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.
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