Population dynamics of

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Population dynamics of infectious diseases
Arjan Stegeman
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Introduction to the population dynamics of
infectious diseases

• Getting familiar with the basic models
• Relation between characteristics of the model and
  the transmission of pathogens

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Modelling population dynamics of infectious
diseases

• model simplified representation of reality
• mathematical using symbols and methods to
  manipulate these symbols

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Why mathematical modeling ?

• Factors affecting infection have a non-linear
  dependence
• Insight in the importance of factors that affect
  the spread of infectious agents
• Provide testable hypotheses
• Extrapolation to other situations/times

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SIR Models
Population consists of
Susceptible
Infectious
Recovered
individuals
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SIR Models

• Dynamic model
• S, I and R are variables (entities that change)
  that change with time,
• parameters (constants) determine how the
  variables change

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Greenwood assumption

• Constant probability of infection (Force of
  infection)

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SIR Model with Greenwood assumption
?I
IRS
S
I
R
IR Incidence rate ? recovery rate parameter
(1/infectious period)
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Transition matrix Markov chain
P probability to go from a state at time t to a
state at time t1
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Markov chain modeling
Starting vector
number of S, I and R at the start of the
modeling
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Example Markov chain modeling
Starting vector
number of S, I and R at the start of the
modeling
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Results of Markov chain model
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Example Markov chain modeling
Starting vector
number of S, I and R at the end of time step 1
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Results of Markov chain model
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Course of number of S, I and R animals in a
closed population (Greenwood assumption)
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Drawback of the Greenwood assumption

• Number of infectious individuals has no influence
  on the rate of transmission

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SIR model with Reed Frost assumption

• Probability of infection upon contact (p)
• Contacts are with rate e per unit of time
• Contacts are at random with other individuals
  (mass action assumption), thus probability that
  an S makes contact with an I equals I/N

p
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SIR Model with Reed Frost assumption
Rate of infection of susceptibles depends on the
number of infectious individuals
I
S
R
?SI/N
?I
? infection rate parameter (Number of new
infections per infectious individual per unit of
time) ? recovery rate parameter (1/infectious
period) N total number of individuals (mass
action)
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SIR Model with Reed Frost assumption

• (formulation in text books, pseudomass action)
• (formulation according to mass action)

It1 number of new infectious individuals at
t1 q probability to escape from infection
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Example Classical Swine Fever virus transmission
among sows housed in crates

• ? 0.29 Susceptible has a probability of
• to become infected in one time step
• ? 0.10 Infectious has a probability of
• to recover in one time step

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Course of number of S, I and R animals in a
closed population (reed Frost assumption with
mass action)
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Deterministic - Stochastic

• Deterministic models all variables have at each
  moment in time for a particular set of parameter
  values only one value
• Stochastic models stochastic variables are used
  which at each moment in time can have many
  different values each with its own probability

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Course of number of S, I and R animals in a
closed population (reed Frost assumption with
mass action)
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Course of number of S, I and R animals in a
closed population (1 run stochastic SIR model)
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Course of number of S, I and R animals in a
closed population (1 run stochastic SIR model)
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Stochastic models

• Preferred above deterministic models because they
  show variability in outcomes that is also present
  in the real world. This is especially important
  in the veterinary field, because we often work
  with populations of limited size.

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Transmission between individuals
b/a Basic Reproduction ratio, R0
Average number of secondary cases caused by 1
infectious individual during its entire
infectious period in a fully susceptible
population
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Reproduction ratio, R0
R0 3
R0 0.5
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Stochastic threshold theoremThe probability of a
major outbreak
Prob major 1 - 1/R0
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Final size distribution for R0 0.5
infection fades out after infection of 1 or a few
individuals (minor outbreaks only)
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Final size distribution for R0 3
R0 gt 1 infection may spread extensively (major
outbreaks and minor outbreaks)
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Deterministic threshold theorem Final size as
function of R0
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Transmission in an open population
I
R
S
mN
?I
?SI/N
mS
mI
mR
b infection rate parameter a recovery rate
parameter
m replacement rate parameter
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Courses of infection in an open population
2 Major outbreak (R0 gt 1)
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3 Endemic infection (R0 gt 1)
1 Minor outbreak (R0 lt 1 of R0 gt 1)
S
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Infection can become endemic when the number of
animals in a herd is at least
M. paratuberculosis a 0.003 m 0.0009 R0
10 Nmin 5 BHV1 a 0.07 m 0.0009, R0
3.5 Nmin 110
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Transmission in an open population
At endemic equilibrium (large population)
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Assumptions

• Mass action (transmission rate depends on
  densities)
• Random mixing
• All S or I individuals are equal (homogeneous)

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SIR model can be adapted to

• SI model
• SIS model
• SIRS model
• SLIR model
• etc.

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Population dynamics of infectious diseases

• Interaction between agent - host contact
  structure between hosts determine the
  transmission
• Quantitative approach R0 plays the central role