong Flu 34,000 deaths in USI 2003 SARS Epidemic 800 deaths world-wide
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Size distributions
Size distributions are important elsewhere:I earthquakes (Gutenberg-Richter law)I city sizes, forest fires, war fatalitiesI wealth distributionsI popularity (books, music, websites, ideas)I Epidemics?
Power laws distributions are common but not obligatory...
Really, what about epidemics?I Simply hasnt attracted much attention.I Data not as clean as for other phenomena.
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Feeling Ill in Iceland
Caseload recorded monthly for range of diseases inIceland, 1888-1990
1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 19900
0.01
0.02
0.03
Date
Freq
uenc
y
Iceland: measlesnormalized count
I Treat outbreaks separated in time as noveldiseases.
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Really not so good at all in Iceland
Epidemic size distributions N(S) forMeasles, Rubella, and 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.
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Measles & Pertussis
0 0.025 0.05 0.075 0.10
1
2
3
4
5
75
N (
)
A
105
104
103
102
101
100
101
102
N>(
)
0 0.025 0.05 0.0750
1
2
3
4
5
75
B
105
104
103
102
101
100
101
102
N>(
)
Insert plots:Complementary cumulative frequency distributions:
N( > ) +1
Limited scaling with a possible break.
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Power law distributions
Measured values of :I measles: 1.40 (low ) and 1.13 (high )I pertussis: 1.39 (low ) and 1.16 (high )
I Expect 2 < 3 (finite mean, infinite variance)I When < 1, cant normalizeI Distribution is quite flat.
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Resurgenceexample 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
I Epidemic slows...then an infective moves to a new context.
I Epidemic discovers new pools of susceptibles:Resurgence.
I Importance of rare, stochastic events.
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The challenge
So... can a simple model produce1. broad epidemic distributions
and2. resurgence ?
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Size distributions
0 0.25 0.5 0.75 10
500
1000
1500
2000A
N(
)
R0=3
Simple modelstypically producebimodal or unimodalsize distributions.
I This includes network models:random, small-world, scale-free, ...
I Exceptions:1. Forest fire models2. Sophisticated metapopulation models
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Burning through the population
Forest fire models: [9]
I Rhodes & Anderson, 1996I The physicists approach:
if it works for magnets, itll work for people...
A 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 forest fire model not completely understood.
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Size distributions
From Rhodes and Anderson, 1996.
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Sophisticated metapopulation models
I Community based mixing: Longini (two scales).I Eubank et al.s EpiSims/TRANSIMScity
simulations.I Spreading through countriesAirlines: Germann et
al., Corlizza et al.I Vital work but perhaps hard to generalize from...I Create a simple model involving multiscale travelI Multiscale models suggested by others but not
formalized (Bailey, Cliff and Haggett, Ferguson et al.)
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Size distributions
I Very big question: What is N?I Should we model SARS in Hong Kong as spreading
in a neighborhood, in Hong Kong, Asia, or the world?I For simple models, we need to know the final size
beforehand...
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Improving simple models
Contexts and IdentitiesBipartite networks
c d ea b
2 3 41
a
b
c
d
e
contexts
individuals
unipartitenetwork
I boards of directorsI moviesI transportation modes (subway)
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Improving simple models
Idea for social networks: incorporate identity.
Identity is formed from attributes such as:I Geographic locationI Type of employmentI AgeI Recreational activities
Groups are crucial...I formed by people with at least one similar attributeI Attributes Contexts Interactions
Networks. [11]
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Infer interactions/network from identities
eca
high schoolteacher
occupation
health careeducation
nurse doctorteacherkindergarten
db
Distance makes sense in identity/context space.
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Generalized context space
100
eca b d
geography occupation age
0
(Blau & Schwartz [1], Simmel [10], Breiger [2])
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A toy agent-based model
Geographyallow people to move betweencontexts:
I Locally: standard SIR model with random mixingI discrete time simulationI = infection probabilityI = recovery probabilityI P = probability of travelI Movement distance: Pr(d) exp(d/)I = typical travel distance
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A toy agent-based model
Schematic:b=2
i j
x ij =2l=3
n=8
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Model output
I Define P0 = Expected number of infected individualsleaving initially infected context.
I Need P0 > 1 for disease to spread (independent ofR0).
I Limit epidemic size by restricting frequency of traveland/or range
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Model output
Varying :
I Transition in expected final size based on typicalmovement distance (sensible)
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Model output
Varying P0:
I Transition in expected final size based on typicalnumber of infectives leaving first group (alsosensible)
I Travel advisories: has larger effect than P0.
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Example model output: size distributions
0 0.25 0.5 0.75 10
100
200
300
400
1942
N(
)
R0=3
0 0.25 0.5 0.75 10
100
200
300
400
683
N
()
R0=12
I Flat distributions are possible for certain and P.I Different R0s may produce similar distributionsI Same epidemic sizes may arise from different R0s
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Model outputresurgence
Standard model:
0 500 1000 15000
2000
4000
6000
t
# N
ew c
ases D R
0=3
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Model outputresurgence
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
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The upshot
Simple multiscale population structure+stochasticity
leads to
resurgence+broad epidemic size distributions
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Conclusions
I For this model, epidemic size is highly unpredictableI Model is more complicated than SIR but still simpleI We havent even included normal social responses
such as travel bans and self-quarantine.I The reproduction number R0 is not terribly useful.I R0, however measured, is not informative about
1. how likely the observed epidemic size was,2. and how likely future epidemics will be.
I Problem: R0 summarises one epidemic after the factand enfolds movement, the price of bananas,everything.
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Conclusions
I Disease spread highly sensitive to populationstructure
I Rare events may matter enormously(e.g., an infected individual taking an internationalflight)
I More support for controlling population movement(e.g., travel advisories, quarantine)
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Conclusions
What to do:I Need to separate movement from diseaseI R0 needs a friend or two.I Need R0 > 1 and P0 > 1 and sufficiently large
for disease to have a chance of spreading
More wondering:I Exactly how important are rare events in disease
spreading?I Again, what is N?
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Simple disease spreading models
Valiant attempts to use SIR and co. elsewhere:I Adoption of ideas/beliefs (Goffman & Newell, 1964)I Spread of rumors (Daley & Kendall, 1965)I Diffusion of innovations (Bass, 1969)I Spread of fanatical behavior (Castillo-Chvez &
Song, 2003)I Spread of Feynmann diagrams (Bettencourt et al.,
2006)
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Predicting social catastrophe isnt easy...
Greenspan Concedes Error on RegulationI . . . humbled Mr. Greenspan admitted that he had put
too much faith in the self-correcting power of freemarkets . . .
I Those of us who have looked to the self-interest oflending institutions to protect shareholders equity,myself included, are in a state of shocked disbelief
I Rep. Henry A. Waxman: Do you feel that yourideology pushed you to make decisions that you wishyou had not made?
I Mr. Greenspan conceded: Yes, Ive found a flaw. Idont know how significant or permanent it is. But Ivebeen very distressed by that fact.
New York Times, October 23, 2008 ()
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Economics, Schmeconomics
Alan Greenspan (September 18, 2007):
Ive been dealing with these bigmathematical models of forecasting theeconomy ...
If I could figure out a way to determinewhether or not people are more fearfulor changing to more euphoric,
I dont need any of this other stuff.
I could forecast the economy better thanany way I know.
http://wikipedia.org
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Economics, SchmeconomicsGreenspan continues:The trouble is that we cant figure that out. Ive been inthe forecasting business for 50 years. Im no better than Iever was, and nobody else is. Forecasting 50 yearsago was as good or as bad as it is today. And the reasonis that human nature hasnt changed. We cant improveourselves.
Jon Stewart:
You just bummed the @*!# out of me.
wildbluffmedia.com
I From the Daily Show () (September 18, 2007)I The full inteview is here ().
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Economics, Schmeconomics
James K. Galbraith:NYT But there are at least 15,000 professional
economists in this country, and youre saying onlytwo or three of them foresaw the mortgage crisis?[JKG] Ten or 12 would be closer than two or three.
NYT What does that say about the field of economics,which claims to be a science? [JKG] Its anenormous blot on the reputation of the profession.There are thousands of economists. Most of themteach. And most of them teach a theoreticalframework that has been shown to be fundamentallyuseless.
From the New York Times, 11/02/2008 ()
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References I
[1] P. M. Blau and J. E. Schwartz.Crosscutting Social Circles.Academic Press, Orlando, FL, 1984.
[2] R. L. Breiger.The duality of persons and groups.Social Forces, 53(2):181190, 1974. pdf ()
[3] E. Hoffer.The True Believer: On The Nature Of MassMovements.Harper and Row, New York, 1951.
[4] E. Hoffer.The Passionate State of Mind: And OtherAphorisms.Buccaneer Books, 1954.
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References II
[5] W. O. Kermack and A. G. McKendrick.A contribution to the mathematical theory ofepidemics.Proc. R. Soc. Lond. A, 115:700721, 1927. pdf ()
[6] W. O. Kermack and A. G. McKendrick.A contribution to the mathematical theory ofepidemics. III. Further studies of the problem ofendemicity.Proc. R. Soc. Lond. A, 141(843):94122, 1927.pdf ()
[7] W. O. Kermack and A. G. McKendrick.Contributions to the mathematical theory ofepidemics. II. The problem of endemicity.Proc. R. Soc. Lond. A, 138(834):5583, 1927.pdf ()
http://www.uvm.eduhttp://www.uvm.edu/~pdoddshttp://www.uvm.edu/~pdodds/research/papers/others/1927/kermack1927a.pdfhttp://www.uvm.edu/~pdodds/research/papers/others/1933/kermack1933a.pdfhttp://www.uvm.edu/~pdodds/research/papers/others/1932/kermack1932a.pdf
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References III
[8] J. D. Murray.Mathematical Biology.Springer, New York, Third edition, 2002.
[9] C. J. Rhodes and R. M. Anderson.Power laws governing epidemics in isolatedpopulations.Nature, 381:600602, 1996. pdf ()
[10] G. Simmel.The number of members as determining thesociological form of the group. I.American Journal of Sociology, 8:146, 1902.
[11] D. J. Watts, P. S. Dodds, and M. E. J. Newman.Identity and search in social networks.Science, 296:13021305, 2002. pdf ()
http://www.uvm.eduhttp://www.uvm.edu/~pdoddshttp://www.uvm.edu/~pdodds/research/papers/others/1996/rhodes1996a.pdfhttp://www.uvm.edu/~pdodds/research/papers/others/2002/watts2002b.pdf
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