Figure 1: Model structure and transitions through the model
compartments. Susceptible individuals (S) become exposed by effective
contact with any infectious (red compartments). Exposed individuals
progress through three stages, E1, E2,
and E3, before illness onset. By testing and clinical
diagnosis, both pre-symptomatic (E3) and unknown (U)
infectious individuals can be detected and thereby enter compartment I.
Severe cases requiring hospitalization (H) or intensive care (C) are
included. Infectious individuals are removed from the chain of
transmission when they recover (R) or die (D) from the disease.
Compartments covered by the green shadow are detected and reported in
data. Unknown cases (U) lead to unknown recoveries.
All the scenarios described below assume that (i) high incidence numbers
lead to automatic contact reduction by making individuals on average
more careful, (ii) effective contact rates are higher in winter than in
spring/summer due to more frequent indoor activities [6], (iii) high
prevalence leads to higher under-reporting due to limited test and
contact tracing capabilities, and that (iv) immunity acquired by having
contracted and recovered from the disease does not wane as evidence on
loss of immunity is yet debated [7,8].
In what follows possible scenarios for reducing infectious contacts are
considered. It should be emphasized that none of the scenarios simulates
a shutdown similar to the one in spring 2020. The term “shutdown” is
used here for lack of a better word to concisely describe the measures
taken. Moreover, the term severity of restrictions can be more
properly interpreted as amount of reduction in effective
contacts .
- Scenario 1 (single shutdown ): one shutdown period of four weeks
(Nov 2 to Dec 1, 2020) is added to the earlier contact reduction
measures (wearing masks, keeping 1.5 meter distance from others,
washing hands, locally applied restrictions on opening hours for bars
and restaurants). The impact of the planned measures being hard to
estimate, we project in Fig. 2 (first row) three possible levels
(weak, moderate, strong effect), and compare with a model that does
not introduce any further restrictions.
- Scenario 2 (wave breaker ): Scenario 1 is enriched with two
further shutdowns during the Christmas holidays (Dec 23 to Jan 11,
2021) and the carnival period (Feb 1 to 21, 2021). We project in Fig.
2 (second row) four possible impact levels (weak, moderate, strong),
possibly combining intervention packages of different severity.
- Scenario 3 (continuous intervention ): In this scenario moderate
restrictions having partially started at the end of October 2020 are
maintained until spring 2021, coupled with four weeks of more
intensive restrictions in November 2020 (black curve in Fig. 2, third
row).ResultsMathematical models similar to the ones used here were employed in the
early phase of the pandemic to follow and predict the outcome of
initial intervention measures. One of the most salient
predictions of these simulations was the inevitability of a second
wave of the epidemic if the measures imposed during the first shutdown
period in Spring of 2020 were to be relaxed too much [5]. This
outcome can be witnessed not only in Germany, but in many countries
all over Europe and world-wide where strict interventions have been
lifted stepwise during the summer. Now, a similar prediction
concerning a third wave can be deduced from the simulations presented
here.
As shown in scenario 1, no matter how effective the shutdown in
November is in reducing contact rates and hence the incidence of new
infections, returning to contact rates close to those prevalent in
late summer will most likely lead to a third wave of rising case
numbers that, if left unchecked, will lead to an even higher demand on
the health care system as observed in either April or November of
2020.
Scenario 2 combines the November shutdown with additional shutdowns
over Christmas and during the Carnival period, either predetermined at
some time in fall or as spontaneous reaction to newly rising case
numbers predicted for scenario 1. This alternation of periods with
moderate restrictions and stricter contact reduction phases
(shutdowns) allows to keep both the incidence and the required number
of ICU beds under control, assuming the contact reduction during the
shutdown periods is sufficiently strong.
Simulations of Scenario 3 indicate that continued moderate
interventions over the whole winter might be a successful strategy to
keep the number of new reported cases under control. Coupling these
measures with a short, more restrictive period like the shutdown in
November could allow to significantly dampen the current second wave
for a few months from now.
Instead of planned wave breaker interventions, fixed for established
periods of the year, one might also think of applying and relaxing
intervention measures based on the reported case incidence. We have
tested (simulations not shown here) also such an intervention strategy
with triggered measures, assuming e.g. that an incidence of 50 cases per
7 days and 100,000 inhabitants triggers severe restrictions leading to
strongly reduced contact rates, whereas lifting of these restrictions is
triggered by the incidence dropping to 8 cases per 7 days and 100,000
inhabitants. One obvious assumption to include in the model is that
control measures cannot be put in place instantaneously, but require a
few days to be effectively introduced or relaxed. This leads to dynamics
similar to that in scenario 2, but with shutdown periods not occurring
at predefined intervals.