  The outcome of SIR epidemics with heterogeneous infective lifetimes, or
heterogeneous susceptibilities, can be mapped onto a directed percolation
process on the underlying contact network. In this paper we study SIR models
where heterogeneity is a result of the degree dependence of disease
transmission rates. We develop numerical methods to determine the epidemic
threshold, the epidemic probability and epidemic size close to the threshold
for configuration model contact networks with arbitrary degree distribution and
an arbitrary matrix of transmission rates (dependent on transmitting and
receiving node degree). For the special case of separable transmission rates we
obtain analytical expressions for these quantities. We propose a categorization
of spreading processes based on the ratio of the probability of an epidemic and
the expected size of an epidemic, and demonstrate that this ratio has a complex
dependence on the degree distribution and the degree-dependent transmission
rates.
  For scale-free contact networks and transmission rates that are power
functions of transmitting and receiving node degrees, the epidemic threshold
may be finite even when the degree distribution powerlaw exponent is below
$\gamma < 3$. We give an expression, in terms of the degree distribution and
transmission rate exponents, for the limit at which the epidemic threshold
vanishes. We find that the expected epidemic size and the probability of an
epidemic may grow nonlinearly above the epidemic threshold, with exponents that
depend not only on the degree distribution powerlaw exponent, but on the
parameters of the transmission rate degree dependence functions, in contrast to
ordinary directed percolation and previously studied variations of the SIR
model.
