  We analytically address disease outbreaks in large, random networks with
heterogeneous infectivity and susceptibility. The transmissibility $T_{uv}$
(the probability that infection of $u$ causes infection of $v$) depends on the
infectivity of $u$ and the susceptibility of $v$. Initially a single node is
infected, following which a large-scale epidemic may or may not occur. We use a
generating function approach to study how heterogeneity affects the probability
that an epidemic occurs and, if one occurs, its attack rate (the fraction
infected). For fixed average transmissibility, we find upper and lower bounds
on these. An epidemic is most likely if infectivity is homogeneous and least
likely if the variance of infectivity is maximized. Similarly, the attack rate
is largest if susceptibility is homogeneous and smallest if the variance is
maximized. We further show that heterogeneity in infectious period is
important, contrary to assumptions of previous studies. We confirm our
theoretical predictions by simulation. Our results have implications for
control strategy design and identification of populations at higher risk from
an epidemic.
