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

cannot make inferences based on the probability of single events, the accumulated
evidence is a strong case against the assumption of homogeneous infectiousness (Figure
2A).
For diseases with different degrees of individual variation we can now predict
the frequency of SSEs (Figure 2C), a standing challenge in emerging disease
epidemiology (4). Once the threshold number of cases Z (99) has been defined for a 99 th -
percentile SSE under effective reproductive number R, then for any k one can calculate
from Z~NegB(R,k) the proportion of individuals ΨR,k expected to generate Z>Z (99) . In a
homogeneous population (k→∞), ΨR,∞≤0.01 by definition (where the less-than arises
because the Poisson distribution is discrete (see appendix A)). When heterogeneity is
accounted for, ΨR,k>ΨR,∞ and varies strongly with both R and k, peaking between k=0.1
and k=1 for the low R values of interest for emerging diseases. Because the variance-
to-mean ratio is fixed at 1 for the Poisson distribution but increases linearly with R for
the NB model (see appendix A), for moderate k values ΨR,k increases strongly with R as
the relative density of Z>Z (99) increases. The influence of SSEs on a particular outbreak
can be summarized intuitively by the superspreading load, LSSE, defined as the factor by
which the number of SSEs exceeds what is expected under homogeneity (For 99th-
percentile SSEs, we suggest LSSE=ΨR,k/0.01. Strictly, this is the factor by which the
frequency of SSEs exceeds the 1% target frequency under homogeneity as laid out in
our SSE definition. The true frequency of SSEs under homogeneity is ΨR,∞=0.01, so
LSSE=ΨR,k /ΨR,∞. ΨR,k /ΨR,∞ is not a desirable summary measure because it varies
erratically with R, as ΨR,∞ fluctuates in response to integer changes in Z (99) .).
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