The COVID-19 outbreak in Germany – Models and Parameter Estimation

Authors

  • Peter Heidrich Mathematical Institute, University of Koblenz–Landau, 56070 Koblenz
  • Moritz Schäfer Mathematical Institute, University of Koblenz–Landau, 56070 Koblenz
  • Mostafa Nikouei Mathematical Institute, University of Koblenz–Landau, 56070 Koblenz
  • Thomas Götz Mathematical Institute, University of Koblenz–Landau, 56070 Koblenz

DOI:

https://doi.org/10.5614/cbms.2020.3.1.5

Keywords:

COVID–19, Epidemiology, Disease dynamics, SEIRD–model, Parameter estimation, Adjoint equations, Metropolis algorithm

Abstract

Since the end of 2019 an outbreak of a new strain of coronavirus, called SARS–CoV–2, is reported from China and later also from other parts of the world. Since 21 January 2020, World Health Organization (WHO) reports daily data on confirmed cases and deaths from both China and other countries [1]. The Johns Hopkins University [2] collects those data from various sources worldwide on a daily basis. For Germany, the Robert–Koch–Institute (RKI) also issues daily reports on the current number of infections and infection related fatal cases and also provides estimates of several disease-related parameters [3]. In this work we present an extended SEIRD–model to describe these disease dynamics in Germany. The model takes into account the susceptible, exposed, infected, recovered and deceased fractions of the population. Epidemiological parameters like the transmission rate, lethality or the detection rate of infected individuals are estimated by fitting the model output to available data. For the parameter estimation itself we compare two methods: an adjoint based approach and a Monte–Carlo based Metropolis algorithm.

References

World Health Organization (WHO), Novel Coronavirus (2019-nCoV) situation reports. www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports, last visited: 18 June 2020

Johns Hopkins University, Time series of confirmed COVID-19 cases globally, github.com/CSSEGISandData/COVID-19/blob/master/csse COVID 19 data/csse COVID 19 time series/time series COVID19 confirmed global.csv, last visited: 18 June 2020.

Robert-Koch-Institute, Daily situation reports. www.rki.de/DE/Content/InfAZ/N/Neuartiges Coronavirus/Situationsberichte/Gesamt.html, last visited: 18 June 2020.

Federal Government of Germany, Guidelines for reducing social contacts. www.bundesregierung.de/breg-de/themen/coronavirus/besprechung-der-bundeskanzlerin-mit-den-regierungschefinnenund-regierungschefs-der-laender-1733248 (in German), last visited: 18 June 2020.

Federal Government of Germany, Contact Restrictions Extended. https://www.bundesregierung.de/breg-en/news/fahrplan-corona-pandemie-1744276, last visited: 18 June 2020.

Chambers, M., Carrel, P., Germany eases lockdown, with 'emergency brake' on hand if needed. https://www.reuters.com/article/us-health-coronavirus-merkel-idUSKBN22I24E, Accessed on May 6, 2020 last visited: 18 June 2020.

Kermack, W.O., McKendrick, A.G., Contributions to the mathematical theory of epidemics"I, Bltn. Mathcal. Biology., 53, 33 55, 1991.

Martcheva, M., An introduction to mathematical epidemiology, Springer, 2015.

Read, J. M., Bridgen, J.R.E., Cummings, D.A.T., Ho, A., Jewell, C.P., Novel coronavirus 2019-nCoV: early estimation of epidemiological parameters and epidemic predictions. medRxiv www.medrxiv.org/content/10.1101/2020.01.23.20018549v2, last visited: 18 June 2020.

Statistisches Bundesamt (Germany), Bevolkerungsstand (31.12.2018). https://www.destatis.de/DE/Home/ inhalt.html, 2020.

Robert-Koch-Institute, Modellierung von Beispielszenarien der SARS-CoV-2-Epidemie 2020 in Deutschland. https://www.rki.de/DE/Content/InfAZ/N/Neuartiges Coronavirus/Modellierung Deutschland.pdf? blob=publicationFile, last visited: 22 June 2020.

Gotz, T., Heidrich, P., Novel Coronavirus (2019-nCoV) situation reports. doi: https://doi.org/10.1101/2020.04.23.20076992

Robert-Koch-Institute, Corona fact sheet. https://www.rki.de/DE/Content/InfAZ/N/Neuartiges Coronavirus/Steckbrief.html, last visited: 18 June 2020.

Lenhart, S., Workman, J.T., Optimal control applied to biological models,. CRC Press, 2007.

Nocedal, J., Wright, S., Numerical Optimization, Springer, 2006.

Armijo, L., Minimization of functions having Lipschitz continuous first partial derivatives, Pacific J. Math., 16 (1), 1 3, 1966.

Schafer, M., Gotz, T., Modelling Dengue Fever Epidemics in Jakarta, Int. J. Appl. Comput. Math, 6, 2020.

Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.W., Teller, A.H., Teller,E., Equation of State Calculations by Fast Computing Machines, J.Chem. Phys., 21, 1087 1092, 1953.

Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B., Bayesian Data Analysis, 2nd Edition, Chapman and Hall, London, 1996.

Gilks, W.R., Richardson, S., Spiegelhalter, D.J., Markov chain Monte Carlo in Practice, Chapman and Hall/CRC, 1996.

Rusatsi, D.N., Bayesian analysis of SEIR epidemic models, Dissertation, Lappeenranta University of Technology. http://lutpub.lut.fi/, last visited: 18 June 2020.

Downloads

Published

2020-07-10

Issue

Section

Articles