A BRIEF THEORY OF EPIDEMIC KINETICS
François Louchet
Biology 2020, 9(6), 134; https://doi.org/10.3390/biology9060134
Open access. Received: 11 April 2020 / Revised: 7 June 2020 / Accepted: 17 June 2020 / Published: 22 June 2020
PDF Version: https://www.mdpi.com/2079-7737/9/6/134/pdf
On the basis of the Theory of Dynamical Systems, we propose a simple theoretical approach for the expansion of contagious diseases. The infection develops through contacts between contagious and exposed people, with a rate proportional to the number of contagious and of non-immune individuals, to contact duration and turnover, inversely proportional to the efficiency of protection measures, and balanced by the average individual recovery response. The initial exponential growth is readily hindered by the increasing recovery rate, and also by the size reduction of the exposed population. The system converges towards a stable attractor whose value is expressed in terms of the "reproductive rate" Ro,
Two main predictions of this study were already confirmed in by statistical data from "Our World in Data" (Johns Hopkins University):
i) Cumulative infection curves of most countries exhibit characteristic straight segments (signature of the predicted attractors) connected by transients (corresponding to changes in disease management policies)
ii) The study predicts a possible transition at high R0 values to a chaotic behaviour.through a bifurcating hierarchy of stable cycles starting with bi-stable oscillations in the number of sick people. This is exactly what is observed in Luxemburg in november and december 2020 and in France in january-february 2021 2020.
3. DYNAMICAL PROPERTIES OF EPIDEMICS:
from steady states to "electrocardiograms"
Francois Louchet
SciTech conference 18-19 may 2021
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ABSTRACT
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, the box content 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 "ECGs" 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 R0 values between different countries.
Keywords: Epidemic evolution, Reproduction factor, Attractor, Instability, Chaos