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

integrated with theoretical modeling to demonstrate the universality and practical
relevance of individual variation in infectiousness.
2. Evidence of individual-level variation
Let Z represent the number of secondary cases caused by a given infectious
individual in an outbreak. Z is a random variable with an “offspring distribution”
Pr(Z=k) that incorporates influences of stochasticity and variation in the population.
Stochastic effects in transmission are modeled using a Poisson process, as is
conventional (Diekmann and Heesterbeek 2000), with intensity given by the individual
reproductive number, ν i.e. Z~Poisson(ν). In conventional models neglecting individual
variation, all individuals are characterized by the population mean, yielding
Z~Poisson(R0). Another common approach, motivated by models with constant
recovery or death rates and homogeneous transmission rates, is to assume that ν is
exponentially distributed, yielding Z~geometric(R0). (For clarity we express all
distributions using the mean as the scale parameter; see appendix A for correspondence
with standard notation.) We introduce a more general formulation, in which ν follows a
gamma distribution with mean R0 and shape parameter k, yielding Z~negative
binomial(R0,k) (henceforth abbreviated Z~NegB(R0,k)) (Taylor and Karlin 1998). The
negative binomial model includes the Poisson (k→∞) and geometric (k=1) models as
special cases.
Realized offspring distributions can be determined from detailed contact tracing
of particular outbreaks, or from surveillance data covering multiple introductions of a
disease. The above candidate models can then be challenged with these data using
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