  In this work, we adapt the epidemiological SIR model to study the evolution
of the dissemination of COVID-19 in Germany and Brazil (nationally, in the
State of Paraiba, and in the City of Campina Grande). We prove the well
posedness and the continuous dependence of the model dynamics on its
parameters. We also propose a simple probabilistic method for the evolution of
the active cases that is instrumental for the automatic estimation of
parameters of the epidemiological model. We obtained statistical estimates of
the active cases based the probabilistic method and on the confirmed cases
data. From this estimated time series we obtained a time-dependent contagion
rate, which reflects a lower or higher adherence to social distancing by the
involved populations. By also analysing the data on daily deaths, we obtained
the daily lethality and recovery rates. We then integrate the equations of
motion of the model using these time-dependent parameters. We validate our
epidemiological model by fitting the official data of confirmed, recovered,
death, and active cases due to the pandemic with the theoretical predictions.
We obtained very good fits of the data with this method. The automated
procedure developed here could be used for basically any population with a
minimum of extra work. Finally, we also propose and validate a forecasting
method based on Markov chains for the evolution of the epidemiological data for
up to two weeks.
