  The commonly observed negative correlation between the number of species in
an ecological community and disease risk, typically referred to as "the
dilution effect", has received a substantial amount of attention over the past
decade. Attempts to test this relationship experimentally have revealed that,
in addition to the mean disease risk decreasing with species number, so too
does the variance of disease risk. This is referred to as the "variance
reduction effect", and has received relatively little attention in the
disease-diversity literature. Here, we set out to clarify and quantify some of
these relationships in an idealized model of a randomly assembled multi-species
community undergoing an epidemic. We specifically investigate the variance of
the community disease reproductive ratio, a multi-species extension of the
basic reproductive ratio R_0, for a family of random-parameter meta-community
SIR models, and show how the variance of community $R_0$ varies depending on
whether transmission is density or frequency-dependent. We finally outline
areas of further research on how changes in variance affect transmission
dynamics in other systems.
