Covid-19: Qualitative Change in the Behavior of the “Virus vs Human” System - From Limit Cycle to Sustained Focus

Abstract

The high contagiousness of the latest strains of Covid-19 qualitatively changes the behavior of the "virus vs human" system. Numerical experiments with a model of the Covid-19 epidemic in Moscow have shown that a reproduction number R0 of about 4 is critical, defining a qualitative change in the dynamics of the epidemic. Below this value (observed until 2022), the long-term forecast tends to undamped oscillations; above this value, it is described by damped oscillations: amplitudes of the epidemic waves get smaller and smaller, with a constant, very high background level of morbidity (or high-intensity vaccination) that maintains the state of natural immunity at a level close to 100% (reaching 93.7% for the current R0 value of about 16). At the limit, the system tends to a stable equilibrium point. Here we consider a reduced model of epidemic dynamics. Its study (search for equilibrium solutions, analysis of their stability, construction a bifurcation diagram and a phase portrait) confirms the presence of points of qualitative change in the behavior of the "virus vs human" system (bifurcation points). Some practical results for Moscow are given. A further increase in the contagiousness of the virus does not change the picture significantly, thus more infectious strains are not to be feared. The key parameter of the study is the function of the immunity level depending on the time after the disease. The damping of omicron waves (oscillations), observed recently in many countries, is a confirmation of the correctness of the accepted hypotheses.