  In this paper we explore a time-depended SEIR model, in which the dynamics of
the infection in four groups from a selected target group (population), divided
according to the infection, are modeled by a system of nonlinear ordinary
differential equations. Several basic parameters are involved in the model:
coefficients of infection rate, incubation rate, recovery rate. The
coefficients are adaptable to each specific infection, for each individual
country, and depend on the measures to limit the spread of the infection and
the effectiveness of the methods of treatment of the infected people in the
respective country. If such coefficients are known, solving the nonlinear
system is possible to be able to make some hypotheses for the development of
the epidemic. This is the reason for using Bulgarian COVID-19 data to first of
all, solve the so-called "inverse problem" and to find the parameters of the
current situation. Reverse logic is initially used to determine the parameters
of the model as a function of time, followed by computer solution of the
problem. Namely, this means predicting the future behavior of these parameters,
and finding (and as a consequence applying mass-scale measures, e.g.,
distancing, disinfection, limitation of public events), a suitable scenario for
the change in the proportion of the numbers of the four studied groups in the
future. In fact, based on these results we model the COVID-19 transmission
dynamics in Bulgaria and make a two-week forecast for the numbers of new cases
per day, active cases and recovered individuals. Such model, as we show, has
been successful for prediction analysis in the Bulgarian situation. We also
provide multiple examples of numerical experiments with visualization of the
results.
