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Probability of a Major Outbreak for Heterogeneous Populations

Probability of a Major Outbreak for Heterogeneous Populations. Math. Biol. Group Meeting 26 April 2005 Joanne Turner and Yanni Xiao. Previously for 1-Group Model (Homogeneous Case). Roger showed that 4 different threshold conditions are equivalent i.e. where

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Probability of a Major Outbreak for Heterogeneous Populations

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  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. equivalent to z = g(z) in Roger’s slides generating function 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. average number of new infs Poisson distribution dictates this form taking the expectation removes the condition on T 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)

  5. average infectious period 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

  6. 1-Group Model (Homogeneous Case) • We find that • probability of a major outbreak (when R0> 1) where a = initial number of infectious individuals This is NOT true for multigroup models

  7. 4-group dairy model deter, N = 112 stoch, N = 11200 stoch, N = 1120 stoch, N = 112 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.

  8. stochastic level prop. sims with prev > 0 stoch prev (t = 1500) prob major outbreak (est) deterministic equilibrium N = 112 0 / 100 0 0 0.086 N = 1120 11 / 100 0.0065 N = 11200 14 / 100 0.0105 0.138 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.

  9. need q and a for each group According to Damian Clancy, • prob. of major outbreak = • (aU, aW, aD, aL) are numbers of infecteds in each group at time t = 0. • is the unique solution in [0,1)4 of • generating function is • are numbers of new infections in each group caused by an infected individual that was initially in group i. • are variables of generating function f. 4-Group Model: Theory of Probability of Major Outbreak

  10. 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 Yanni’s paper for full details].

  11. prop. sims with prev > 0 prev (t = 1500) prob major outbreak (est) upper limit prob major outbreak deterministic equilibrium prevalence upper limit prevalence x = N = 112 0 / 100 0 0 0.086 N = 1120 11 / 100 0.0065 N = 11200 14 / 100 0.0105 0.138 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

  12. 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 Yanni’s paper

  13. 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

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