  The fundamental models of epidemiology describe the progression of an
infectious disease through a population using compartmentalized differential
equations, but do not incorporate population-level heterogeneity in infection
susceptibility. We show that variation strongly influences the rate of
infection, while the infection process simultaneously sculpts the
susceptibility distribution. These joint dynamics influence the force of
infection and are, in turn, influenced by the shape of the initial variability.
Intriguingly, we find that certain susceptibility distributions (the
exponential and the gamma) are unchanged through the course of the outbreak,
and lead naturally to power-law behavior in the force of infection; other
distributions often tend towards these "eigen-distributions" through the
process of contagion. The power-law behavior fundamentally alters predictions
of the long-term infection rate, and suggests that first-order epidemic models
that are parameterized in the exponential-like phase may systematically and
significantly over-estimate the final severity of the outbreak.
