  In the absence of other tools, monitoring the effects of protective measures,
including social distancing and forecasting the outcome of outbreaks is of
immense interest. Real-time data is noisy and very often hampered by systematic
errors in reporting. Detailed epidemic models may contain a large number of
empirical parameters, which cannot be determined with sufficient accuracy. In
this paper, we show that the cumulative number of deaths can be regarded as a
master variable, and the parameters of the epidemic such as the basic
reproduction number, the size of the susceptible population, and the infection
rate can be determined. In the SIR model, we derive an explicit single variable
differential equation for the evolution of the cumulative number of fatalities.
We show that the epidemic in Spain, Italy, and Hubei Province, China follows
this master equation closely. We discuss the relationship with the logistic
growth model, and we show that it is a good approximation when the basic
reproduction number is less than $2.3$. This condition is valid for the
outbreak in Hubei, but not for the outbreaks in Spain, Italy, and New York. The
difference is in the shorter infectious period in China, probably due to the
separation policy of the infected. For more complex models, with more internal
variables, such as the SEIR model, the equations derived from the SIR model
remain valid approximately, due to the separation of timescales.
