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It is of vital importance to understand and track the dynamics of rapidlyunfolding epidemics. The health and economic consequences of the currentCOVID-19 pandemic provide a poignant case. Here we point out that since theyare based on differential equations, the most widely used models of epidemicspread are plagued by an approximation that is not justified in the case of thecurrent COVID-19 pandemic. Taking the example of data from New York City, weshow that currently used models significantly underestimate the initial basicreproduction number ($R_0$). The correct description, based on integralequations, can be implemented in most of the reported models and it much moreaccurately accounts for the dynamics of the epidemic after sharp changes in$R_0$ due to restrictive public congregation measures. It also provides a novelway to determine the incubation period, and most importantly, as we demonstratefor several countries, this method allows an accurate monitoring of $R_0$ andthus a fine-tuning of any restrictive measures. Integral equation based modelsdo not only provide the conceptually correct description, they also have morepredictive power than differential equation based models, therefore we do notsee any reason for using the latter.

Why integral equations should be used instead of differential equations to describe the dynamics of epidemics