key: cord-1044052-tqfbs568
authors: Gardner, Billy J.; Kilpatrick, A. Marm
title: Contact tracing efficiency, transmission heterogeneity, and accelerating COVID-19 epidemics
date: 2021-06-17
journal: PLoS Comput Biol
DOI: 10.1371/journal.pcbi.1009122
sha: 8c435bb0afcee51d676f82944d518515e497a8ab
doc_id: 1044052
cord_uid: tqfbs568

Simultaneously controlling COVID-19 epidemics and limiting economic and societal impacts presents a difficult challenge, especially with limited public health budgets. Testing, contact tracing, and isolating/quarantining is a key strategy that has been used to reduce transmission of SARS-CoV-2, the virus that causes COVID-19 and other pathogens. However, manual contact tracing is a time-consuming process and as case numbers increase a smaller fraction of cases’ contacts can be traced, leading to additional virus spread. Delays between symptom onset and being tested (and receiving results), and a low fraction of symptomatic cases being tested and traced can also reduce the impact of contact tracing on transmission. We examined the relationship between increasing cases and delays and the pathogen reproductive number R(t), and the implications for infection dynamics using deterministic and stochastic compartmental models of SARS-CoV-2. We found that R(t) increased sigmoidally with the number of cases due to decreasing contact tracing efficacy. This relationship results in accelerating epidemics because R(t) initially increases, rather than declines, as infections increase. Shifting contact tracers from locations with high and low case burdens relative to capacity to locations with intermediate case burdens maximizes their impact in reducing R(t) (but minimizing total infections may be more complicated). Contact tracing efficacy decreased sharply with increasing delays between symptom onset and tracing and with lower fraction of symptomatic infections being tested. Finally, testing and tracing reductions in R(t) can sometimes greatly delay epidemics due to the highly heterogeneous transmission dynamics of SARS-CoV-2. These results demonstrate the importance of having an expandable or mobile team of contact tracers that can be used to control surges in cases. They also highlight the synergistic value of high capacity, easy access testing and rapid turn-around of testing results, and outreach efforts to encourage symptomatic cases to be tested immediately after symptom onset.

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Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) emerged in late 2019, spread globally in early 2020, and resulted in rapidly growing local epidemics, large scale mortality, and strains on hospital capacity in many countries [1] [2] [3] [4] . Initial outbreaks in most countries were limited only by severe control measures including closing all but essential businesses as well as schools, churches, and other organizations [5, 6] . Only a few countries were able to limit transmission with less disruptive public health measures [7] [8] [9] . The combination of severe disease control measures and the disease itself have had devastating impacts on economies and societies [10, 11] . Following control with strict lockdown measures, most countries attempted to re-open as many business sectors and activities as possible while avoiding a rapid rise in infections.

Although self-isolation of symptomatic individuals, social distancing, and mask wearing have reduced the transmission of SARS-CoV-2, additional interventions, including business closures and working from home, have often been required to keep the pathogen reproductive number R t below 1 [10, 12, 13] . One public health strategy that has been used to reduce transmission in some countries is testing symptomatic individuals, tracing their contacts to people they may have infected, isolating infected individuals, and quarantining people that may have become infected but have yet to show symptoms or test positive for the virus (hereafter abbreviated TTIS) [8, 13, 14] . If contacts of cases can be found and quarantined or isolated before or during their infectious period, this can limit onward spread of the virus [15] .

Numerous studies have examined the effectiveness and limitations of TTIS on transmission of SARS-CoV-2 and other pathogens [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] . Many studies have shown that TTIS can substantially reduce the pathogen reproductive number, R t , but its efficacy depends on the importance of pre-symptomatic and asymptomatic transmission, delays between symptom onset and being tested, and the fraction of infections that are tested and traced [14, 19, 22, 24] . Previous studies have explored various parameter values for contact tracing efficacy by varying the fraction isolated, the fraction symptomatic, and the contribution to transmission of undetected infections [14, 19, 22] . A key unexplored challenge in implementing TTIS is that tracing contacts and ensuring they can safely quarantine or isolate is a time-consuming process which results in only a fraction of contacts being reached if cases and their contacts exceed case investigation and contact tracing capacity. This reduces the effectiveness of contact tracing as cases increase. Previous studies have assumed fixed values for contact tracing parameters, or have simulated epidemics with models that do not describe the links between cases and their contacts, or use models that don't include the interactions between rising cases, delays between symptom onset and tracing, and reductions in the pathogen reproductive number R t .

Our aim was to examine the relationship between increasing cases, contact tracing efficacy, and the pathogen reproductive number, R t , and to examine the potential outcomes for disease dynamics. We built a compartment model of SARS-CoV-2 transmission, parameterized it with data from the literature, and examined how R t varied with number cases traced, delays between symptom onset and the start of contact tracing, the numbers of contacts per case, and different fractions of symptomatic cases being tested and traced. We also simulated a stochastic version of the model with variable numbers of initial infections and with and without contact tracing to examine how reductions in R t and initial conditions affected variation in the timing of epidemics.

We built a susceptible-exposed-infected-recovered (SEIR) compartment model of SARS-CoV-2 that included two compartments for infected individuals that reflect the presence of symptoms (pre-symptomatic, I ps , and symptomatic, I s ; S1 Fig) . For simplicity, we omitted asymptomatically infected individuals (i.e. those that never develop any symptoms) because meta-analyses [28, 29] , as well as subsequent studies [30] , showed that asymptomatic individuals infect a much smaller fraction of their contacts than individuals that eventually develop symptoms. We also omitted severely symptomatic individuals because they are likely to be hospitalized and thus cause very few infections in the community. In addition, contact tracing severely symptomatic cases is ineffective because, by the time an infection progresses to severe symptoms 6-8 days after symptom onset [31, 32] , many of their contacts will already have finished most of their infectious period, and quarantining or isolating these contacts would have little effect. We note that models that included both of these classes of infected hosts produced results that were very similar to those described below. We also omitted more complex population structure (e.g. household clustering) in order to develop a more transparent relationship between rising cases, delays, and contact tracing efficacy.

The equations of the model (S1 Fig) 

Parameter values are given in Table 1 . κ is a social distancing factor between 0 and 1 that scales the transmission rate β (and directly scales the reproductive rate R t ). σ is the relative infectiousness for the two I classes. q are the transition rates between classes given by the subscripts separated by the arrow (q E!Ips is the transition rate between the E and I ps classes). τ Is is the rate of testing and removal of symptomatic infected individuals I s . The terms f tr � ε are the removal rate of individuals by contact tracing, with the ε values being the maximum rate of removal by contact tracing from the first subscripted class, due to infections that were caused by the second subscript (e.g. ε E Ips is the rate that individuals in the E class are removed by contact tracing that were infected by I ps individuals), and f tr is the fraction between 0 and 1 that are actually removed, due to limited contact tracing capacity (see below). Q is the quarantine/isolation class, α are the disease-caused death rates, and γ is the recovery rate to the R class.

The fraction of contacts that can be reached and placed in quarantine each day, f tr , is simply the contact tracing capacity (the number of contact tracers, N CT multiplied by the number of calls they can make per day N CCTD ), divided by the number of contacts that need to be reached each day (the number of symptomatic cases detected by testing, I s τ Is multiplied by the number of close contacts per case, N cpc ):

If tracing capacity exceeds contacts requiring tracing, all contacts are traced, so max f tr = 1. We note that contact tracing includes both case investigation to obtain information on contacts, and successfully reaching contacts and ensuring they can safely quarantine or isolate themselves. The capacity of a given public health jurisdiction reflects whichever part of this process is limiting.

The maximum contact tracing removal rates ε are derived from the fact that all infected individuals in the E, I ps , and I s classes were infected by individuals in the I ps and I s classes, at some time in the past. The maximum removal rates ε are the product of three terms: 1. the fraction of symptomatic cases I s that are tested, f te , before they recover:

2. the fraction of individuals in a given class that were infected by pre-symptomatic, I ps , or symptomatic, I s individuals, f Ips ¼ ðs Ips =q Ips!Is Þ=½s Ips =q Ips!Is þ s Is =ðt Is þ g Is Þ� f Is ¼ ½s Is =ðt Is þ g Is Þ�=½s Ips =q Ips!Is þ s Is =ðt Is þ g Is Þ�; ½Eq 4� and 3. the fraction of individuals that are still in infected class x (E, I ps , or I s ) by the time they are traced (S2 Fig) , f x y = e -λδ , where y is the class they were infected by (I ps or I s ), λ is the average rate that individuals leave a class (or a series of host classes), and δ is the delay from infection until being traced, which is the sum of the average time to test and trace individuals after symptom onset, 1/τ Is , the average time to trace contacts (0.5 days, since each day a new set of contacts arise), and, for infections caused by I ps individuals, the delay between I ps individuals infecting the contacts, and when I ps individuals are identified by testing in the I s class, 1/q Ips!Is :

The fraction remaining in each of the 3 infected classes, E, I ps , I s , that were infected by each of the two infectious classes, I ps and I s , respectively, is thus given by six equations:

The fraction of infections remaining in each class for variable delays in case symptom onset to testing positive, τ Is , is shown in S2 Fig. The maximum removal rates ε are simply the product of these three quantities: We parameterized the model with data from the literature (Table 1) . We note that spatial or temporal variability in contact rates (e.g. from social distancing or other non-pharmaceutical interventions) have direct linear impacts on R t and shift the curves in Figs 1 and 2 vertically (S3 Fig) , but do not change their shape and thus don't affect the proportional impact of contact tracing on R t . Increased transmissibility of viral variants such as B.617.2, B.1.1.7 or P.1 [33, 34] would similarly shift the curves vertically without changing their shape unless the temporal dynamics of infectiousness are substantially different [35] ; at present, data on viral load dynamics for viral variants are sparse [36] . Although the compartmental model above results in exponential distributions for the duration that hosts remain in each host class, the combination of an exposed class E and a pre-symptomatic class I ps results in an incubation period that is approximately lognormally distributed with mean and dispersion similar to the empirically observed values [37] . We explored models with multiple compartments for other classes (e.g. I s ) to investigate the implications of non-exponentially distributed durations and obtained very similar results to those described below. We note that while notifying individuals that they have had contact with a case can be done quickly (especially with using a cell-phone tracing app; [22] ), successfully ensuring a contact has their needs met (including food, medicine, clothing) to quarantine in a safe space where they won't infect their household members requires substantially more time (https://www.cdc. gov/coronavirus/2019-ncov/php/notification-of-exposure.html). Thus, we assumed the approximate duration required for a successful contact tracing call was 40 min, resulting in the number of contacts reached by a contact tracer per day, N CCTD , of 12.

We derived an expression for the pathogen reproductive number R t for the equations above using the next generation matrix [38] :

½½s Ips =ðq Ips!Is þ f tr ε Ips Ips þ f tr ε Ips Is Þ�þ ½ðs Is Þ=ðg Is þ t Is þ f tr ε Is Ips þ f tr ε Is Is þ a Is Þ��½q Ips!Is =ðq Ips!Is þ f tr ε Ips Ips þ f tr ε Ips Is Þ�� ½Eq 8�

This expression can be understood as the fraction of the population that is susceptible, S/N, multiplied by the contact rate β (which is scaled by the social distancing factor κ), multiplied by the fraction of hosts that pass through the E class to the I ps class without being removed by contact tracing (1 st row), multiplied by the sum of the infections arising from I ps (row 2) and the infections arising from I s (row 3). The infections arising from I ps (row 2) are the product of their infectiousness, σ Ips , and the infectious period of the I ps class (the inverse of the losses from the I ps class). The infections arising from I s are the product of their infectiousness, σ Is , and the infectious period of the I s class (the inverse of the losses from the I s class) all multiplied by the fraction of hosts in the I ps class that pass into the I s class without being contact traced.

We examined how R t varied with different numbers of new symptomatic cases detected (τ s I s ), delays of 1 to 10 days (which captures the range observed during the epidemic) between symptom onset and the start of contact tracing (1/τ Is in Eq 2), and numbers of contacts per case (N cpc in Eq 2) [39] . Rapid tests taken on the day of symptom onset, as used by the United Kingdom starting in April 2021, could result in delays as short as 1 day, whereas the time from symptom onset until test results were returned often exceeded 10 days in the US in 2020 [24, 40] . In some figures we used the baseline contact tracing capacity standards suggested by the US National Association of County and City Health Officials (15 contact tracers per 100,000 people; https://www.naccho.org/uploads/full-width-images/Contact-Tracing-Statement-4-16-2020.pdf), but note that because the fraction of contacts traced (Eq 2) is a ratio of four quantities, any combination of values that produce the same number of contacts per contact tracer calls per day will produce the same value of R t . Thus, the results are not geographically specific; all that is needed to apply the results to a new setting is the ratio of contacts needing tracing to tracing capacity. We performed a simple sensitivity analysis of this relationship by determining how much R t varied with a ten percent increase or decrease in each model parameter (S3 Fig) .

We examined the effect of decreasing contact tracing efficiency as infections increased on disease dynamics and R t by simulating a deterministic version of the model in Eq 1 and plotted R t in real time as an epidemic swept through a population over one year. We note that SARS-CoV-2 epidemics in most countries have consisted of multiple surges or waves [15, 41] as restrictions have limited contact rates. However, the relative relationships we show between the reproductive number R t and contact tracing demand relative to capacity are time-insensitive and depend only on infections, contacts and contact tracing capacity, and can be scaled by adjusting the fraction of the population that is susceptible.

Finally, we explored the implications of stochastic variability and contact tracing on infection dynamics in a scenario based on a moderate size city (100,000 people) with partly effective non-pharmaceutical interventions/social distancing (κ = 0.6), resulting in R t = 1.33 with moderately effective testing and contact tracing (equivalent to 1/τ Is = 10 days which results iñ 25% of infected individuals being tested or quarantined), or R t = 1.57 without contact tracing. We simulated a stochastic version of the model given by Eq 1 where the number of new infections was drawn from a negative binomial distribution with mean equal to R t and dispersion parameter 0.16 which is intermediate between available estimates for COVID-19 [42] [43] [44] . We examined different initial numbers (5 and 50) of latently infected individuals, E, at the start of the epidemic to understand how stochastic variation in transmission could impact the timing of epidemics when the number of initial infections was small or moderately large. R code to reproduce all results is available from: https://github.com/marmkilpatrick/ Contact-Tracing-Efficiency

The effectiveness of contact tracing in reducing the pathogen reproductive number, R t , was dependent on synergistic interactions among three factors: the number of cases being traced (given a fixed number of contact tracers), the delay between symptom onset and the start of tracing, 1/τ Is , (including getting tested and receiving result), and the fraction of symptomatic cases that get traced (Figs 1 and 2) .

First, the relationship between R t and the number of cases per contact tracer calls per day was approximately sigmoid (Figs 1 and 2) ; at both high and low case numbers adding or removing contact tracers had smaller effects, whereas at intermediate case numbers relative to capacity, shifting contact tracers had a much larger impact. When the ratio of contacts needing tracing to capacity was high (i.e. 10 or higher), contact tracing had relatively little effect in reducing R t no matter how long the delay was between symptom onset and the start of tracing, 1/τ Is (right side of Fig 1; this pattern is also evident in Fig 2) . This is because <10% of the contacts needing tracing and isolation were reached and isolated.

Contact tracers in regions with very high new case numbers relative to contact tracing capacity (>10 on Fig 1) would have a larger reduction on R t if they were tracing calls in locations with intermediate numbers of cases (ratios of contacts needing tracing to capacity of 1-5). In order to reverse an increase in cases (i.e. to reduce R t <1), the analyses in Figs 1 and 2 suggest that when new case burdens are high relative to capacity, population-wide interventions (e.g. social distancing or different levels of shelter-in-place or lockdown orders which reduce contact rates, β or κ, and shift the entire curves in Figs 1 and 2 downward proportionately and S3 Fig) , or orders of magnitude increases in contact tracing capacity are needed until R t can be effectively reduced by contact tracing. Conversely, and intuitively, when there is excess contact tracing capacity (to the left of 1 on the x-axis in Fig 1) , contact tracing was as effective as it could be in reducing R t , but excess capacity is underutilized. Shifting a subset of contact tracers to areas where cases and contacts needing tracing exceed capacity could substantially reduce R t in those areas.

Second, increasing delays, 1/τ Is , between symptom onset and the start of tracing had a synergistic effect on the efficacy of contact tracing (Fig 1) . With a 10 day delay, contact tracing could only reduce R t by 20% (from 2.6 to 2.1 in Fig 1; compare the right and left heights of the yellow curve). In contrast, if all symptomatic individuals got tested within 4 days of symptom onset and results were returned within the next 24 hours (1/τ Is = 5 days), contact tracing could reduce R t by 40% from 2.2 to 1.3 (Fig 1 green curve) . The maximum benefit of contact tracing (the difference between the maximum and minimum of each curve) is relatively constant for short delays of 1-3d between symptom onset and testing (1/τ Is ) because most infected contacts are still in the E class or have just entered the I ps class (S2 Fig). As these delays increase further, more contacts have moved into the infectious classes and have transmitted the pathogen to other hosts (S2 Fig). With a 10 day delay, 96% of contacts have left the E class, and 49% have already reached the R class (S2 Fig). The number of contacts per case obviously also influences the time required to trace these contacts (Fig 2) . If allowable (or illegal) gathering sizes increase, this increases the number of contacts per case, which reduces contact tracing efficacy if contact tracing capacity is exceeded.

Thirdly, if contacts for a substantial fraction of all symptomatic cases do not get traced and quarantined, TTIS is far less effective. Figs 1 and 2 primarily showed optimistic scenarios where the fraction of symptomatic infections that are tested and traced is determined only by the delay between symptom onset and testing results being returned (1/τ Is ) and the rate of recovery. With a 5 day delay (1/τ Is = 5) (Fig 2) , this results in 50% of infections being detected by testing in the mildly symptomatic state (I s ) which is higher than some estimates of case under-ascertainment based on seroprevalence studies, especially early in the pandemic [47, 48] . In contrast, if only half of symptomatic cases are tested (and their contacts traced), this is similar to a 10 day delay (1/τ Is = 10) between symptom onset and tracing, which has a far lower impact in reducing R t at both high and low case burdens (the yellow curve in Fig 1) . If quarantining contacts is only partly effective (e.g. they stay home but infect household members who go on to infect others) this will similarly reduce the effectiveness of contact tracing.

Limited contact tracing can also produce unexpected dynamics. Reduced contact tracing efficiency with increasing cases results in a transient accelerating epidemic where R t actually increases over time leading to a much larger epidemic size and an accelerated epidemic ( Fig  3E and 3F) . A decrease in contact tracing efficiency as cases rise can, initially, outweigh the depletion of susceptible individuals which leads to a spike in R t over time, until depletion of susceptibles overwhelms this effect (compare rightmost panels, Fig 3E and 3F , which have limited contact tracing, to leftmost panels, Fig 3A and 3B , where social distancing reduces R t to the same initial value as contact tracing). With effectively unlimited contact tracing this phenomenon does not arise (compare middle panels, Fig 3C and 3D , which show effectively unlimited contact tracing, to rightmost panels Fig 3E and 3F) .

Finally, the impact of contact tracing in reducing the pathogen reproductive number R t has two consequences on the temporal timing and establishment of epidemics. First, as is well known, reducing R t delays and reduces the peak of the epidemic (Fig 4 top vs bottom panels) . Second, and less appreciated during the current pandemic, stochastic variation in R t can lead show dynamics with effectively unlimited contact tracing (1500 contact tracers making 12 calls/day; 10 contacts per case) but no social distancing (κ = 1), with an identical value of R 0 as in panels A, D. Right panels (C, F) show dynamics with the same parameter values as (B, E) except with limited contact tracing (15 contact tracers). R 0 is the same value as in panels D and E (R 0 = 1.35), but R t increases as cases increase and contact tracing becomes inefficient, which overwhelms the decrease in the fraction of the population that is susceptible. In all panels, the delay from symptom onset to receiving test results, 1/τ Is , is 5d. All populations start with 100,000 individuals. Note the identical epidemic sizes (final fraction susceptible 0.53) for panels A (social distancing) and B (effectively unlimited contact tracing), but much larger epidemic size for limited contact tracing in panel C (final fraction susceptible 0.14).

https://doi.org/10.1371/journal.pcbi.1009122.g003

Contact tracing efficiency, transmission heterogeneity and accelerating COVID-19 epidemics to very different timing of epidemics if the initial number of infected individuals is low (Fig 4A  and 4C ; epidemic peaks can vary by 6 months by chance), and variation is larger if R t is lower (Fig 4A vs Fig 4C and Fig 4B vs Fig 4D) . Finally, heterogeneity in individual transmission can result in local fadeout of the pathogen and fadeout is more likely when contact tracing reduces R t closer to 1, and if the number of initially infected individuals is lower (compare the Fraction of epidemics established in panels A vs B-D).

The two main strategies that have been used to control SARS-CoV-2 transmission before the development of vaccines were TTIS and society-wide social distancing interventions (including closing businesses, banning gatherings, wearing masks, etc.) [12] . Closing businesses has had devastating impacts on employment and economies, as well as cascading impacts on society. TTIS has far smaller economic and societal costs, but its efficacy in controlling epidemics is not fully understood, and some studies suggest that it is insufficient to keep R t below 1 in many settings, especially without widespread testing and digital contact tracing [13, 14, 22, 49] . We examined how the efficacy of contact tracing decreases with increasing case burden. As case burdens increased relative to contact tracing capacity, contact tracing reached a small fraction of contacts and had little effect on R t , leading to accelerating epidemics. Conversely, when case numbers were very low relative to contact tracing capacity, there was excess capacity and, all else being equal, contact tracers could be used more effectively in higher case burden settings with negligible impacts on local transmission. We note that the exact number of contact tracers needed to reduce R t depends on the number of contacts per case and the number of calls each tracer can make each day (Fig 2) . However, this key quantity appears as a ratio of case-contacts per contact tracer calls per day. Thus, each contact tracing team (e.g. a county or local jurisdiction) can use local estimates of contacts per case and the number of calls each tracer can make each day to determine where they are on the modelled relationships (Figs 1  and 2) .

Allocating contact tracers among populations to most effectively reduce total infections is a non-trivial problem. A smaller reduction in R t (e.g. 10%) in one population can prevent more infections (especially over multiple generations of transmission) than a larger (e.g. 20%) reduction in R t in a second population if R t in the second location is lower (especially when R t >1 in the first population), or when there is a larger number of infected individuals in the first population. Thus, transferring contact tracers from a region with a high case burden relative to contact tracing capacity to maximize their efficacy in reducing R t should only be done if other measures (e.g. social distancing) can be put into place to reduce R t where case numbers are high. More generally, allocation of contact tracers to maximize the number of cases prevented given an array of tools would require a complex dynamic analysis beyond that examined here. However, the results shown here show the quantitative impact that shifts in contact tracers can have in reducing R t , which can be critical, especially if the goal is to use contact tracing to reduce R t <1 in a given population.

We also found that the efficacy of contact tracing itself, regardless of capacity, was strongly influenced by delays between the onset of symptoms and the beginning of tracing, as well as the fraction of symptomatic infections that were traced. Unless delays were short and the fraction of symptomatic cases that were traced was high, contact tracing had limited effects in reducing R t . This finding of synergistic effects between testing delays and contact tracing efficiency parallels results from other studies demonstrating the large effects of delays in reducing efficacy of isolating infections by testing alone [50] . We note that in the model considered here, only symptomatic individuals were removed by testing (pre-symptomatic individuals were not detected by testing) which leads to a smaller impact of testing on R t than is possible if all individuals are tested [40] . Our results emphasize the importance of encouraging people to get tested as soon as possible after mild symptom onset, and having sufficient testing capacity to return their results quickly [40] . Similarly, the fraction of symptomatic infections that get tested and traced is poorly known, but if the ratio of infections to cases from seroprevalence studies in some locations is approximately correct (e.g. 10:1 to 4:1; [47, 48] ), then contact tracing will have limited effects in reducing transmission.

Allocation of contact tracing resources can be most efficiently deployed in two ways. First, contact tracing is much more effective when infections are detected soon after symptom onset. One should prioritize these individual cases for tracing since their contacts are likely to be earlier in their infections and quarantining/isolating them will cut off most or all of their infectious period (S2 Fig). If one knows the date of contact between the case and the contact, one could also prioritize tracing more recent contacts and those that had contact with the case during the case's days of peak infectiousness just before and after symptom onset [51, 52] . Second, if one is attempting to limit transmission in multiple regions (e.g. counties within a state) one could deploy contact tracers to counties where they will be able to have the most impact: from places with excess capacity to those with intermediate numbers of cases per contact tracer calls per day. Conversely, if contact tracers cannot quarantine the contacts of cases within 10-12 days of the case's symptom onset, they will be unlikely to effectively reduce transmission from those contacts.

Our results also demonstrate two phenomena observed in COVID-19 epidemics. First, epidemic dynamics sometimes differ enormously between places that seem otherwise similar which is well understood by mathematical modelers [15, [53] [54] [55] , but sometimes forgotten in considering spatial variation in epidemic outcomes. Spatial variation in disease dynamics may be due to differences in social behavior or contact patterns, but we showed that stochastic chance may also play a large role in shifting the timing of epidemics by up to six months. The impact of stochasticity is largest when the initial number of cases is low and R t is close to 1 (i.e. when lockdowns are first lifted), because this results in populations spending longer periods of time with few cases where stochastic variation is most important. Second, staged business reopenings and the emergence of new virus variants have sometimes led to accelerating or runaway epidemics. These may be due to sudden changes in social behavior, but we showed that accelerating epidemics can also result from decreases in contact tracing efficiency [26] . Increasing contact tracing capacity could limit this epidemic acceleration as cases increase, which suggests that training a reserve capacity of tracers and being able to deploy a mobile tracing force could help limit runaway epidemics.

More broadly, contact tracing could play an important role in limiting transmission of SARS-CoV-2 and other pathogens [49] . However, we found that its efficacy depends on participation in seeking testing immediately following symptom onset, quick return of test results, and sufficient contact tracing capacity if case numbers surge. Shortcomings in each of these factors greatly limit its efficacy, especially as cases increase, which could necessitate much more damaging measures to control transmission, including widespread business and school closures. Investments in public health, including testing, contact tracing, and public outreach to encourage health seeking when symptomatic, is likely a much more cost-effective approach to control COVID-19, and other diseases. Table 1 for parameter values. Boxes represent Susceptible (S), Exposed (E), Infected (I), recovered (R), and Quarantined/Isolated (Q) classes. There are two compartments for infected individuals that reflect the presence of symptoms (pre-symptomatic, I ps , and symptomatic, I s ). κ is a social distancing factor between 0 and 1 that modifies the contact rate β, σ are infectiousness for each of the I ps and I s classes, q are transition rates between classes given by the subscripts separated by the arrow (e.g. q E!ps is the transition rate between the E and I ps classes), ε are the rates of removal by contact tracing from the E or I classes to the quarantined class Q based on which class infected those individuals (e.g. ε E Ips is the contact tracing removal rate of E individuals that were infected by I ps individuals), τ Is is the removal rate by testing of symptomatic infected individuals, α is the disease-caused death rate, and γ are the recovery rates to the R class. The dashed lines indicate that both classes of infected individuals contribute to transmission. (TIF)

For example, if the delay between infection and quarantine/isolation is 6 days, then 15% of infected individuals will still be in the latently infected class, E, 28% in the pre-symptomatically infected class, I ps , 33% in the symptomatically infected class, I s , and 24% will have already recovered, R. (TIF) S3 Fig. Sensitivity analysis. The plot shows how much R t changes from a 10% increase (red) or 10% decrease (blue) in that model parameter relative to values in Table 1 (with τ Is = 0.2 and κ = 1). R t scales linearly with β and κ, whereas transition parameters q E!Ips , q Ips!Is , the testing rate τ Is , and pre-symptomatic infectiousness σ Ips have approximately 50-75% as large an effect as β or κ. The recovery rate, γ Is , and symptomatic infectiousness σ Is are less influential, and the death rate α Is has very little effect on R t . (TIF)

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We thank the dozens of scientists and public health practitioners who have informed this work through comments and discussions, and helpful comments from Odo Diekmann and HansHeesterbeek.

Conceptualization: Billy J. Gardner, A. Marm Kilpatrick.