Before revision:  Taking the example of COVID-19 data for New York City, we show that our model yields a significantly larger estimate for the initial basic reproduction rate than other models, much more accurately accounts for the dynamics of the epidemic after  restrictive public congregation measures were introduced, and provides a novel way to determine the incubation period . We suggest that no decisions about public health or economic measuresshould be based on any unimproved model .
After revision:  Taking the example of COVID-19 data for New York City, we show that our model yields a significantly larger estimate for the initial basic reproduction rate than other models, much more accurately accounts for the dynamics of the epidemic after restrictive public congregation measures were introduced, and provides a novel way to determine the incubation period , and most importantly, as we demonstrate for several countries, this method allows an accurate monitoring of R_0 and thus a fine-tuning of any restrictive measures. Integral equation based models do not only provide the conceptually correct description, they also have more predictive power than differential equation based models, therefore we do not see any reason for using the latter .