K - College of Natural Resources - University of California, Berkeley

Z=2 cases is then equal to binomial(N=33,p=1/33), from which exact percentile values
could be obtained as above. Since all other Z values in the bootstrap dataset will equal
zero, these are easily converted to Rˆ and k ˆ using maximum-likelihood estimation.
For all methods of bootstrapping, datasets with very few non-zero values of Z
generated significant proportions of bootstrapped datasets with all zeros. The negative
ˆ ( ) k −
binomial shape parameter k cannot be calculated from
pˆ
= 1+
Rˆ
0 k in this case.
Because all-zero datasets are completely homogeneous and have variance equal to their
mean value, we assumed ˆ →∞ for these datasets. When 5% or more of bootstrapped
k pz
datasets had all zero values, then one limit of the 90% CI was determined by this
assumption. We marked such instances with a superscript u in Tables 1 and 2.
Proportion of transmission due to top 20%
The proportion of transmission due to a given proportion of the population,
plotted in Figure 1B, was calculated as follows. First we estimated R0 and k, which
specify the pdf fν(x) and cdf Fν(x) of the gamma-distribution describing the individual
reproductive number ν for a given disease and population. We then calculated the
cumulative distribution function for transmission of the disease:
F
trans
x
1
( x)
= ∫ u fν
( u)
du
R
such that Ftrans(x) is the proportion of all transmission due to infectious individuals with
ν < x. The proportion of transmission due to individuals with ν > x is thus 1-Ftrans(x),
0
while the proportion of individuals with ν > x is 1-Fν(x). These quantities were plotted
parametrically as a function of x to make Figure 1B. Similarly, the proportion of
0
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