Low case numbers enable long-term stable pandemic control without lockdowns.

TLDR: This work analytically derives the existence of a third, viable solution: a stable equilibrium at low case numbers, where test-trace-and-isolate policies partially compensate for local spreading events, and only moderate contact restrictions remain necessary, and minimises lockdown duration and hence economic impact.

Abstract: The traditional long-term solutions for epidemic control involve eradication or population immunity. Here, we analytically derive the existence of a third viable solution: a stable equilibrium at l...

Figures

Figure 2: (a, b) In the stable and metastable regimes, daily new cases approach an equilibrium value N̂obs∞ that depends on contact reduction kt and external influx of new cases Φt. a: The equilibrium value N̂obs∞ increases with weaker contact reduction kt or higher influx Φt. No equilibrium is reached, if either kt or Φt are above the respective (critical) threshold values. b: The critical value kcritt represents the minimal contact reduction that is required to reach equilibrium and stabilise case numbers. If case numbers are below the TTI capacity limit, lower values of kcritt are required for stabilisation (blue) than if cases exceed TTI (grey). Confidence intervals originate from error propagation of the uncertainty of the underlying model parameters. (c–f) In the unstable dynamic regime (kt = 20%), a tipping point is visible when exceeding TTI capacity. We observe a self-accelerating increase of case numbers after crossing the TTI limit (c) and a subsequent increase of the reproduction number (d). Furthermore, the absolute number (e) and the proportion — “dark figure” (f) — of cases that remain unnoticed increase over time.

Figure 4: On a long-term perspective, recurrent lockdowns are not required if the subsequent contact reduction knLD is sufficient to reach equilibrium. A two-week lockdown of default strength (kLD = 75%) is either enacted when the test-trace-and-isolate (TTI, yellow lines) or the hospital (red lines) capacity limits are crossed. a: Given only a mild contact reduction while not in lockdown knLD = 20%, the imposed lockdowns need to reoccur with a high frequency. Moreover, the frequency at which they reoccur is higher at high case numbers (aiming at hospital capacity) than when aiming at the TTI capacity limit because of the self-accelerating effect described previously. b: If the contact reduction remains moderate knLD = 40%, lockdowns only need to reoccur if they fail to bring case numbers below TTI capacity. c, d: The total cost (in terms of cumulative cases and total lockdown duration) depends on the level of contact reduction and the lockdown policy: The cost is low if lockdowns are initiated at the TTI capacity limit and if knLD is high. e, f: In the future, the minimal contact reduction required for stability kcritt may be lowered by the implementation of large-scale random testing (e) or by immunisation of the population (f).

Figure 3: The effectiveness of a lockdown depends on three main parameters: its duration, stringency (strength), and starting time. (a–c) Observed daily new cases for a lockdown (abbreviated as LD) which is enacted after the TTI capacity has been exceeded. Reference parameters are a lockdown duration of 4 weeks, contact reduction during lockdown of kLD = 75% and a start time at 4 weeks after exceeding TTI capacity. We vary lockdown duration (a), lockdown strength (b) and lockdown starting time (c) to investigate whether stable case numbers can be reached. (d–f) Total cases after three months, if the lockdown is parameterised as described in panels a–c, respectively. (g, h:) The minimal required duration of lockdown to reach equilibrium depends both on strength and start time. g: Heavy contact reduction and timely lockdown enacting can create effective short lockdowns (≤ 2 weeks, lower left, dark region). Whereas with mild contact reduction and very late start times, lockdowns become ineffective even when they last indefinitely (Upper left, bright region). h: Horizontal slices through the colour-map (g). Here, colours match panels (c,f) and correspond to the lockdown start time.

Figure 1: Spreading dynamics depend not only on the balance between destabilising and stabilising contributions, but also on the level of case numbers itself. a: Among the factors that destabilise the spread, we find the basic reproduction number R0 and the external influx of infections (and possibly seasonality). On the other hand, various contributions can stabilise it, including increased hygiene, test-trace-isolate (TTI) strategies, contact reduction, but also immunity. We specifically investigated how contact reduction kt and limited TTI capacity determine the stabilisation of case numbers. b: Assuming only mild contact reduction (kt = 20% compared to preCOVID-19 times), TTI is not sufficient to prevent increasing case numbers — even when TTI capacity is still available; case numbers are increasing. c: At moderate contact reduction (kt = 40%), a metastable equilibrium emerges (grey dots) to which case numbers converge — if case numbers do not exceed the TTI capacity. However, destabilizing events (as, e.g., a sudden influx of infections), can push a previously stable system above the TTI capacity and lead to an uncontrolled spread (black line as an example). d: Assuming heavy contact reduction (kt = 60%), case numbers decrease even if the TTI capacity is exceeded.

Full text

Low case numbers enable long-term stable pandemic
control without lockdowns
Sebastian Contreras
1
, Jonas Dehning
1
, Sebastian B. Mohr
1
, F. Paul Spitzner
1
, and Viola
Priesemann
1,2
∗
1
Max Planck Institute for Dynamics and Self-Organization, Am Faßberg 17, 37077 Göttingen, Germany.
2
Department of Physics, University of Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany.
All authors contributed equally.
Abstract
The traditional long-term solutions for epidemic control involve eradication or herd immu-
nity
1;2
. Neither of them will be attained within a few months for the COVID-19 pandemic.
Here, we analytically derive the existence of a third, viable solution: a stable equilibrium at
low case numbers, where test-trace-and-isolate policies partially compensate for local spread-
ing events, and only moderate contact restrictions remain necessary. Across wide parameter
ranges of our complementary compartmental model
3
, the equilibrium is reached at or below
10 daily new cases per million people. Such low levels had been maintained over months
in most European countries. However, this equilibrium is endangered (i) if contact restric-
tions are relaxed, or (ii) if case numbers grow too high. The latter destabilisation marks a
novel tipping point beyond which the spread self-accelerates because test-trace-and-isolate
capacities are overwhelmed. To reestablish control quickly, a lockdown is required. We show
that a lockdown is either effective within a few weeks, or tends to fail its aim. If effective,
recurring lockdowns are not necessary — contrary to the oscillating dynamics previously
presented in the context of circuit breakers
4;5;6
, and contrary to a regime with high case
numbers — if moderate contact reductions are maintained. Hence, at low case numbers,
the control is easier, and more freedom can be granted. We demonstrate that this strategy
reduces case numbers and fatalities by a factor of 5 compared to a strategy focused only
on avoiding major congestion of hospitals. Furthermore, our solution minimises lockdown
duration, and hence economic impact. In the long term, control will successively become
easier due to immunity through vaccination or large scale testing programmes. International
coordination would facilitate even more the implementation of this solution.
As SARS-CoV-2 is becoming endemic and knowledge about its spreading is accumulated, it becomes clear
that neither global eradication nor herd immunity will be achieved soon. Eradication is hindered by the world-
wide prevalence and by asymptomatic spreading. Reaching herd immunity without an effective vaccine or
medication would take several years and cost countless deaths, especially among the elderly
1;7
. Moreover,
evidence for long-term effects (“long COVID") is surfacing
8;9;10;11;12
. Hence, we need long-term, sustainable
strategies to contain the spread of SARS-CoV-2. The common goal, especially in countries with an ageing
population, should be to minimise the number of infections and, thereby, allow reliable planning for indi-
viduals and the economy – while not constraining individuals’ number of contacts too much. Intuitively, a
regime with low case numbers would benefit not only public health and psychological well-being but would
also profit the economy
13;14
.
However, control of SARS-CoV-2 is challenging. Many infections originate from asymptomatic or pre-
symptomatic cases
15
or indirectly through surfaces and aerosols
16
, rendering mitigation measures difficult.
Within test-trace-and-isolate (TTI) strategies, the contribution of purely symptom-driven testing is limited,
but together with contact tracing, it can uncover asymptomatic chains of infections. Additional challenges
∗
viola.priesemann@ds.mpg.de
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preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
The copyright holder for thisthis version posted December 11, 2020. ; https://doi.org/10.1101/2020.12.10.20247023doi: medRxiv preprint
NOTE: This preprint reports new research that has not been certified by peer review and should not be used to guide clinical practice.

Low case numbers enable long-term stable pandemic control without lockdowns
are the potential influx of SARS-CoV-2 infections (brought in by travellers or commuting workers from
abroad), imperfect quarantine, limited compliance, and delays in TTI and case-reporting. Lastly, any coun-
tries capacity to perform TTI is limited, so that spreading dynamics change depending on the level of case
numbers. Understanding these dynamics is crucial for informed policy decisions.
Analytical framework: Overview
We analytically show the existence of a stable regime at low case numb ers, where control of SARS-CoV-2 is
much easier to achieve and sustain. In addition, we investigate mitigation strategies and long term control
for COVID-19, where we build on our past work to understand the effectiveness of non-pharmaceutical
interventions, particularly test-trace-and-isolate (TTI) strategies
3;17;18
. For quantitative assessments, we
adapt an SEIR-type compartmental model
19
to explicitly include a realistic TTI system that considers the
challenges above.
A central parameter for our analysis is the effective reduction of contagious contacts k
t
(relative to pre-
COVID-19). More precisely, k
t
does not simply refer to the reduction of contacts a person has, but rather to
the encounters that bear a potential for transmission. Apart from direct contact reduction, contributions to
k
t
also come from improved hygiene, mandatory face-mask policies, frequent ventilation of closed spaces, and
avoiding indoor gatherings, among other precautionary measures. As the latter measures are relatively fixed,
direct contact reduction remains the central free variable, which is also the one tuned during lockdowns. All
other parameters (and their references) are listed in Table S1.
Equilibrium at low case numbers
Figure 1: Spreading dynamics depend not only on the balance between destabilising and stabilising
contributions, but also on the level of case numbers itself. a: Among the factors that destabilise the spread,
we find the basic reproduction number R
0
and the external influx of infections (and possibly seasonality). On the
other hand, various contributions can stabilise it, including increased hygiene, test-trace-isolate (TTI) strategies,
contact reduction, but also immunity. We specifically investigated how contact reduction k
t
and limited TTI capacity
determine the stabilisation of case numbers. b: Assuming only mild contact reduction (k
t
= 20 % compared to pre-
COVID-19 times), TTI is not sufficient to prevent increasing case numbers — even when TTI capacity is still available;
case numbers are increasing. c: At moderate contact reduction (k
t
= 40 %), a metastable equilibrium emerges (grey
dots) to which case numbers converge — if case numbers do not exceed the TTI capacity. However, destabilizing
events (as, e.g., a sudden influx of infections), can push a previously stable system above the TTI capacity and lead to
an uncontrolled spread (black line as an example). d: Assuming heavy contact reduction (k
t
= 60 %), case numbers
decrease even if the TTI capacity is exceeded.
2
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Low case numbers enable long-term stable pandemic control without lockdowns
Between the scenarios of eradication of the disease or uncontrolled spreading, we find a regime where the
spread reaches an equilibrium at low daily case numbers. The main control parameter that determines
whether the system can reach an equilibrium is the contact reduction k
t
.
If the contact reduction k
t
is mild, case numbers grow exponentially, as measures could not counterbalance
the basic reproduction number (R
0
≈ 3.3 for SARS-CoV-2
20;21;22
) (Fig. 1b). In contrast, if the contact
reduction is strong and (together with hygiene and TTI) outweighs the drive by the basic repro duction
number, case numbers decrease to a low equilibrium value (Fig. 1d).
Importantly, if the contact reduction is moderate (and just-about balances the drive by the basic reproduction
number), we find a metastable regime: The spread is stabilised if and only if the overall case numbers are
sufficiently low to enable fast and efficient TTI (Fig. 1c). However, this control is lost if the limited TTI
capacity is overwhelmed. Beyond that tipping point, the number of cases starts to grow exponentially as
increasingly more infectious individuals remain undetected
3
.
The capacity of TTI determines the minimal required contact reduction for controlling case numbers around
an equilibrium. If case numbers are sufficiently below the TTI capacity limit, the required contact reduction
to maintain the (meta-)stable regime is only k
crit
t
= 39 % (95% confidence interval (CI):[24, 53]). However, if
case numbers exceed the TTI capacity limit, a considerably stronger contact reduction of k
crit
t
= 58 % (95%
CI: [53, 62]) is required to reach the stable regime (Fig. 2 b, Fig. S4 and Table S3).
Equilibrium depends on influx and contact reduction
Figure 2: (a, b) In the stable and metastable regimes, daily new cases approach an equilibrium value
ˆ
N
obs
∞
that depends on contact reduction k
t
and external influx of new cases Φ
t
. a: The equilibrium value
ˆ
N
obs
∞
increases with weaker contact reduction k
t
or higher influx Φ
t
. No equilibrium is reached, if either k
t
or Φ
t
are
ab ove the respective (critical) threshold values. b: The critical value k
crit
t
represents the minimal contact reduction
that is required to reach equilibrium and stabilise case numbers. If case numbers are below the TTI capacity limit,
lower values of k
crit
t
are required for stabilisation (blue) than if cases exceed TTI (grey). Confidence intervals originate
from error propagation of the uncertainty of the underlying model parameters. (c–f) In the unstable dynamic
regime (k
t
= 20%), a tipping point is visible when exceeding TTI capacity. We observe a self-accelerating
increase of case numb ers after crossing the TTI limit (c) and a subsequent increase of the reproduction number (d).
Furthermore, the absolute numb er (e) and the proportion — “dark figure” (f) — of cases that remain unnoticed
increase over time.
If an equilibrium is reached, the precise value of daily new cases
ˆ
N
obs
∞
at which the system stabilises depends
on both the reduction of contagious contacts k
t
and the external influx of new cases Φ
t
(Fig. 2a). In general,
for realistic low values of influx Φ
t
, the equilibrium level
ˆ
N
obs
∞
is low. However,
ˆ
N
obs
∞
increases steeply
(diverges) when the contact reduction k
t
approaches the tipping point to unstable dynamics (Fig. 2a,b).
3
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Low case numbers enable long-term stable pandemic control without lockdowns
Such a divergence near a critical point k
crit
t
is a general feature of continuous transitions between stable
and unstable dynamics
23;24
. As a rule of thumb, in an analytical mean-field approximation,
ˆ
N
obs
∞
would be
proportional to Φ
t
and diverge when k
t
approaches its critical value k
crit
t
from above:
ˆ
N
obs
∞
∝ Φ
t
/(k
t
−k
crit
t
)
24
.
Robust control of the pandemic requires maintaining a sufficient safety-margin from the tipping point (and
the subsequent transition to instability) for two reasons. First, small fluctuations in k
t
, Φ
t
(or other model
variables) could easily destabilise the system. Second, near the critical value k
crit
t
, reductions in k
t
are
especially effective: already small further reductions below k
crit
t
lead to significantly lower stable case numbers
(Fig. 2a). Already with moderate contact reduction (40 % < k
t
< 50 %), the spread can be stabilised to a
regime of case numbers clearly below 10 per million (Fig. S1 b, lower right region).
Limited TTI and self-acceleration
If mitigation measures are insufficient, case numbers rise and eventually surpass the TTI capacity limit.
Beyond it, health authorities are not able to efficiently trace contacts and uncover infection chains, thus the
control of the spread becomes more difficult. We start our scenario with a slight increase in case numbers
over a few months, as seen in many European countries throughout the summer 2020 (Fig. S5 and Fig. S6).
A tipping point is then visible in the following observables (Fig. 2 c–f):
First, when case numbers surpass the TTI capacity, the increase in daily new observed cases
ˆ
N
obs
becomes
steeper, growing even faster than the previous exponential growth (Fig. 2 c, full versus faint line). The
spread self-accelerates because increasingly more contacts are missed, which, in turn, infect more people.
Importantly, in this scenario the accelerated spread arises solely because of exceeding the TTI limit —
without any underlying behaviour change among the population.
Second, after case numbers surpass the TTI limit, the observed reproduction number
ˆ
R
obs
t
, which had been
only slightly above the critical value of unity, increases significantly by about 20 % (Fig. 2 d). This reflects a
gradual loss of control over the spread and explains the faster-than-exponential growth of case numbers. The
initial dip in
ˆ
R
obs
t
is a side-effect of the limited testing: As increasingly many cases are missed, the observed
reproduction number reduces transiently.
Third, compared to the infectious individuals who are quarantined I
Q
, the number of infectious individuals
who are hidden I
H
(i.e. those who are not isolated or in quarantine) increases disproportionately (Fig. 2 e)
which is measured by the “dark figure” (I
H
/I
Q
) (Fig. 2 f). The hidden infectious individuals are the silent
drivers of the spread as they, unaware of being infectious, inadvertently transmit the virus. This implies
a considerable risk, especially for vulnerable people. At low case numbers, the TTI system is capable to
compensate the hidden spread, because it uncovers hidden cases through contact tracing. However, at high
case numbers, the TTI becomes inefficient: If the TTI measures are “slower than the viral spread”, many
contacts cannot be quarantined before they become spreaders.
Re-establishing control with lockdowns
Once the numb er of new infections has overwhelmed the TTI system, re-establishing control can be chal-
lenging. A recent suggestion is the application of a circuit breaker
4;5;6
, a short lockdown to significantly
lower the number of daily new infections. Already during the first wave, lockdowns have proven capable to
lower case numbers by a factor 2 or more every week (corresponding to an observed reproduction numb er of
ˆ
R
obs
t
≈ 0.7). With the knowledge we now have acquired about the spreading of SARS-CoV-2, more targeted
restrictions may yield a similarly strong effect.
Inspired by the lockdowns installed in many countries
25
, we assume a default lockdown of four weeks, starting
four weeks after case numbers exceed the TTI capacity limit, and a strong reduction of contagious contacts
of k
LD
= 75 % (which corresponds to an
ˆ
R
eff
t
≈ 0.85, see Table S4). We further assume that during lockdown
the external influx of infections Φ
t
is reduced by a factor ten, and that after the lockdown, a moderate contact
reduction (k
nLD
= 40 %) is maintained. By varying the parameters of this default lockdown, we show in the
following that the lockdown strength, duration, and starting time determine whether the lockdown succeeds
or fails to reach equilibrium.
In our scenario, a lockdown duration of four weeks is sufficient to reach the stable regime (Fig. 3 a). However,
if lifted too early (before completing four weeks), then case numbers will rise again shortly after. The shorter
an insufficient lockdown, the faster case numbers will rise again. Also, it is advantageous to remain in
4
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preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
The copyright holder for thisthis version posted December 11, 2020. ; https://doi.org/10.1101/2020.12.10.20247023doi: medRxiv preprint

Low case numbers enable long-term stable pandemic control without lockdowns
Start time of lockdown
(weeks after exceeding TTI)
Time (weeks after exceeding
TTI capacity limit)
Time (weeks after exceeding
TTI capacity limit)
Time (weeks after exceeding
TTI capacity limit)
(per million inhabitants) (per million inhabitants)
Lockdown duration (weeks)
Required
lockdown duration (weeks)
Total casesTotal cases Total cases
Lockdown starting time
2
8
6
4
Observed
new cases
Lockdown
duration
Lockdown
strength
Lockdown
start time
Total cases
after three months
a d g
h
b e
c f
40%
50%
60%
75%
85%
no LD.
no LD.
1
0
2
3
4
5
no LD.
no
LD.
Start time
(weeks)
Start
time (weeks)
Contact
reduction
Duration
(weeks)
2
3
4
5
6
7
Minimum required
lockdown duration
Contact reduction
during lockdown
Contact reduction
during lockdown
Contact reduction
during lockdown
Lockdown duration
(weeks)
80% 100%60%40%
100%80%
20% 60%40%
440 16128
40 16128
40 16128
9 14
TTI
capacity
Hospital
capacity
no LD.
no
LD.
100%80%60%
0
2
4
6
8
unstable
2
3
5
6
200
200
200
0
0 0
0
0
0
0 2 4 6 8
2 4 6
50.000 4
50.000
50.000
100.000
8
100.000
100.000
0
0
400
400
400
1
5
Figure 3: The effectiveness of a lockdown depends on three main parameters: its duration, stringency
(strength), and starting time. (a–c) Observed daily new cases for a lockdown (abbreviated as LD) which is
enacted after the TTI capacity has been exceeded. Reference parameters are a lockdown duration of 4 weeks, contact
reduction during lockdown of k
LD
= 75 % and a start time at 4 weeks after exceeding TTI capacity. We vary lockdown
duration (a), lockdown strength (b) and lockdown starting time (c) to investigate whether stable case numbers can
b e reached. (d–f) Total cases after three months, if the lo ckdown is parameterised as described in panels a–c,
resp ectively. (g, h:) The minimal required duration of lockdown to reach equilibrium depends both on strength and
start time. g: Heavy contact reduction and timely lockdown enacting can create effective short lockdowns (≤ 2 weeks,
lower left, dark region). Whereas with mild contact reduction and very late start times, lockdowns become ineffective
even when they last indefinitely (Upper left, bright region). h: Horizontal slices through the colour-map (g). Here,
colours match panels (c,f ) and correspond to the lockdown start time.
lockdown for a short time even after case number have fallen below the TTI limit — in order to establish
a safety-margin, as shown above. Overall, the major challenge is not to ease the lockdown too early, as
otherwise the earlier success is soon squandered.
During lockdown, it is necessary to reduce contagious contacts k
t
severely in order to decrease case numbers
below the TTI capacity limit (Fig.
3 b). In our scenario, the contacts have to be reduced by at least
k
LD
= 75 % to bring the system back to equilibrium. A lo ckdown that is only slightly weaker cannot reverse
the spread to a decline of cases. Furthermore, increasing the lockdown strength decreases both the required
lockdown duration (Fig. 3 g,h) and the total number of cases accumulated over three months (Fig. 3 e). This
shows that stricter lockdowns imply shorter-lasting social and economical restrictions.
The earlier a lockdown begins after exceeding the TTI capacity limit, the faster control can be re-established
and constraints can be lo osened again (Fig. 3 c). If started right after crossing the threshold, in principle,
only a few days of lockdown are necessary to bring back case numbers below TTI capacity limit. On the
other hand, if the lockdown is started weeks later, its duration needs to increase (Fig. 3 c,d) and the total
number of cases will be significantly larger (Fig. 3 f).
In conclusion, to re-establish control, a lockdown needs to be strong enough to reach equilibrium within a
few weeks, or it fails almost completely.
The parameter regime between these two options is quite narrow; it is not likely that equilibrium can
eventually be reached as a lockdown exceeds many weeks (cf. Fig. 3 h). For practical p olicies, this means
5
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Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Low case numbers enable long-term stable pandemic control without lockdowns" ?

The authors show that a lockdown is either effective within a few weeks, or tends to fail its aim. The authors demonstrate that this strategy reduces case numbers and fatalities by a factor of 5 compared to a strategy focused only on avoiding major congestion of hospitals. Preprint ( which was not certified by peer review ) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. This preprint reports new research that has not been certified by peer review and should not be used to guide clinical practice. Low case numbers enable long-term stable pandemic control without lockdowns are the potential influx of SARS-CoV-2 infections ( brought in by travellers or commuting workers from abroad ), imperfect quarantine, limited compliance, and delays in TTI and case-reporting. The authors analytically show the existence of a stable regime at low case numbers, where control of SARS-CoV-2 is much easier to achieve and sustain. In addition, the authors investigate mitigation strategies and long term control for COVID-19, where they build on their past work to understand the effectiveness of non-pharmaceutical interventions, particularly test-trace-and-isolate ( TTI ) strategies3 ; 17 ; 18. For quantitative assessments, the authors adapt an SEIR-type compartmental model19 to explicitly include a realistic TTI system that considers the challenges above. The authors specifically investigated how contact reduction kt and limited TTI capacity determine the stabilisation of case numbers. Preprint ( which was not certified by peer review ) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Furthermore, their solution minimises lockdown duration, and hence economic impact. More precisely, kt does not simply refer to the reduction of contacts a person has, but rather to the encounters that bear a potential for transmission. 

Q2. How can the authors estimate the value of in the integration routine?

The explicit value of φ can be obtained numerically in the integration routine, or estimated trough the use of the equilibrium values of the differential equations, φ = IH,s ∞ IH∞(as implemented in their code). 

Q3. How do the authors determine if a system can reach an equilibrium at low case numbers?

Low case numbers enable long-term stable pandemic control without lockdownsBetween the scenarios of eradication of the disease or uncontrolled spreading, the authors find a regime where the spread reaches an equilibrium at low daily case numbers. 

Q4. How do the authors determine if a lockdown is necessary to re-establish?

In conclusion, to re-establish control, a lockdown needs to be strong enough to reach equilibrium within a few weeks, or it fails almost completely. 

Q5. How much contact reduction is required to maintain the (meta-)stable regime?

If case numbers are sufficiently below the TTI capacity limit, the required contact reduction to maintain the (meta-)stable regime is only kcritt = 39 % (95% confidence interval (CI):[24, 53]). 

Q6. What is the effect of TTI measures on case numbers?

As the effectiveness of TTI measures depends on daily infections, case numbers can seemingly explode when the (hard-to-estimate) TTI capacity limit is exceeded. 

Q7. how many infections are found in the symptom-driven and random testing?

(35)If both symptom-based and random testing take place simultaneously, the number of discovered infections is given byN test = λrIH + λsIH,s (36)Further, assuming that after reaching N testmax, the testing rates at the overhead pool-sizes would decrease to λ′s and λ′r, respectively, for symptom-driven and random testing. 

Q8. How can the authors express Rt in terms of the basic reproduction number?

It can be expressed in terms of the effective reproduction number Rt:ϱ = 1 − 1 Rt . (27)In the context of their model, Rt can be expressed in terms of the reduction of contagious contacts kt and the basic reproduction number R0; Rt = (1 − kt) R0. 

Q9. How much contact reduction is necessary to stay within the metastable regime?

maintaining a moderate contact reduction while not in lockdown (knLD = 40 %) is sufficient to stay within the metastable regime — if lockdowns are enacted such that case numbers stay below the TTI capacity (Fig. 4 b, yellow line). 

Q10. How many days of lockdown are necessary to bring back cases below TTI capacity limit?

If started right after crossing the threshold, in principle, only a few days of lockdown are necessary to bring back case numbers below TTI capacity limit. 

Q11. How did you study the stability of the governing differential equations?

S2.2 Linear stability analysis and uncertainty propagationFor analysing the stability of the governing differential equations, namely, whether an outbreak could be controlled, the authors studied the linear stability of the system. 

Q12. How long does the lockdown DL take?

The duration of the lockdown DL, namely, the time-frame between the start of the restrictive measures and the beginning of their relaxation, is measured in weeks, and its default length – for analysis purposes – is four weeks. 

Q13. What is the definition of random testing?

Random testing is defined here as applying tests to individuals irrespective of their symptom status, or whether they belonging to the contact-chain of other infected individuals. 

Q14. How much contact reduction is required to maintain the stable regime?

if case numbers exceed the TTI capacity limit, a considerably stronger contact reduction of kcritt = 58 % (95% CI: [53, 62]) is required to reach the stable regime (Fig. 2 b, Fig. S4 and Table S3). 

Q15. What is the simplest way to determine the tipping point between controlled and uncontrolled outbreaks?

To derive the tipping point between controlled and uncontrolled outbreaks (e.g. critical, minimal required contact reduction kcritt for both stability and metastability), and to plot the stability diagrams, the authors used the @fzero MATLAB function, and the linear approximation of the system of DDE (2)–(6) for the SM ≈ 1 limit. 

Q16. How does the model transfer infected individuals from the hidden to the quarantined infectious pools?

In their model, random testing transfers infected individuals from the hidden to the quarantined infectious pools with fixed rate λr, irrespective of them showing symptoms or not.