  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.
