  This paper is an exploratory study of two epidemiological questions on a
worldwide basis. How fast is the disease spreading? Are the restrictions
(especially mobility restrictions) for people bring the expected effect? To
answer the first question, we propose a tool for estimating the reproduction
number of epidemic (the number of secondary infections $R_t$) based on the SEIR
model and compare it with an non-model $R_t$ estimation. To measure the $R_t$
of COVID-19 for different countries, real-time data on coronavirus daily cases
of infections, recoveries, deaths are retrieved from the Center for Systems
Science and Engineering (CSSE) at Johns Hopkins University. To assess the
effectiveness of mobility restrictions for the COVID-19 pandemic in 2020, the
correlations between the $R_t$ and people's mobility (based on the Apple
mobility index) are presented. The correlations were considered for 12
countries and for most of them, the correlations are negative. This shows a
delay in the implementation of mobility restrictions - the countries imposed
them in response to growth of new COVID-19 cases, rather than preventively.
