COVID-19 Simulations with an Expanded SIR-Model
At the beginning of an epidemic typically an exponential spread of the disease or infection
occurs, which means that hospitals can quickly become overloaded . The growth essentially
depends on the transmission rate and the duration of an infection.
To reduce the spreading speed of Corona, consequently in almost all countries of the world
more or less radical measures like mask duty, contact restrictions, curfews or even total lock-
downs were applied. To find the most acceptable ways and the best decisions during an
epidemic or even pandemic mathematical models can help to better understand the
exponential growth under different conditions and disposed measures. The Robert Koch
Institute also uses computer models for influenza simulations and now for COVID-19 to consult
health departments and politicians about an expected spread of the virus. This can answer
What is the maximum number of people infected at the same time and
when will this be?
What is the proportion of the population that will be infected in the worst case?
What can be useful measures to reduce the spread?
What is the influence of vaccination and what vaccination readiness is required?
How many deaths have to be expected depending on the measures?
First mathematical models go back to the work of Ogilvy Kermack and Anderson Gray
McKandrick in the 1927 , who had developed the Susceptible-Infected-Removed Model (SIR-
model), which got its name from the grouping of the population in Susceptible Individuals (S),
Infected Individuals (I) and Resistant Individuals (R) after an infection. Despite this
comparatively simple model they could well model the data of a plague epidemic in Bombay in
The following considerations are based on the general SIR model, which is used to simulate the
COVID-19 pandemic, but which has been expanded to include some additional functions and
calculations. This includes:
Consideration of the death rate and display of the total number of deceased at the end
of the calculation period,
Impact of vaccinations under the influence of the population's willingness to vaccinate,
Lock-down simulation with a reduced infection rate,
Simulation of virus mutations with changed infection or incubation time.
All calculations were done with an Excel spreadsheet program, which can be downloaded from
this web-site together with a description of the theoretical background and a manual for
executing own simulations.
The calculations are displayed as graphs for the total population (N = S+I+R) and the three
groups Susceptibles (S), Infected (I) and Resistants (R) over time in the left diagram, the New
Infections per Day and the 7-Days-Incidence are shown on the right.
COVID-19 Calculations using the SIR-Model
H. Harde, March 2021
Download-English, Download-German, Excel-File
O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. A,
Band 115, 1927, S. 700–721
Michael Li, An introduction to mathematical modeling of infectious diseases, Springer, 2018, Abschnitt 2.1,
Dr. Hermann Harde
22043 Hamburg, Germany
Last update: April 17, 2023
Physics & Climate