Authors:
Reinhard Schuster
1
;
Klaus-Peter Thiele
2
;
Thomas Ostermann
3
and
Martin Schuster
4
Affiliations:
1
Chair of Department of Health Economics, Epidemiology and Medical Informatics, Medical Advisory Board of Statutory Health Insurance in Northern Germany (MDK Nord), 23554 Lübeck, Germany
;
2
Medical Director of the Medical Advisory Service Institution of the Statutory Health Insurance in North Rhine (MDK Nordrhein), 40212 Düsseldorf, Germany
;
3
Chair of Research Methodology and Statistics in Psychology, Witten/Herdecke University, Alfred-Herrhausen-Straße 50, 58448 Witten, Germany
;
4
Faculty of Epidemiology, Christian-Albrechts University Kiel, 24105 Kiel, Germany
Keyword(s):
COVID-19, SIR Model, Torus, Differential Equation, Laplace Operator, Mean Value Operator.
Abstract:
The ongoing COVID-19 pandemic threatens the health of humans, causes great economic losses and may
disturb the stability of the societies. Mathematical models can be used to understand aspects of the dynamics of
epidemics and to increase the chances of control strategies. We propose a SIR graph network model, in which
each node represents an individual and the edges represent contacts between individuals. For this purpose, we
use the healthy S (susceptible) population without immune behavior, two I-compartments (infectious) and two
R-compartments (recovered) as a SIR model. The time steps can be interpreted as days and the spatial spread
(limited in distance for a singe step) shell take place on a 200 by 200 torus, which should simulate 40 thousand
individuals. The disease propagation form S to the I-compartment should be possible on a k by k square (k=5
in order to be in small world network) with different time periods and strength of propagation probability in
the two I co
mpartments. After the infection, an immunity of different lengths is to be modeled in the two R
compartments. The incoming constants should be chosen so that realistic scenarios can arise. With a random
distribution and a low case number of diseases at the beginning of the simulation, almost periodic patterns
similar to diffusion processes can arise over the years. Mean value operators and the Laplace operator on the
torus and its eigenfunctions and eigenvalues are relevant reference points. The torus with five compartments is
well suited for visualization. Realistic neighborhood relationships can be viewed with a inhomogeneous graph
theoretic approach, but they are more difficult to visualize. Superspreaders naturally arise in inhomogeneous
graphs: there are different numbers of edges adjacent to the nodes and should therefore be examined in an
inhomogeneous graph theoretical model. The expected effect of corona control strategies can be evaluated by
comparing the results with various constants used in simulations. The decisive benefit of the models results
from the long-term observation of the consequences of the assumptions made, which can differ significantly
from the primarily expected effects, as is already known from classic predator-prey models.
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