Mathematical Models for Infectious Diseases

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Mathematical Models for Infectious
Diseases Alun Lloyd Biomathematics
Graduate Program Department of Mathematics North
Carolina State University
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2001 Foot and Mouth Outbreak in the UK

• February 19th, 2001 clinical signs of FMD
  spotted at an ante mortem examination of pigs at
  a slaughterhouse
• January 14th, 2002 final county in the UK
  declared FMD free
• Over epidemic animals on 9900 premises were
  culledincluding 594 000 cattle, 3 315 000 sheep,
  142 000 pigs
• Impact to UK economy estimated at 4 to 7
  billion
• Use of models to interpret incidence data and to
  guide policy decisions (following experience of
  BSE)

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2001 Foot and Mouth Outbreak in the UK
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UK FMD Data, Models and Policy

• Control measures introduced rapidly
• Movement restrictions
• Culling animals on infected farms and nearby
  farms
• Models used to interpret time series (incidence)
  data on a day-by-day basis
• How rapidly is infection spreading?
• How well are the control measures working?
• Models used to predict impact of different
  control measures
• Vaccination of animals?
• Culling on nearby but uninfected farms
  (contiguous cull)

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UK FMD Data, Models and Policy

• More than one modeling group, employing different
  approaches
• Model validation (did predictions agree or
  differ?)
• Recommendations following epidemic
• Rapid response
• Usefulness of modeling approaches
• Need for data to be made available
• (Report into Infectious Diseases in Livestock
  Royal Society)

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What is a Model?

• Simplest explanation consistent with reality
• Models are widely used in science
• abstraction of real situation
• simplification
• more amenable to experimentation or
  analysis(think about a lab rat or cell culture)
• A mathematical model is really no different to a
  lab model
• Just describe some system in mathematical terms
• Designed so that the model captures important
  features of the system

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Why Do We Model?

• To make predictions
• (weather forecast)
• To examine scenarios
• do experiments that we couldnt do in
  reality(compare impact of different control
  measures)
• To guide data collection
• what do we need to know in order to make
  predictions?
• How much data do we need? (c.f. sample-size issue
  in stats)
• To make sense of data
• Are there patterns that underlie data?

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Why Do We Model?

• To infer biological processes from
  epidemiological patterns
• Why do number of measles cases rise and fall over
  the course of a year?
• To provide a quantitative framework within which
  we can ask questions
• This sometimes helps us pose questions in a more
  precise and meaningful way

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A Fundamental Principle of Modeling

• START WITH A SIMPLE MODEL
• If it works great!
• If it doesnt work improve it!make it more
  realistic include more complexity

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The Modeling Process
Data disease incidence record
Epidemiological Processes
Model Predictions
Mathematical Model
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Simplicity Versus Complexity

• A general theme of modeling
• Simple models are easy to understand, but might
  not reflect reality too closely (or at all)
• Complex models are more likely to be more
  realistic, but are too difficult for us to
  understand
• Type of model needed depends on the question
  being asked and how much we know
• What scale map do we need? State/County/City/Subdi
  vision

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Types of Epidemiological Models

• Compartmental model
• divide population according to infection status
  SIR model susceptible, infectious, recovered
• Makes strong assumptions about
  population(nature of mixing, averaging over
  individuals)
• Individual based model (agent based)
• collection of individuals and rules specifying
  behavior

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Simplest Epidemiological Model

• Rate at which new infections arise is
  proportional to number of infectious
  individuals
• This is just an exponential growth model
• Why doesnt an epidemic continue to grow
  exponentially?Depletion of susceptible
  population
• Transmission depends on both the number of
  infectives and the number of susceptible
  individuals

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The SIR model in a closed population
Mass-actionassumption
S
Infection
I
Recovery
R
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The SIR model in a closed population

• If bS gt g , number of infectives increases
• Single epidemic spreads through population
• Number of infectives falls when S falls below b
  / g
• We call this ratio the basic reproductive number,
  R0

S
I
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The Basic Reproductive Number

• THE central concept of mathematical epidemiology
• Basic reproductive number givesthe average
  number of secondary infections that result from
  the introduction of a single infective individual
  into an entirely susceptible population
• If R0 is greater than one, an epidemic can
  occur(each case leads to more than one secondary
  case)
• Invasion criterion for infection
• Epidemic ends when S falls below b / g
• We call this ratio the basic reproductive number,
  R0

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The SIR model with demography

• Demography (births and deaths) can be added
• Susceptible population is replenished by births
• If R0 gt 1 , system goes to an endemic
  equilibrium
• Birth rate balances infection rate
• Infection rate balances recovery

S
Birth
Death
Infection
Death
I
Recovery
R
Death
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The SIR model with demography

• Endemic equilibrium approached via damped
  oscillations

S
I
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The Basic Reproductive Number

• R0 gt 1 is also a persistence criterion
• R0 tells us how easy or difficult it is to
  eradicate an infection
• In well mixed situationsCritical vaccination
  fraction pc 1 - 1/R0
• Easier to eradicate an infection with low R0 than
  high R0(e.g. smallpox R0 ? 5, measles R0 ? 15)

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Comparison to Measles Data

• Oscillations are maintained in the measles
  incidence time seriesbut not in the model. WHY?
• Seasonal forcing transmission rates are
  higher during school terms than vacations
• This can be incorporated into the modeling
  framework refine model
• Seasonality in childhood diseases leads to
  multi-annual oscillations

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More Complex Models

• Multitude of ways to make more complex models
• More realistic ways of describing timecourse of
  infection
• Stochastic effects (population consists of
  individuals)
• Persistence becomes a more delicate question
• Relax mixing assumption not all individuals
  are equally likely to interact with each other
• Particularly important in models for STIs

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Deterministic versus Stochastic

• Differential equations are deterministic
• They ignore randomness
• In reality, epidemics are stochastic (involve
  randomness)
• Re-run an introduction wont get exactly the
  same outcome
• Distribution of Epidemic Sizes

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Why is Modeling Difficult?

• Complexity of the system
• Lack of information
• might not know all the details of the biology
• might not observe all of the relevant variables
  data usually tells us about prevalence or
  incidence, less often about susceptible
  population
• Parameter estimation issues
• can we estimate parameters?
• independently of the data set of interest
  (avoid circularity!)

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Limitations of Models

• Modelers must keep in mind the limitations of
  their models
• Many assumptions are made in their
  formulationwhich of them have important
  effects? Sensitivity and structural stability
• Can we trust the answer that a model gives us?
• Garbage in, garbage out
• Does the model address the whole story?
  Particularly important when guiding policy
  decisions (e.g. economic aspects)
• Modelers must be particularly careful when
  communicating model results to non-specialists

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Successes and Failures of Models

• Successes many examples where models have been
  highly informative
• Dynamics of childhood diseases (e.g. measles)
• Models explain temporal patterns
• Models explain spatial patterns (city to city
  spread)
• Within-host dynamics of HIV and drug treatment
• Models, when used to analyze data from drug
  treatment studies, revealed a highly dynamic
  picture of ongoing infection during the long
  asymptomatic phase

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Successes and Failures of Models

• Failures? Where models have been less informative
• Prediction of variant CJD in UK, following BSE
  epidemic
• Models gave an incredibly wide range of
  predictions of the potential number of cases
  (e.g. between 29 and 10 million cases)
• Insufficient information to parameterize
  model(in particular, CJD has a long and variable
  incubation period, so there is tremendous
  uncertainty in the early stages of an epidemic)
• Modelers did acknowledge the weaknesses of their
  approach
• Models highlighted information that would be
  needed

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Summary

• Epidemiological models provide a useful framework
  within which we can understand epidemiological
  processes
• Increasing use of modeling in public health
  settings
• Importance of confronting models with data,
  understanding and challenging assumptions and
  realizing their limitations
• Infectious Disease Modeling course next semester

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UK Foot and Mouth Disease Movie from Grenfell et
al. Science 2001