Mathematical modeling of the dynamics of the COVID-19 pandemic

Abstract

The detailed review of mathematical models represented by the systems of ordinary differential equations (ODE) and used to study the complex interconnected mechanisms and pathways of the spread of infection diseases over a country, a region or worldwide is presented. The most popular models are based on functional interconnections between the groups of susceptible, exposed, infected and recovered individuals in the community (SIRS and SEIRS models). It is shown that in some cases the number of ODE can be decreased. The SEIRS models and their recent modifications are discussed. The generalized SEIRS-model, which has six parameters and takes into account the noticeable decrease in the availability and quality of medical care with a rapid increase in the number of infected patients, is considered. The asymptotic stability of the stationary solution of the system is studied, and some analytical estimates of the basic reproduction number (BRN) of infection R₀ are presented. It is shown that, depending on the model parameters, a bifurcation in the behaviour of the dynamical system is possible. In those cases the dynamical system could transfer from the conditions with one stability criterion R₀₁<0 to another R₀₂<0 during the same “wave” of the pandemic in the country. The model has been validated on the data for Indonesia obtained from the open sources. The numerical calculations by using the statistical data for a number of European countries, including Ukraine, have been carried out. It is shown that the stability parameter for the model system of ODE corresponds to the BRN index of secondary infection calculated by the statistical analysis of the epidemiological data. The obtained results can be used in the decision making about the measures against the COVID-19 propagation in a given country or region.