Urban Scaling of COVID-19 epidemics

Susceptible-Invective-Recovered (SIR) mathematical models are in high demand due to the COVID-19 pandemic. They are used in their standard formulation, or through the many variants, trying to fit and hopefully predict the number of new cases for the next days or weeks, in any place, city, or country. Such is key knowledge for the authorities to prepare for the health systems demand or to apply restrictions to slow down the infectives curve. Even when the model can be easily solved ---by the use of specialized software or by programming the numerical solution of the differential equations that represent the model---, the prediction is a non-easy task, because the behavioral change of people is reflected in a continuous change of the parameters. A relevant question is what we can use of one city to another; if what happened in Madrid could have been applied to New York and then, if what we have learned from this city would be of use for S\~ao Paulo. With this idea in mind, we present an analysis of a spreading-rate related measure of COVID-19 as a function of population density and population size for all US counties, as long as for Brazilian cities and German cities. Contrary to what is the common hypothesis in epidemics modeling, we observe a higher {\em per-capita} contact rate for higher city's population density and population size. Also, we find that the population size has a more explanatory effect than the population density. A contact rate scaling theory is proposed to explain the results.