Before revision:  We demonstrate how to incorporate this heterogeneity into a wide class of epidemiological models , and derive a non-linear dependence of the effective reproduction number Re on the susceptible population fraction S. Persistent heterogeneity has three important consequences compared to the effects of short-term overdispersion: (1) It results in a major modification of the early epidemic dynamics;
After revision:  We demonstrate how to incorporate them phenomenologically into a wide class of epidemiological models , and derive a non-linear dependence of the effective reproduction number Re on the susceptible population fraction S. Persistent heterogeneity has three important consequences compared to the effects of short-term overdispersion: (1) It results in a major modification of the early epidemic dynamics;