Background: The purpose of our study is to develop a quite precise mathematical model which describes epidemics spread in a country with non-uniform population density. This model gives explanation of quite long duration of the peak of a respiratory infection such as the coronavirus disease 2019 (COVID-19).

Methods: The theory of kinetic equations and fractal analysis are used in our mathematical model. According to our model, COVID-19 spreading takes the form of several spatio-temporal waves developing almost independently and simultaneously in areas with different population density. The intensity of each wave is described by a power-law dependence. The parameters of the dependence are determined by real statistical data at the initial stage of the disease spread.

Results: The results of the model simulation were verified using statistical data for the Republic of Belarus. Based on the developed model, a forecast calculation was made at the end of May, 2020. It was shown that the epidemiological situation in the Republic of Belarus is well described by three waves, which spread respectively in large cities with the highest population density (the first wave), in medium-sized cities with a population of 50−200 thousands people (the second wave), in small towns and rural areas (the third wave). It was shown that a new wave inside a subpopulation with a lower density was born 20−25 days after the appearance of the previous wave. Comparison with actual data several months later showed that the accuracy of forecasting the total number of cases for a period of 3 months for total population in the proposed approach was approximately 3%.

Conclusions: The high accuracy mathematical model is proposed. It describes the development of a respiratory epidemic in a country non-uniform population density without quarantine. The model is useful for predicting the development of possible epidemics in the future. An accurate forecast allows to correctly allocating available resources to effectively withstand the epidemic.