Modeling the spread of Covid-19 under active management

Abstract

Classical approaches to modeling the spread of epidemics are based on two assumptions: the exponential growth of the total number of infections and the saturation due to the herd immunity. With Covid-19, the growth is essentially power-type, especially during the middle stages, and the saturation is currently mostly due to the protective measures. Focusing on these features and the role of epidemic management, we obtain differential equations for the total number of detected cases of Covid-19, which describe the actual curves in many countries almost with the accuracy of physics laws. The two-phase solution we propose works very well almost for the whole periods of the spread practically in all countries we analyzed that reached the saturation during the first waves. Bessel functions play the key role in our approach. Due to a very small number of parameters, namely, the initial transmission rate and the intensity of the hard and soft measures, we obtain a convincing explanation of the surprising uniformity of the curves of the total numbers of detected infections in many different areas. This theory can serve as a tool for forecasting the epidemic spread and evaluating the efficiency of the protective measures, which is very much needed for epidemics. As its practical application, the computer programs aimed at providing projections for late stages of Covid-19 proved to be remarkably stable in many countries, including Western Europe, the USA and some in Asia. We provide a projection for the saturation of the 3rd wave in the USA: the corresponding number of total, detected or not, cases can presumably reach then the herd immunity levels (G-strains). This can be used to analyze the efficiency of the vaccinations.