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 [1–4]. 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 [7–9]. 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 [14–27]. 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) are: dS/dt=−κβS/N(σIpsIps+σIsIs)dE/dt=κβS/N(σIpsIps+σIsIs)‐(qE→Ips+ftrεE←Ips+ftrεE←Is)EdIps/dt=qE→IpsE‐(qIps→Is+ftrεIps←Ips+ftrεIps←Is)IpsdIs/dt=qIps→IsIps‐(τIs+γIs+ftrεIs←Ips+ftrεIs←Is+αIs)IsdQ/dt=ftr(εE←Ips+εE←Is)E+ftr(εIps←Ips+εIps←Is)Ips+(τIs+ftr(εIs←Ips+εIs←Is))Is‐γQQ‐αQQdR/dt=γsIs+γQQ [Eq 1]

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_sτ_Is multiplied by the number of close contacts per case, N_cpc): ftr=NCTNCCTD/IsτIsNcpc [Eq 2]

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:

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]: Rt=S/N[κβ]*[qE→Ips/(qE→Ips+ftrεE←Ips+ftrεE←Is]*[[σIps/(qIps→Is+ftrεIps←Ips+ftrεIps←Is)]+[(σIs)/(γIs+τIs+ftrεIs←Ips+ftrεIs←Is+αIs)]*[qIps→Is/(qIps→Is+ftrεIps←Ips+ftrεIps←Is)]] [Eq 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_sI_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 in ~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 [42–44]. 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 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 [15,53–55], 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 re-openings 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.