A Local and Time Resolution of the COVID-19 Propagation—A Two-Dimensional Approach for Germany Including Diffusion Phenomena to Describe the Spatial Spread of the COVID-19 Pandemic

Günter Bärwolff

doi:10.3390/physics3030033

Abstract

The understanding of factors that affect the dissemination of a viral infection is fundamental to help combat it. For instance, during the COVID-19 pandemic that changed the lives of people all over the world, one observes regions with different incidences of cases. One can speculate that population density might be one of the variables that affect the incidence of cases. In populous areas, such as big cities or congested urban areas, higher COVID-19 incidences could be observed than in rural regions. It is natural to think that if population density is such an important factor, then a gradient or difference in population density might lead to a diffusion process that will proceed until equilibrium is reached. The aim of this paper consists of the inclusion of a diffusion concept into the COVID-19 modeling. With this concept, one covers a gradient-driven transfer of the infection next to epidemic growth models (SIR-type models). This is discussed for a certain period of the German situation based on the quite different incidence data for the different federal states of Germany. With this ansatz, some phenomena of the actual development of the pandemic are found to be confirmed. The model provides a possibility to investigate certain scenarios, such as border-crossings or local spreading events, and their influence on the COVID-19 propagation. The resulting information can be a basis for the decisions of politicians and medical persons in charge of managing a pandemic.

Highlights

The curves of infected, susceptible and recovered people, which can be found in every newspaper, describe the global pandemic behavior of the whole country, e.g., Italy, France or Germany
Based on the local-dependent density of people and a diffusion model, the COVID-19 propagation is considered here to be resolved in a finer manner
The result of the diffusion model coupled with the stochastic SIRmodel (ν = 0.25) over the period of April 26th to May 5th is shown in Figure 7 (∆t = 1 day, δt = ∆t /10)

Summary

Introduction

The curves of infected, susceptible and recovered people, which can be found in every newspaper, describe the global pandemic behavior of the whole country, e.g., Italy, France or Germany. The mathematical modeling of COVID-19 with susceptible-infected-recovered (SIR) type models [1,2,3,4,5,6,7] leads to averaged results and does not take into account unequal population numbers or population densities. Based on the local-dependent density of people and a diffusion model, the COVID-19 propagation is considered here to be resolved in a finer manner.

The Mathematical Diffusion Model

Data of the Different Federal States of Germany

The Numerical Solution of the Initial Boundary Value Problem

The Qualitative Behavior of the Diffusion Model

Discussion and Conclusions