COVID-19: From Limit Cycle to Stable Focus

The study aims at investigating a new fundamental property of infectious diseases with natural adaptive immunity that weakens over time—qualitative change (bifurcation) in the behavior of the “virus vs. human” system with an increase in contagiousness. Numerical experiments with a model of the COVID-19 epidemic in Moscow have demonstrated that when the reproduction number R 0 is about 4, a qualitative change (bifurcation) occurs in the behavior of the virus–human system. Below this value, the long-term forecast tends toward undamped oscillations; above it, the forecast shows damped oscillations: the amplitudes of epidemic waves decrease gradually, with a constant, very high background level of morbidity that keeps natural immunity near 100%. To confirm this result analytically, we use an original modification of the Euler–Lotka renewal equation, which describes the dynamics of infected patients distributed by disease duration (time since infection) and accounts for immunity. To construct a bifurcation diagram, which illustrates the dependence of the equilibrium stability on the parameter R 0 , we linearize the equation in the vicinity of the equilibrium point and examine its numerical approximation (discrete form). This approximation can be interpreted as a Leslie model, with the matrix elements dependent on the parameter R 0 . By examining the roots of the corresponding Lotka polynomial, we can assess the stability of the equilibrium point and verify the basic assumption about the change in the properties of the system with increasing R 0 —about the transition from undamped oscillations to damped ones. For the bifurcation diagram, we use the functions obtained from the simulation of the COVID-19 epidemic in Moscow. However, observations of the epidemic in other cities and countries support the primary finding of our study regarding the attenuation of epidemic waves.