Mathematical and computer modeling of COVID-19 transmission dynamics in Bulgaria by time-depended inverse SEIR model

Since the end of 2019, with the outbreak of the new virus COVID-19, the world changed entirely in many aspects, with the pandemia affecting the economies, healthcare systems and the global socium. As a result from this pandemic, scientists from many countries across the globe united in their efforts to study the virsus's behavior and are attempting to predict mathematically its infection model in order to limit its impact and developing new methods and models to achieve this goal. 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.