This saturation of NPIs generates a positive feedback between the number of infected individuals and disease growth rate, which has critical consequences in the disease dynamics, particularly at low numbers. This feedback loop connects the saturation of NPIs with the Allee effect, a central concept in population biology that explains the fate of several populations at low numbers 12, 13 . Indeed, the Allee effect involves a positive feedback between population abundance and population growth rate, which may determine a transition from a negative to a positive growth rate at a given abundance threshold 12, 13 . In disease dynamics, this is interpreted as an epidemic breakpoint, a tipping point below which the outbreak tends to diminish (R e <1), but above which the outbreak grows (R e >1) 8 . However, this phenomenon is not captured in the many variants of SIR and logistic models widely used for the analysis and forecasting of COVID-19 dynamics 2, 3, 6, 7, 19, 23 . Here we aim to fill this gap, formalizing the connection between saturating NPIs and disease dynamics. We show how the saturation of NPIs induce a positive feedback in disease dynamics, from which abrupt transitions in spread rate are expected after a threshold in the number of infected individuals is crossed. Finally, we look for traces of this footprint in actual data from the COVID-19 pandemic.
In the absence of contact tracing and quarantine, a randomly chosen infected individual will produce an average number of second infections R e nq (nq: non quarantine). For the sake of clarity in our model we will use quarantine to encapsulate the concepts of 'quarantine' and 'case isolation' without losing explanatory power. This value is determined by the number of social links accumulated in the infectious time (L max ), the probability of disease transmission in each link (b link ), and the probability of having a link with a susceptible individual (P susceptible ) thus R e nq =P susceptible ·b link ·L max (Fig. 1A ).
Strong enough NPIs will determine a R e <1 controlling the epidemic 2, [9] [10] [11] . While face mask use, hygienic measures, and physical distancing reduce b link 18 , the number of social links L max can be reduced through contact tracing and subsequent quarantine 10 . Individuals detected by the tracing system are contacted and then put on quarantine. However, only a fraction f q of the infected individuals is detected (D=f q ·I) by the contact tracing, which is determined by the maximum number of cases (K) that can be processed in a day. Moreover, because of the saturation of tracing systems, as the number of infected individuals increases, the fraction f q decreases-here modeled as f q =K/(I 50,D +I), with I 50,D being the number of infected cases at which half the maximum detection is reached 24 .
The saturation of the contact tracing system increases the time lag between infection and detection. This is modeled considering that the tracing system can make a fixed number of calls (N calls ) each day and that each contact attempt has a probability of success P find . Consequently, the expected number of alerts received per individual on a given day is N calls /D, and the probability of finding and quarantining him is P q =1-(1-P find ) Ncalls/D . The probability that an individual is contacted and quarantined on day d after the infection onset follows the geometric distribution Integrating all of the above, we can express the effective reproduction number R e (I) determined by the second infections generated by both fractions of quarantined f q and non quarantined f nq infected individuals as:
[1] R e (I)= R e q · f q + R e nq · f nq =P susceptible · b link · (L · f q +L max · f nq )
This equation's plot in Figure 1a shows that NPIs induce a positive relationship between the infected individuals in a population and the epidemic reproduction number. In particular, it is expected that the epidemic does not take-off at low infection numbers (R e (I)<1), requiring a minimum level of infections (I*) to cause an outbreak (R e (I)>1).
The strength of the different components of NPIs-contact tracing capacity and speed, social distance, and transmission barriers-have nonlinear relationships with the outbreak threshold I* (Fig. 1b) . Alternative states 8,17 of growing and diminishing epidemic was considered elsewhere for COVID-19 but as a response to the strength of NPIs 2, [9] [10] [11] . Here, we show that for a fixed level of saturating NPIs, the transition from contention to outbreak (i.e. R e (I)<1⇒ R e (I)>1) may be determined by the number of infected individuals (Fig. 1a, b) .
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The copyright holder for this this version posted September 15, 2020. . 13 . When NPIs are in place (red line) R e (I) first increases with the proportion of infected individuals due to the saturation of NPIs and then decreases approaching the dynamics in absence of NPIs (equation [1] ). This determines a positive feedback at low numbers (NPI-Allee effect) and a threshold for epidemic outbreak in the number of infections (I*). (b) The alternative epidemic states of growth (R e (I)>1, red) or containment (R e (I)<1, blue) are determined by both the proportion of the infected population (vertical axis) and the strength of the different NPIs (horizontal axis) (c) SIR dynamics without NPIs (right) and with NPIs of limited capacity (left), inducing an Allee effect that can generate abrupt transitions in disease spread rate (equation [2] ). (d) Relaxation of NPIs and epidemic containment. In trajectory i→ii→iii→iv an abrupt relaxation surpasses the epidemic threshold provoking an increase in infections. Return to previous strength of NPIs fails to contain the outbreak. In trajectory i→v the gradual relaxation of NPIs follows the decrease in the infected population keeping the epidemic under control.
In a deterministic system with a saturation of NPIs that induces an Allee effect (hereafter NPI-Allee effect), infections are expected to either vanish or take-off. However, at low numbers, pulses of immigration of infected individuals and super spreading events could determine the transition from a negative to a positive disease growth rate if the threshold is exceeded 26 . In this section, we assess the qualitative consequences of the saturation of NPIs on disease dynamics through a SIR model considering stochasticity and infection pulses with a heavy tailed distribution.
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The NPI-Allee effect is introduced in the daily rate of disease transmission through a saturating function b t (I)= b max ·I(t)/(I 50.b +I(t)) and also in the average duration of the infection period, g t (I)= g max ·I(t)/(I 50.g +I(t)):
Where = b max and , g= g max (Fig. 1c and Extended Data Figure 1 ).
The logarithm of the cumulative number of infections grows linearly in time in exponential dynamics 13, 24 . However, as observed in COVID-19, sub-exponential dynamics grow linearly with the logarithm of time 7 , as described by the relationship log(Ct)=µ·log(t). We use this time scale to visualize our simulations, where the parameter µ describes the disease dynamics. The SIR model without the NPI-Allee effect presents a consistent trend with minor changes fueled by the stochastic effects of imported cases and super spread events (Fig. 1c, Extended Data Figs. 1) . On the other hand, when the NPI-Allee effect is considered, we observe a qualitative transition in the dynamics: from a slow rate of case accumulation to a markedly larger one, suggesting that these random events may determine the surpass of a tipping point from containment to outbreak. We fitted a segmented regression to examine the differences in dynamics with and without NPI-Allee effect (Fig. 1c, Extended Data Fig 1) . Both simulations present breaking points and may look similar when this point is rapidly reached. In order to further explore their difference, we plotted the relative increment of the slope after the breaking point (µ2/µ1) as a function of the initial slope (µ1) for both kinds of simulations ( Fig. 2a and 2c) . Now, it becomes apparent that including . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
The copyright holder for this this version posted September 15, 2020. . https://doi.org/10.1101/2020.09.14.20194159 doi: medRxiv preprint saturating NPIs expands the range of observed dynamics, promoting slower rates during the initial phase and, after the tipping point, an accelerated dynamic capturing the loss in effectiveness of the NPIs (Figs. 2a and 2b, Extended Data Fig. 1 ). Our model results show how NPIs saturation may induce an Allee effect that produces abrupt transitions, with periods of disease containment and periods of rapid growth. In this sense, the role of a heavy-tailed distribution in new infections markedly differs between the two models, because super spread events, and/or imported cases, may cause the infection number to exceed the threshold. Fig. 4) .
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Observed dynamics of cumulative cases of COVID-19 and SIR models simulations with and without NPI-Allee effect. Trajectories are described by segmented regressions on log(Ct)=µ·log(t) with the slopes µ1 and µ2 representing the rate of cases accumulation before and after the breakpoint. (a) COVID-19 dynamics in U.S. counties (blue circles) frequently shown an initial phase with low rate of increase in cumulative cases followed by a transition to a faster growth rate. The initial slope (µ1) is shown in the horizontal axis as log 10 (1+ µ1) and the relative increment in slope after the breaking point (µ2/ µ1) is shown in the vertical axis as log 10 (1+ µ2/µ1). This dynamic is frequently observed in the SIR model with NPI-Allee effect (red contours) but not in models without this effect (green contours). Red and green density plots were estimated from 1000 simulated SIR dynamics with and without NPI-Allee effect, with parameters: b max =0.5, g max =4, p=0.8, m=1, I 50.g =I 50.b =10 and s=0.2, with population sizes sampled from a log-Normal distribution with equal mean and variance than U.S. counties populations. Simulations covering a wide range of plausible parameters for COVID-19 consistently support the congruence of observed dynamics with the NPI-Allee The observed dynamics show several cases with a striking similarity to the NPI-Allee effect simulations, with abrupt transitions from slow to fast growth rate and a delayed tipping point.
When we project the US-county data and worldwide country/region data of COVID-19, the results are eloquent (Fig. 2b and 2d and Extended Data Figs. 2 and 3) . Indeed, the area covered with actual data is remarkably similar to the simulated cases with NPI-Allee effect. On the other hand, many . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint
The copyright holder for this this version posted September 15, 2020. . regions do not show this behaviour, either because the breaking point occurred early in the epidemic spread--e.g., surpassing I* before NPIs were fully implemented--or because the NPIs have not determined the I* threshold in these locations.
Amidst an ongoing worldwide outbreak of COVID-19, studies connecting NPIs and disease dynamics are all the more needed 1, [4] [5] [6] [7] 16 . Our results link NPIs with the substantial knowledge built around the Allee effect and tipping points in ecology 12, 15, 17, 26, 28 , and previous studies that related the saturation of NPIs, reactive individual behaviour, and positive feedbacks with disease dynamics 8, 21, 22, 29 . The recognition of an NPI-Allee effect and the existence of tipping points contributes to the understanding of several features of disease dynamics 12, 13, 26 .
An immediate consequence is the understanding that the timing of NPIs is crucial.
Interventions implemented when the disease has had enough time to surpass the correspondent threshold will fail in its containment 8, 10, 22 . Similarly, the relaxation of NPIs has to be done in attention to the epidemic threshold for avoiding outbreak resurgence. The idea of relaxing interventions and then returning them to former levels if the number of infected individuals increases may be futile after the tipping point. NPIs may be relaxed cautiously following the decrease in the number of infections with attention to potential tipping points (Fig.1d) , a context where early warning indicators become crucial. These indicators were related to systems resilience and increased trajectory variance 17 , but more specific indicators for diseases under NPIs management are probably required. For example, in a contained state of the disease, the fraction of reported infections with no epidemiological link (not detected by the tracing system), and the time between individual exposure and quarantine should decrease through time. A systematic increase in these indicators may be an adequate early warning signal that containment could be compromised.
Another important feature is that NPIs could operate as exogenous variables synchronizing disease dynamics along large geographic scales 28, 30 , being more likely to simultaneously control the outbreaks or to enable global resurgence 14, 15 . The success of proposed spatial synchrony of NPIs for controlling COVID-19 16 and vaccination strategies may depend on the associated NPI-Allee effect's strength. More generally, the spatial spread of diseases with NPI-Allee effect may be guided by local surpass of epidemic threshold and neighbor contagion 15, 26 . In diseases around . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint
The copyright holder for this this version posted September 15, 2020. an outbreak tipping point, spatial and temporal mosaic of R e >1 and R e <1 may arise, determining local outbreaks, and fostering disease persistence. Finally, we showed how the dynamic consequences of super spread events are contingent to its effect on saturating NPIs.
Recognizing the NPIs-induced Allee effect has large consequences for the understanding and management of COVID-19 and other diseases and may significantly expand the theoretical framework for the application of NPIs in ongoing and future epidemics. This knowledge is of particular relevance for explaining the past dynamics of COVID-19 in different regions of the world and, more importantly, as input for guiding NPIs relaxation strategies and preventing new outbreaks.
Country and regional COVID-19 daily data and demographic data were obtained from Johns Hopkins University COVID-19 Data Repository 31 .
All the code used in this work is available at https://github.com/almadana/alle.covid/.