  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.
