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 –R 0 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.

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 p G = prob that 1 infected produces G new infections, then equivalent to z = g(z) in Roger’s slides generating function

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) Poisson distribution dictates this form taking the expectation removes the condition on T average number of new infs

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 average infectious period

1-Group Model (Homogeneous Case) We find that probability of a major outbreak (when R 0 > 1) where a = initial number of infectious individuals This is NOT true for multigroup models

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. stoch, N = 1120 deter, N = 112 stoch, N = stoch, N = group dairy model

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: stochastic level deterministic equilibrium Further increases in N indicate that the prob. major outbreak tends to a limit of approx prop. sims with prev > 0 stoch prev (t = 1500) prob major outbreak (est) N = 112 N = 1120 N = / / / results for t = 1500

4-Group Model: Theory of Probability of Major Outbreak According to Damian Clancy, prob. of major outbreak = –(a U, a W, a D, a L ) 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. need q and a for each group

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 –X j * = disease-free equilibrium value for group j –T j (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].

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 = –upper limit for prevalence = prop. sims with prev > 0 prev (t = 1500) prob major outbreak (est) N = 112 N = 1120 N = / / / upper limit prevalence upper limit prob major outbreak deterministic equilibrium prevalence = x results for t = 1500

4-Group Model: 1 – q W versus 1 – 1/R 0 1-group model with a = 1:1 – q = 1 – 1/R 0 4-group model with a W = 1 and a U = a D = a L = 0: 1 – q W  1 – 1/R 0 e.g. from Yanni’s paper

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/R 0 4-group model with a W = 1 and a U = a D = a L = 0: 1 – q W  1 – 1/R 0