In 2020, the COVID-19 pandemic has influenced human life very quickly and globally to an enormous extent. The discussion about appropriate actions to control it has stimulated me to perform own calculations on base of the SEIR model.

## The SEIR model

The SEIR model (https://en.wikipedia.org /wiki/Compartmental_models_in_ epidemiology#The_SEIR_model) is a mathematical model for the description of epidemics. It is able to realistically reproduce the course of an epidemic.

The letters SEIR stand for a total of four fractions,

*S*(susceptible, not yet infected),*E*(exposed, infected, but not yet infectious),*I*(infectious), and*R*(recovered as well as removed from the infection scenario by isolation or death),

*S* + *E* + *I* + *R* = 1.

These states can be run through sequentially. The time derivatives of these quantities are described by

*S'* = -*βSI*,

*E'* = *βSI* - *aE*,

*I'* = *aE* - *γI*,

and

*R'* = *γI*.

These four equations represent a system of ordinary differential equations, which can be solved numerically as an initial value problem e.g. using Runge-Kutta methods (https://en.wikipedia.org/wiki/Runge–Kutta_methods). The parameters of the system are the three rates

*β*(infection rate),*γ*(rate of recovery, including rates of isolation and death) and*a*(inverse average latency period, when infected are not yet infectious; not controllable).

The important **basic reproduction number** *R*_{0}
(https://en.wikipedia.org
/wiki/Basic_reproduction_number)
indicates how many individuals would be infected by an infectious person
on average, if everyone else was not yet infected (*S* = 1).
In the framework of the SEIR model it is given by the ratio of the
two controllable rates:

*R*_{0} = *β* / *γ*.

This relation results from the condition that the time derivative of the total
part of infected (*E* + *I*) vanishes at the climax of the
epidemic:

*E'* + *I'* = *βSI* - *γI* = 0,

*βS* / *γ* = *R*_{eff} = 1

with the effective reproduction number

*R*_{eff} = *R*_{0}*S*,

which at this point is exactly 1 (in the average then an infectious person infects exactly one person). For COVID-19, the basic reproduction number is estimated to be in the range from 2.4 to 3.3.

For very large rates *a* and *E* = 0 the SEIR model
transform to the simpler
SIR model.

### Model calculations

Here first of all the results of a calculation for uncontrolled parameters are shown (parameters and initial values as used in the calculation on https://de.wikipedia.org/wiki/SEIR-Modell):

**Figure 1:** Resulting fractions in dependence of time
*t* in days with *R*_{0} = 2.4,
*γ* = 1/3 d^{-1},
*β* = *γR*_{0} = 0.8 d^{-1},
*a* = 1/5.5 d^{-1},
*E*_{0} = *E*(*t* = 0) = 4.8077·10^{-4},
and
*I*_{0} = *I*(*t* = 0) = 1.2019·10^{-4}. The maximum in *E* + *I*
is obtained at a time when
*S* = 1 / *R*_{0} and
*R*_{eff} = 1, respectively. The maxima in *E* and
*I*, however, are obtained a little earlier and later, respectively.

#### Control of parameters

For control of an epidemic it is attempted to reduce the basic
reproduction number
*R*_{0} = *β* / *γ*.
This can be achieved by either reducing the infection rate *β*
(e.g. by restriction of contacts and improvement of hygiene) or
by increasing the rate *γ* (by means of large
numbers of infection tests and prompt isolation of infectious).

It is interesting to compare the effect of an exclusive reduction
of *β* with that of an exclusive increase of *γ*.
For this purpose in the following results of appropriate calculations
are shown:

For the range of time
0 ≤ *t* ≤ *t*_{1} the parameters
*β* = *β*_{0} and
*γ* = *γ*_{0}, respectively,
are used; for
*t*_{1} ≤ *t* ≤ *t*_{2}
the parameters *β* = *β*_{1} and
*γ* = *γ*_{1}, respectively, and for
*t* ≥ *t*_{2} again the parameters
*β* = *β*_{0} and
*γ* = *γ*_{0}, respectively.

**Figure 2:** Resulting fraction *I* of
infectious in dependence on the time
*t* in days with *R*_{0} = 2.4,
*γ*_{0} = 1/3 d^{-1},
*β*_{0} = 0.8 d^{-1},
*a* = 1/5.5 d^{-1},
*E*_{0} = 4.8077·10^{-4}, and
*I*_{0} = 1.2019·10^{-4}.
In the range
*t*_{1} ≤ *t* ≤ *t*_{2},
*β* is decreased by a factor of 3/5 (corresponding to
*R*_{0} = 1.44) and by a factor of 1/3
(corresponding to *R*_{0} = 0.8), respectively.
For comparison, the red curve shows the result for an unchanged parameter.

**Figure 3:** Resulting fraction *I* of
infectious in dependence of time
*t* in days with *R*_{0} = 2.4,
*γ*_{0} = 1/3 d^{-1},
*β*_{0} = 0.8 d^{-1},
*a* = 1/5.5 d^{-1},
*E*_{0} = 4.8077·10^{-4} and
*I*_{0} = 1.2019·10^{-4}.
In the range
*t*_{1} ≤ *t* ≤ *t*_{2},
*γ* is increased by a factor of 5/3 (corresponding to
*R*_{0} = 1.44) and a factor of 3 (corresponding
to *R*_{0} = 0.8), respectively. For comparison,
the red curve shows the result for an unchanged parameter.

A comparison of the figures shows that an exclusive increase of the
rate *γ* by means of massive testing for
infection and prompt isolation of infectious (Figure 3) results in
a faster decrease of *I* in comparison to a corresponding
exclusive decrease of the infection rate (Figure 2), at least in
the framework of the SEIR model with the parameters chosen here.
In Figure 2 the number of infectious even increases immediately after
*t* = *t*_{1} for some time.

In both cases one has to expect of a second epidemic wave, if, for
a significant reduction of *R*_{0} by a factor of 1/3
(blue curves), the actions are canceled completely to early (here after
90 days).

It should be kept in mind, that the SEIR model certainly cannot describe exactly real epidemics. In particular, the implicit assumption of exponential distributions of the transition times in this model is questionable.

### Online calculator

You can perform own calculations with this online calculator for the SEIR model, implemented in JavaScript. First the parameters are set to that used for Figure 1. The button "Calculate" is used to apply changes of their values in the entry fields for recalculation of the diagram.

The results can be exported as a SVG graphic and as a CSV file. The corresponding buttons lead to the display of dialogs, their content can be used by copy (usually Ctrl+A and Ctrl+C) and paste.