  We study a general epidemic model with arbitrary recovery rate distributions.
This simple deviation from the standard setup is sufficient to prove that
heterogeneity in the dynamical parameters can be as important as the more
studied structural heterogeneity. Our analytical solution is able to predict
the shift in the critical properties induced by heterogeneous recovery rates.
Additionally, we show that the critical value of infectivity tends to be
smaller than the one predicted by quenched mean-field approaches in the
homogeneous case and that it can be linked to the variance of the recovery
rates. We then illustrate the role of dynamical--structural correlations, which
allow for a complete change in the critical behavior. We show that it is
possible for a power-law network topology to behave similarly to a homogeneous
structure by an appropriate tuning of its recovery rates, and vice versa.
Finally, we show how heterogeneity in recovery rates affects the network
localization properties of the spreading process.
