Why are Epidemics so Unpredictable?

1
Why are Epidemics so Unpredictable?

• Duncan Watts, Roby Muhamad, Daniel Medina, Peter
  Dodds
• Columbia University

2
An Obvious Question

• Whenever a novel outbreak of infectious disease
  is announced (SARS, Avian Influenza, Ebola,
  etc.), one of the most pressing questions is
  How big will it get?
• One could also ask an analogous question for
  existing epidemics (HIV, TB, Malaria)
• Amazingly, mathematical epidemiology currently
  has no way to answer these questions

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Mathematical Epidemiology

• Started with Daniel Bernoullis analysis of
  smallpox epidemic (1760s)
• Has been developed extensively since late 1920s
  (Kermack and McKendrick)
• Now hundreds of models deal with many variations
  of human, animal, and plant diseases
• Incredible diversity of models, which can be
  extremely complex, but most are variants of the
  original

4
The Standard (SIR) Model
b
r
g
(1) Individuals cycle between three
states Susceptible Infected and Removed
(2) Mixing is uniformly random Mass Action
assumption
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Basic Reproduction Number R0

• Mass Action assumption means that epidemics
  depend only total fraction of infectives (I) and
  susceptibles (S)
• Condition for an epidemic is simple R0 gt 1
• R0 is the Basic Reproduction Number
• Average number of new infectives generated by a
  single infected individual in a susceptible
  population
• R0 b/g S depends on
• Infectiousness of disease (??
• Recovery rate (??
• Density of susceptibles near outbreak (S)
• Preventing an epidemic thus becomes equivalent to
  keeping R0 lt 1

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Standard models imply outbreaks are bi-modal

• When R0 lt 1 Epidemics never occur
• When R0 gt 1, Only one of two outcomes possible
• Outbreak fails to achieve epidemic status (left
  peak)
• Outbreak becomes a full-fledged epidemic,
  infecting a significant fraction of the entire
  population (right peak)

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Epidemic size should therefore be predictable
Together, R0 and N (population size)
completely Determine the expected number of cases.
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Also, epidemics should peak only once

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Result Epidemics should follow classic
logistic curve
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Real Epidemics, However

• Differ dramatically in size
• 1918-19 Spanish Flu 500,000 deaths in US
  (20-80 Million world-wide)
• 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
• All these diseases have about same R0!
• How different in size can epidemics of seemingly
  similar diseases be?
• Unfortunately, historical data on large epidemics
  is hard to collect. Thus true size distributions
  are unknown

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Seem to be multi-modal
Size distribution of epidemics for (A) measles
and (B) pertussis (whooping cough) in Iceland,
1888-1990
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Real Epidemics alsoResurgent
Global Daily Case Load for 2003 SARS
Epidemic Epidemic had several peaks,
interspersed with lulls
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Result is unpredictability

• Multi-modal size distributions imply that any
  given outbreak of the same disease can have
  dramatically different outcomes
• Resurgence implies that even epidemics which seem
  to be burning out can regenerate themselves by
  invading new populations

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What makes epidemics unpredictable?

• Key insight from the literature on social
  networks
• populations exhibit structure
• What kind of structure?
• Inhomogeneous population distribution
• Transportation and infrastructure networks
• Social, Organizational, and Sexual Networks
• Result is that
• Uniform mixing occurs only in small, relatively
  confined contexts (where standard model applies)
• Large epidemics are not single events they are
  concatenations of many, small epidemics

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Influenza Pandemic, 1957
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How do network models help?

• Last 20 years has seen rapid growth in network
  epidemiology
• In principal, tremendously appealing
• Problem is that in a SARS-like epidemic, many
  kinds of networks can potentially matter
• Social, organizational, infrastructural,
  transport
• Result is both empirically and analytically
  intractable
• What to include and what to exclude?
• How to estimate parameters?
• How to balance realism with complexity?

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Compromise between realism and complexity

• Assume mass-action assumption holds
  (approximately) in local contexts (schools,
  hospitals, apartment buildings, villages, etc.)
• Then incorporate main insight of population
  structure
• Local contexts are embedded in a series of
  successively larger contexts (neighborhoods,
  cities, counties, states, regions, countries,
  continents)
• Global populations are nested
• Result is a multiscale metapopulation model
• Individuals can escape current local context
  and move to another (with probability p)
• Once escaped, move to another local context,
  chosen from contexts at characteristic length x

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Multiscale metapopulation model
Incorporates essence of population structure
while remaining simple
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Previous metapopulation models

• Metapopulation models hardly new idea
• Some very detailed models worked out using
  airline network data (Longini), and even detailed
  maps of individual cities (Halloran et al.
  Eubanks et al.)
• These models all restricted to two scales (local
  and global)
• Multiscale models have been advocated previously
  (Bailey Cliff and Haggett Ferguson et al.) but
  not formally specified
• Technically, extension to multiple scales is
  trivial, but consequential nonetheless

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Difference with network models

• Critical feature of biological disease (in
  contrast with information spread) is that
  individuals must be physically co-located
• Thus i cannot infect j1 and j2 sequentially
  unless j1 and j2 are also co-located.
• Very different from small world and other
  network models in which individuals can sustain
  multiple long-range contacts simultaneously
• Subtle difference, but turns out to be critical
• Network models also generate bimodal distributions

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What does epidemic size depend on?
Mobility (R0 3)
Range
Average epidemic size vs. P0 (expected number of
infectives escaping a local context)
When P0 gt 1 Average epidemic size vs.
x (typical distance traveled)
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What do these results tell us?

• Still need R0 gt 1 as necessary condition
• Everything has to start locally, and locally,
  mass-action models might be OK
• But also need some non-local mobility
• P0 gt 1 and R0 gt 1 are sufficient for non-local
  epidemic (on average -- remember stochastic)
• Average size of non-local epidemic then depends
  on transport range (x)
• Average size appears more sensitive to range (x)
  than to volume (P0) of transport
• Simply restricting range of travel may be an
  effective intervention strategy
• E.g. issuing travel advisories

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When non-local epidemics do occur Multiscale
populations generate flat distributions of
epidemic sizes

• Very different outcomes possible for same R0
• Very similar distributions for very different R0

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Whats up with R0?

• R0 is supposed to be the one number that accounts
  for almost everything
• It is a necessary condition for an epidemic
• However, as long as R0 gt 1, the value of R0 tells
  us very little about size or duration in a
  multi-scale world
• Similar R0 can lead to very different outcomes
• Very different R0 correspond to similar
  distributions of outcomes
• Result is many-to-many mapping between R0 and
  epidemic size

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Are we just computing it wrong?

• If our model were deterministic, we could compute
  R0 in terms of the eigenvalues of the inter-group
  mixing matrix (i.e. in terms of p and x)
• However, we would still get just one value
• In fact, can estimate its value directly
• Some variance, but very small.
• The ambiguity is real
• Method of computation doesnt eliminate many to
  many mapping problem

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Where does the variance come from?

• Problem is that model is not deterministic
• Stochasticity is well known to cause problems
  when epidemic is small, even for mass-action
  models
• Here, combination of multiple scales and
  individual transport across them, means that
  stochasticity continues to matter throughout
  course of epidemic
• Thus a few individuals (rare events) can have
  huge impact on size and duration

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• Same disease can have very different trajectories
• Resurgence driven by rare events

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Importance of Network Thinking

• Large populations exhibit network structure
• Social, sexual, infrastructure, transportation
• Large epidemics need to be understood as many
  small epidemics linked by networks
• But taken too literally, network models are a
  losing proposition
• Complexity is virtually unlimited
• Empirical estimation impossible
• Modeling impossible
• Need some compromise between tractability and
  realism

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Multi-scale metapopulation models

• Incorporating multi-scale structure of the
  world in epidemic models can explain
  multi-modality and resurgence of epidemics
• Knowledge of disease itself (R0) doesnt help
  predict size or duration of epidemic
• Reason is that rare events (e.g. one person
  getting on a plane) can have big consequences
• Population structure itself can be used as
  control measure (e.g. travel advisories)
• See Watts et al. PNAS, 102(32), 11157-11162 for
  details