German version

The SIR-model

We use calligraphic letters to model the classical SIR-model compartments “Susceptibles” S(t), “Infected” I(t), and “Removed” R(t), especially in order to avoid confusion with the basic reproduction number R0 = R(0) the (effective) reproduction number R = Rt = R(t), which both are widely used in literature and public media.

The original differential equations of the classical SIR-model are:

           S˙    =      - β-IS---               (1)
                           N - 1
           ˙I     =      β-IS---- γI             (2)
                         N - 1
           ˙R     =      γI                      (3)
S (0) = N - 1, I(0) = 1, R(0) = 0                (4)
Here, N = N(t) is the overall population size, which will be assumed to remain constant for all t. Furthermore, β = β(t) denotes the infection rate, and is also dependent on the public health measures (such as mask obligation, distance keeping etc.). γ is the removal rate, its inverse the generation time TG : = γ-1 > 0, i.e. the time it takes for an individual to infect another one, measured since having been infected. In our model, TG is assumed to simulate the duration of the whole infectious period of an individual as well since there is neither any compartment for “infected but not yet infectious” individuals (the so-called “exposed” compartment in SEIR-models) nor any “quarantine” compartment.

The basic resproduction number is R0 : = βTG, i.e. the number of individuals a single individual would infect during all the time TG of its infectious period, thus under the hypothetical assumption that S equaled N- 1 during all the time TG. However, this hypothetical assumption never becomes true in real epidemics since even during the time interval t [0;TG] the “Susceptibles” compartment S(t) strictly monotonically decreases (easily verifiable by (1)).