  The duration of the infectious period is a crucial determinant of the ability
of an infectious disease to spread. We consider an epidemic model that is
network based and non-Markovian, containing classic Kermack-McKendrick,
pairwise, message passing, and spatial models as special cases. For this model,
we prove a monotonic relationship between the variability of the infectious
period (with fixed mean) and the probability that the infection will reach any
given subset of the population by any given time. For certain families of
distributions, this result implies that epidemic severity is decreasing with
respect to the variance of the infectious period. The striking importance of
this relationship is demonstrated numerically. We then prove, with a fixed
basic reproductive ratio (R0), a monotonic relationship between the variability
of the posterior transmission probability (which is a function of the
infectious period) and the probability that the infection will reach any given
subset of the population by any given time. Thus again, even when R0 is fixed,
variability of the infectious period tends to dampen the epidemic. Numerical
results illustrate this but indicate the relationship is weaker. We then show
how our results apply to message passing, pairwise, and Kermack-McKendrick
epidemic models, even when they are not exactly consistent with the stochastic
dynamics. For Poisson contact processes, and arbitrarily distributed infectious
periods, we demonstrate how systems of delay differential equations and
ordinary differential equations can provide upper and lower bounds,
respectively, for the probability that any given individual has been infected
by any given time
