Probability of a Major Outbreak for Heterogeneous Populations

1
Probability of a Major Outbreak for Heterogeneous
Populations

• Math. Biol. Group Meeting
• 26 April 2005
• Joanne Turner and Yanni Xiao

2
Previously for 1-Group Model (Homogeneous Case)

• Roger showed that 4 different threshold
  conditions are equivalent
• i.e.
• where
• R0 is basic reproduction ratio (number of
  secondary cases per primary in an unexposed
  population)
• z? is probability of ultimate extinction
  (probability pathogen will eventually go extinct)
• r is exponential growth rate of incidence i(t)
• s(?) is proportion of the original population
  remaining susceptible.

3
1-Group Model Theory of Probability of Major
Outbreak

• When there are a infecteds at time t 0,
• prob. of ultimate extinction
• prob. of major outbreak
• As Roger showed, q is the unique solution in
  0,1) of
• If G number of new infections caused by 1
  infected individual during its infectious period.
• and pG prob that 1 infected produces G new
  infections,
• then

4
1-Group Model Calculation of Probability of
Ultimate Extinction

• number of new infections created by 1 infectious
  individual
• ? direct transmission parameter
• X disease-free equilibrium value for the
  number of susceptibles
• T infectious period
• Therefore
• where ? ?X(1-q) (i.e. ? is a function of q)

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1-Group Model Calculation of Prob. of Ultimate
Extinction (cont.)

• Infectious period
• ? rate of loss of infected individuals (i.e.
  death rate recovery rate)
• p.d.f. is
• Now need to solve

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1-Group Model (Homogeneous Case)

• We find that
• probability of a major outbreak (when R0 gt 1)
• where a initial number of infectious
  individuals

This is NOT true for multigroup models
7
4-Group Model Prevalence Plots

• Herd size affects persistence of infection and,
  hence, probability of a major outbreak.
• Same is true for 1-group models (previous results
  only true for large N).
• When we start with 1 infected (i.e. invasion
  scenario), average prevalence for stochastic
  model does not tend to deterministic equilibrium.

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4-Group Model Estimate of Probability of Major
Outbreak

• Prob. of major outbreak ?
• Stochastic prevalence level depends on proportion
  of minor outbreaks (long-term zeros
  drag down the average).
• In previous example

results for t 1500
Further increases in N indicate that the prob.
major outbreak tends to a limit of approx 0.14.
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4-Group Model Theory of Probability of Major
Outbreak
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4-Group Model Theory of Probability of Major
Outbreak

• Direct transmission
• Number of new infecteds in group j created by an
  infected initially in group i is
• ?j direct transmission parameter for group j
• Xj disease-free equilibrium value for group j
• Tj(i) time spent in group j by an infected
  initially in group i
• Therefore
• Repeat for indirect transmission (much more
  complicated) and pseudovertical transmission see
  Yannis paper for full details.

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4-Group Model Theoretical Result

• Theory is only true for large N. Therefore, it
  gives the upper limit for the probability of a
  major outbreak.
• For previous example
• upper limit for prob major outbreak q 0.145.
• upper limit for prevalence 0.011.

results for t 1500
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4-Group Model 1 qW versus 1 1/R0

• 1-group model with a 1 1 q 1 1/R0
• 4-group model with aW 1 and aU aD aL 0
  1 qW ? 1 1/R0
• e.g. from Yannis paper

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Conclusions

• Herd size affects persistence of infection and,
  hence, probability of a major outbreak.
• Theory is only true for large N. Therefore, it
  gives the upper limit for the probability of a
  major outbreak.
• 1-group model with a 1 1 q 1 1/R0
• 4-group model with aW 1
• and aU aD aL 0 1 qW ? 1 1/R0