Conference Proceedings
Dynamical Properties of Epidemics: From Steady States to " Electrocardiograms "
Francois Louchet*
Corresponding Author: Dr. Francois Louchet, Grenoble University (retired), Professor Emeritus, 284 chemin du Pre Roudon. 38410 St Martin dUriage, France
Citation: Louchet F. (2021) Dynamical Properties of Epidemics: From Steady States to
Copyrights: ©2021 Louchet F. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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In a previous paper, we established a general evolution equation ruling epidemics infection rates. This differential equation was based on the well-known SEIR (Susceptible Exposed Infected Recovered) assumptions. We showed that, for reproduction factors R0 larger than 1, the system always converges to an "attractor", i.e., a stable state at which the infected people proportion X keeps a constant value X*=1-1/R0, usually ascribed to an "immunity threshold", which it is not. This constant value results from a balance between infection and recovery rates. This means that the people in the "infected box" are never the same ones, being continuously refreshed.

We explore here the various properties of such attractors. One of them is that the constant flux of new infections, balanced by an equal number of recoveries (or deaths), should appear on cumulative contamination curves as straight lines with positive slopes. This prediction is now widely verified in COVID-19 infection data (e.g., Johns Hopkins University), showing several straight segments, connected by transients corresponding to changes in sanitary policies. 

Another interesting property of attractors is the way in which the system travels toward the attractor. We expected a smooth convergence for low R0 values, replaced by a hierarchy of multi-stable cycles as R0 increases, eventually leading to a chaotic behavior. These last features were not observed during the first contamination wave, due to low R0 values, but recent available data corresponding to the second contamination wave, with significantly higher R0 levels, show oscillations reminiscent of those usually found in electrocardiograms (ECG). Such & quot; ECGs & quot; are shown to correspond to bi-stable states followed by 4-stable ones, approximately above resp. 200 and 400 daily contamination rates per million in France and Germany. Such thresholds may be used to renormalize contamination data and associated R 0 values between different countries.

Keywords: Epidemic evolution, Reproduction factor, Attractor, Instability, Chaos