A Discrete SIR Model with Spatial Distribution on a Torus for COVID-19 Analysis using Local Neighborhood Properties

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 compartments. 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.