It is of vital importance to understand and track the dynamics of rapidly unfolding epidemics. The health and economic consequences of the current COVID-19 pandemic provide a poignant case. Here we point out that since they are based on differential equations, the most widely used models of epidemic spread are plagued by an approximation that is not justified in the case of the current COVID-19 pandemic. Taking the example of data from New York City, we show that currently used models significantly underestimate the initial basic reproduction number ($R_0$). The correct description, based on integral equations, can be implemented in most of the reported models and it much more accurately accounts for the dynamics of the epidemic after sharp changes in $R_0$ due to restrictive public congregation measures. It also 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.
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