The -model

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

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

Here, = (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 T

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