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 demonstrate 

that a class of widely used models is fundamentally flawed and cannot 

account for some important features of the viral spread. We suggest an 

integral equation based method that can be implemented in most of the 

reported models. 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 measures should be based on any unimproved model.