Severe acute respiratory syndrome coronavirus 2 (SARS-CoV2, the virus that causes has swept the globe, revealing the strengths and weaknesses of our international, national, state, and local public health systems. Emerging evidence from countries such as Vietnam 1 , Thailand 2 , Singapore 3 , South Korea 4 , New Zealand 5 , China 6 , and others 7 suggests that coordinated, national-level policies can control SARS-CoV-2 transmission. However, in many locations -in particular the United States of America -efforts to stem the spread of SARS-COV2 were instead implemented as a patchwork of self-isolation, school closures, and business restrictions 8 . For example, throughout the months of March and April 2020, US states, counties, and cities often independently implemented stay-at-home orders, mask mandates, limits on gathering sizes, etc. 9 . A potentially dire epidemiological consequence of this lack of coordination is that individuals can easily travel to areas with different control measures and avoid locally disrupted services and gatherings.
Even in countries with more uniform policies, religious activities have been the subject of much debate as the local risks associated with the the activity [10] [11] [12] clashed with protections of the activity as an essential service to individuals and the community 13 . Choirs and large services in particular have led to many superspreading events 14 , with attack rates well-above 50% in some cases 15 . Unfortunately, there have been little efforts devoted to replacing religious services with safe alternatives, leading individuals to defy public health recommendations. As an example, individuals have defied church closures and attended mass gatherings, at times leading to legal prosecution 16, 17 .
Other essential services have seen similar patterns, with public spaces such as urban and suburban parks and trails also being the subject of inconsistent visitation patterns and closures. As other businesses close, there has been increased foot traffic in parks with many reporting overcrowding. When some, but not all, parks and trails close, individuals may travel further to areas remaining open potentially seeding virus to previously uninfected areas 18 .
To compound the risk posed by unintended increased travel to obtain certain essential services, the rush to reopen US states with stay-at-home orders has been similarly haphazard, leading to massive upticks in cases in previously well-controlled states 19 . Taken together, the non-uniform implementation and relaxation of US state-level interventions has left the country with high numbers of cases and potential distrust of public health interventions 20 . Quantification of how movement patterns have changed from one's local closed business to a neighboring open business has been under-explored. Also needed are more general investigations into the impacts of non-uniform implementation of interventions on the transmission dynamics of SARS-COV2 and other pathogens. Here, we first examine online information seeking and physical foot traffic data to see how gathering-specific behavior has varied across the US during the COVID-19 pandemic, and then study a model of epidemics with partial gathering restrictions and partial adoption of said restrictions. Finally, we discuss the implications of these results especially as they relate to current discussions on relaxing or re-implementing stay-at-home orders and allowing gatherings.
We use data from SafeGraph to quantify human mobility after the adoption of physical distancing measures. SafeGraph is a data company that aggregates anonymized location data from numerous applications in order to provide insights about physical places. To enhance privacy, SafeGraph excludes census block group information if fewer than five devices visited an establishment in a month (two devices in a week) from a given census block group. Using these data we use counts of visits and unique visitors to businesses across the US as well as the distance traveled from 'home' (defined as the common nighttime location for the device over a 6 week period where nighttime is 6 pm to 7 am).
What is clear from the data is that many individuals are willing to travel further to attend certain gatherings, in particular church services. We find that despite seeing an overall 56% (95% CI: 40-76) decrease in visits to churches comparing the first to last week of March, individuals that do visit a church travel on average 13% (95% CI: 4-26) further to churches across most states in the country (Fig. 1) .
That individuals are looking and traveling further for churches is also seen in Google search trends (as downloaded from the Google API for Trends), where queries for "churches" have increased since the beginning of March while searches for "parks" have decreased. We compared search volumes for church on Sundays between March 13 and April 13, 2020 to Sundays in the previous 10 years of searches occurring on the same date across all US states. By normalizing to previous years, we are able to capture deviations during 2020 above and beyond typical searching patterns over this period, which encompasses Lent where individuals may have increased interest in attending church. Comparison to previous years should alleviate potential biases as the previous years act as counterfactuals to 2020. Overall, we see 49.6% (95% CI: 46.4-52.9) higher search volumes for churches in 2020 as compared to 2010-2019, with similarly higher volumes across states. Indeed, only two states (D.C. and NV) did not show statistically significantly higher search volumes in 2020, with the others ranging from 10% (95% CI: 7.0-11.8) in MA to 106% (95% CI: 100.5-111.3) in WY (Fig. 2) . Conversely, we fit a series of sinusoidal regressions to estimate the expected number of searches for "parks" and find a 22% (95% CI, -10 to -31) decrease after the declaration of national emergency. These patterns of searches indicate increased information-seeking for churches, potentially because an individual's normal church is closed. From a social network perspective, traveling to a more distant church is an unintended 2/11 . 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)
The copyright holder for this preprint this version posted August 24, 2020. consequence where our intervention might now be increasing contacts between communities. Looking at the average attendance at churches, we find that while the number of visitors has decreased significantly overall, our result on increased travel were specifically driven by states with significant restrictions on visits and large decrease in visits. More specifically, we estimate an increase of about 5 additional km traveled for every 25% decrease in visitors (95% CI, 15-34, p = 0.002). While we can not confirm the exact mechanism, these results are consistent with the idea that some individuals will travel further to seek an open church when their local church closes.
This phenomenon is not limited to religious services. We compared differences in numbers of visits to grocery stores (North American Industry Classification System [NAICS] code 4451) that had increases in visits to churches who had increases in visits. We find a correlation between increases to visits to churches with increases in visits to grocery stores (Pearson's r = 0.44) with increases to grocery stores being higher than to churches (slope = 0.9, 95% CI: 0.37-1.44; Fig. 3 ). Additionally we find increases to churches and grocery stores to be largely independent of whether the state had a stay-at-home order in place, suggesting that the phenomenon is closely related to the local distribution of services and behavior of the local population rather than top-down interventions.
Even ignoring the fact that these additional visitors travel further and are therefore likely to increase the coupling between distinct communities, they also simply increase the number of contacts in their new church. Given that the expected number of contacts is expected to increase non-linearly with the number n of participants in a gathering (i.e. potential contacts are proportional to n(n 1)/2 ⇠ n 2 ), it is unclear whether or not closing some churches might be worth the increased risk in the remaining open churches. To investigate this trade-off, we now design a simple model.
To more broadly explore the potential unintended consequences of inconsistent epidemic control policies, we formulated a simple, mathematical model which we call cloSIR to couple disease dynamics with closure policies. Specifically, we model an epidemic in a population of size N uniformly distributed across M gatherings of size n = N/M. We assume that a fraction X of gatherings are closed at time t c to help contain an outbreak and that a fraction Y of members in closed gatherings then decide to defy the closure by travelling to one of the remaining open gatherings. These open gatherings could be under a different set of rules in a different location or the venue/location itself may be defying government restrictions. Closures therefore protect the local community, which does comply with the closures, but can potentially increase attendance in any open venues.
As a first approximation, we ignore any spatial features and contacts occurring outside of average gatherings. We track Susceptible-Infectious-Recovered (SIR) dynamics within gatherings by assuming that the natural normalized transmission rate of the disease is l (with a recovery rate equal to 1 for time units set to the recovery period). We use S o/c , I o/c and R o/c to denote the number of susceptible, infectious and recovered individual in a typical open/closed gathering, respectively. Applying standard SIR dynamics in open gatherings but removing transmission events in closed gatherings, we write
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The copyright holder for this preprint this version posted August 24, 2020. 
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The copyright holder for this preprint this version posted August 24, 2020. There is a positive correlation between both but not clear distinction with local policy, suggesting that the phenomenon is related to the local distribution of services and behavior of the local population.
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The copyright holder for this preprint this version posted August 24, 2020. Fig.1(c) . This mechanism increases the population of open gatherings to N M ⇥ [1 + XY /(1 X)], and similarly decreases the population assigned to closed gatherings to N M (1 Y ). After closures are implemented, the dynamics of the cloSIR model are still governed by the same set of ordinary differential equations. As the outbreak progresses, we are interested in two key observables: first, the total number of infectious individuals
and, second, the total fraction of recovered individuals
Finally, note that N and M are used to help us write the equations but only act as scale factors in our results and can therefore be set to N = M = 1 for simplicity and without loss of generality. Ultimately, although the dynamics are governed by the classic SIR differential equations for all time, the cloSIR model offers an interesting tradeoff between controlling transmission by closing venues and intensifying transmission by aggregating contacts in a smaller number of still open venues. The question then becomes whether the redistribution of participants among gathering locations, e.g., churches or parks, will have a positive or negative impact on the epidemic. Assuming one cannot ensure the closure of all venues, is closing a certain percentage of venues worth the increase in visitors to those that remain open?
Strikingly, we find that in many scenarios the optimal strategy to minimize the size of the outbreak is often no intervention at all. Figure 4 shows that depending on the proportion of the population that chooses to go to another open business (Y ) the final outbreak size is often minimized when X = 0 (no closures). In fact, close or below the epidemic threshold l c = 1, interventions can spark an outbreak in communities otherwise not at risk by increasing the concentration of susceptible individuals. However, for stronger epidemics (larger l ), although a complete closure of gatherings X = 1 might be the optimal strategy, the expected outbreak size often follows a non-monotonous function of X such that the optimal outcome at X = 1 is next to a worst-case scenario at large values of X just below 1. What this implies is that the outcome is highly dependent on the amount of non-compliance that one can expect in a population (i.e. larger values of non-compliance Y ).
Similarly, poorly-timed interventions can actually lead to additional waves of infection. Figure 5 shows that secondary peaks of infection occur if intervention is initiated too late. Interestingly, stronger interventions tend to dramatically heighten the epidemic peak under many closure scenarios (colored curves) compared to the no-closure baseline (black curve).
The cloSIR model therefore provides the simplest possible example of the potential impact of the collective behaviour observed in the empirical mobility and search data from the US around essential services. Although future research should layer in additional complexity into models of policy interventions, within the idealized scenario considered by the model, one can solve for specific features: E.g., the final outbreak size, the optimal closure percentage X, and the critical value of non-compliance Y such that weak interventions increase outbreak size. Analyses of these different questions are presented in the Supplementary Information document.
Balancing the mental, economic, and social health of populations with the serious risks of COVID-19 means that restricting movement (e.g., cordons sanitaires) should only be considered as a last resort. Indeed, evidence from China suggests that while the cordon sanitaire of Wuhan may have delayed the outbreak, it was local measures that slowed transmission and ultimately controlled the outbreak 6, 21 . As now seen in numerous countries, COVID-19 can be controlled in the absence of strict mobility restrictions and/an effective vaccine with coordinated public health responses 7 . Despite evidence of the efficacy of consistent non-pharmaceutical interventions, many countries (especially the United States of America) continue to implement control measures in a scattered, patchwork manner 8 .
Using real-time mobility and search data in the US, we found that distance traveled and the number of visits to essential services did not correlate strongly with any demographic variables (e.g. population density, average age). However, both of these responses did correlate with community tightness, with tight communities being those "with strong norms and little
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The copyright holder for this preprint this version posted August 24, 2020. . We see that the worst-case scenario is a function of l , X and Y , as there is no consistent ranking in outbreak sizes. Small interventions appear beneficial against very transmissible pathogens but risk lowering the epidemic threshold at high frequency of non-compliant individuals, and larger interventions highlight this effect. The bottom right panel highlights this effect: we see that the impact of the fraction of non-compliant individuals is non-linear close to the epidemic threshold. At low values of Y , i.e., in a population with high compliance to recommendations, closing more gatherings is always beneficial. At the opposite end, for high values of Y , keeping all gatherings open is the optimal intervention. However, at medium values of Y , while closing all gatherings is still the optimal intervention, no closures is better than partial closures. tolerance for deviance" (Fig. S1) . Gelfand et al. (2020) found that countries with both efficient governments and those with tight cultures were the most effective in limiting COVID-19 cases and deaths 22 . However, White & Hébert-Dufresne (2020) found the opposite for the United States, with the tighter states having the faster COVID-19 growth rates early in the pandemic 9 . In the context of our cloSIR model, tight communities are those with strong restrictions (i.e. a high X) and few non-compliant individuals (i.e. a low Y ). Thus, if we know the level of tightness of a community, in this case a state, we can make more targeted policy recommendations. For example, if a government expects a large number of non-compliant individuals and complete lockdown is not possible, it might be best to have no lockdown at all (Figures 4 & 5) .
Given these unintended consequences, instead of blanket closures of churches, parks, and other essential services, leaders can enact measures that provide for the benefits of physical distancing, while balancing the mental health costs of social isolation. For example, more frequent services with fewer members could increase compliance with physical distancing. Without balancing the requirements of the intervention with the needs of the people, partial measures can be worse than no measures at all. In the context of vaccination, such effects have been previously identified. Past work has shown that implementing stricter vaccination rules tends to cluster children that are under-vaccinated 23, 24 . These findings of a policy backfiring effect leading to disease spread are in line with our empirical and modeling results.
Our results are particularly relevant as the loosening of restrictions on businesses and other institutions is also occurring in a disorganized and spatially heterogeneous way across the US 25 . As we say during the inconsistent shutdown, this lack of coordination at the policy-level creates incentives for individuals to travel to those areas which are open to purchase products 7/11 . 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 preprint this version posted August 24, 2020. Colored dots indicate epidemic peaks larger than the no intervention baseline scenario. Intuitively, we find that earlier interventions are always better. We see that delayed and imperfect interventions can cause second epidemic waves. There are several caveats to our study. First, we developed a simple model that was able illustrate the potential unintended consequences of individuals adapting their behavior to seek essential services under inconsistent physical distancing policies. While the simplicity of this model is a strength when trying to isolate the effects of inconsistent control policies on COVID-19
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The copyright holder for this preprint this version posted August 24, 2020. . transmission, future work will be needed before such models could be used to actively inform specific policy decisions. Second, the because the SafeGraph data do not track individual users over long periods of time, those observed in late March are not necessarily the same individuals observed earlier in the month. Moreover, we may expect biases in the diversity and behaviors of individuals tracked by the system. Different types of gatherings attract different types of individuals which will not be sampled homogeneously by any one data source. Altogether, these limitations mean that small geographic regions should not be directly compared to one another, or even to themselves at a different time, and different locations should not be directly compared. This is why we coarse-grained our results over states, why we only compare relative changes and not absolute differences, and why we attempted to correlate our findings with a secondary data source like online searches. Future work is therefore warranted, on both data collection and analysis (comparing changing movement patterns for various other business types) and mathematical modeling (expansion to include more metapopulation structure to explore the interplay of interventions across scales).
As we have shown, it is of key importance that non-pharmaceutical interventions for the fight against COVID-19specifically related to business and venue closure -must be implemented in a uniform way to avoid situations where enacting an intervention is worse than no intervention to begin with. Further, relaxation of such interventions must be done methodically and over time, with a strong emphasis on equity across incomes and geographies to avoid endangering individuals with lower socioeconomic status 28 . Human behavior is a strong driver of the transmission dynamics of SARS-COV2 and care must be taken to reduce the heavy burden imposed by COVID-19 and avoid unintended, negative consequences from inconsistent policies around implementing and relaxing non-pharmaceutical interventions.