  This paper seeks to study the evolution of the COVID-19 pandemic based on
daily published data from Worldometer website, using a time-dependent SIR
model. Our findings indicate that this model fits well such data, for different
chosen periods and different regions. This well-known model, consisting of
three disjoint compartments, susceptible , infected , and removed , depends in
our case on two time dependent parameters, the infection rate $\beta(t)$ and
the removal rate $\rho(t)$. After deriving the model, we prove the local
exponential behavior of the number of infected people, be it growth or decay.
Furthermore, we extract a time dependent replacement factor $\sigma_s(t)
={\beta(t)}s(t)/{\rho(t) }$, where $s(t)$ is the ratio of susceptible people at
time $t$. In addition, $i(t)$ and $r(t)$ are respectively the ratios of
infected and removed people, based on a population of size $N$, usually assumed
to be constant. Besides these theoretical results, the report provides
simulations on the daily data obtained for Germany, Italy, and the entire
World, as collected from Worldometer over the period stretching from April 2020
to June 2022. The computational model consists of the estimation of $\beta(t)$,
$\rho(t)$ and $s(t)$ based on the time-dependent SIR model. The validation of
our approach is demonstrated by comparing the profiles of the collected $i(t),
r(t)$ data and those obtained from the SIR model with the approximated
parameters. We also consider matching the data with a constant-coefficient SIR
model, which seems to be working only for short periods. Thus, such model helps
understanding and predicting the evolution of the pandemics for short periods
of time where no radical change occurs.
