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),
of states of individuals in the total population with
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 R0 (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:
R0 = 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 = Reff = 1
with the effective reproduction number
Reff = R0S,
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 R0 = 2.4, γ = 1/3 d-1, β = γR0 = 0.8 d-1, a = 1/5.5 d-1, E0 = E(t = 0) = 4.8077·10-4, and I0 = I(t = 0) = 1.2019·10-4. The maximum in E + I is obtained at a time when S = 1 / R0 and Reff = 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 R0 = 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 ≤ t1 the parameters β = β0 and γ = γ0, respectively, are used; for t1 ≤ t ≤ t2 the parameters β = β1 and γ = γ1, respectively, and for t ≥ t2 again the parameters β = β0 and γ = γ0, respectively.
Figure 2: Resulting fraction I of infectious in dependence on the time t in days with R0 = 2.4, γ0 = 1/3 d-1, β0 = 0.8 d-1, a = 1/5.5 d-1, E0 = 4.8077·10-4, and I0 = 1.2019·10-4. In the range t1 ≤ t ≤ t2, β is decreased by a factor of 3/5 (corresponding to R0 = 1.44) and by a factor of 1/3 (corresponding to R0 = 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 R0 = 2.4, γ0 = 1/3 d-1, β0 = 0.8 d-1, a = 1/5.5 d-1, E0 = 4.8077·10-4 and I0 = 1.2019·10-4. In the range t1 ≤ t ≤ t2, γ is increased by a factor of 5/3 (corresponding to R0 = 1.44) and a factor of 3 (corresponding to R0 = 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 = t1 for some time.
In both cases one has to expect of a second epidemic wave, if, for a significant reduction of R0 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.