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First 5000 Characters:Previous work has indicated that contact tracing and isolation of index case and quarantine of potential secondary cases can, in concert with physical distancing measures, be an effective strategy for reducing transmission of SARS-CoV-2 (1) . Currently, contacts traced manually through the NHS Test and Trace scheme in the UK are asked to self-isolate for 14 days from the day they were exposed to the index case, which represents the upper bound for the incubation period (2) . However, following previous work on screening strategies for air travellers (3,4) it may be possible that this quarantine period could be reduced if combined with PCR testing. Adapting the simulation model for contact tracing, we find that quarantine periods of at least 10 days combined with a PCR test on day 9 may largely emulate the results from a 14-day quarantine period in terms of the averted transmission potential from secondary cases (72% (95%UI: 3%, 100%) vs 75% (4%, 100%), respectively). These results assume the delays from testing index cases' and tracing their contacts are minimised (no longer than 4.5 days on average). If secondary cases are traced and quarantined 1 day earlier on average, shorter quarantine periods of 8 days with a test on day 7 (76% (7%, 100%)) approach parity with the 14 day quarantine period with a 1 day longer delay to the index cases' test. However, the risk of false-negative PCR tests early in a traced case's infectious period likely prevents the use of testing to reduce quarantine periods further than this, and testing immediately upon tracing, with release if negative, will avert just 17% of transmission potential on average. In conclusion, the use of PCR testing is an effective strategy for reducing quarantine periods for secondary cases, while still reducing transmission of SARS-CoV-2, especially if delays in the test and trace system can be reduced, and may improve quarantine compliance rates.
Following the notation of Kretzschmar et al. (2020) (5) , we consider the following events to be relevant to the tracing of the contacts of an index case -an individual assumed to be newly-symptomatic with COVID-19 and seeking a test (Figure 1 ). Each of the following variables are specific to an individual, but we omit a subscript, , for brevity. An individual is exposed and becomes infected at time
We assume the i . T 0 index case has onset of symptoms at time , lasting until time . For asymptomatic cases, no symptoms T 2 T 2 ′ are ever displayed and hence both and are undefined. For sensitivity, we assume testing of the index T 2 T 2 ′ case occurs at time , 1, 2 or 3 days after symptom onset , with the results of the test available at time ;
those testing positive will go on to isolate for 10 days from their symptom onset (6) . We assume that positive cases are immediately referred to contact tracers, with the index case's contacts' information sourced at , T 4 and these contacts are then traced and quarantined at time For comparison, we consider the baseline for . T 4 ′ index case testing delay to be 2 days. Figure 1 -Example schematic of the contact tracing process and associated timings where an index case causes two secondary cases, one of which is symptomatic and one of which is asymptomatic. Darker shaded regions of each cases' timeline indicate periods of increased infectivity. Arrows indicate transmission events, with red crosses indicating transmission prevented through quarantine of traced contacts. As the asymptomatic secondary case is quarantined (dashed red line) before they become infectious, they never spend any time infectious in the community.
Rather than assuming a specific time at which infectivity begins ( in Kretzschmar's notation) we consider T 1 the infectivity profile, i.e, a distribution of times at which transmission is likely to occur. This distribution is derived by considering the incubation period (time from exposure to onset of symptoms), , and T 2 − T 0 delay from infectiousness to onset of symptoms , and using the corrections by Ashcroft et al. (2020) T 2 − T 1 (7) to the method of He et al. (2020) (8) with incubation periods sampled from the distribution in (9) .
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For the index case we parameterise the delays associated with the contact tracing system (having a test to receiving the results ( ), sourcing contact information ( ), and tracing ( )) according
to the latest NHS Test and Trace data from the week 16 July 2020 to 22 July 2020 (Table 1 ) (10) . These times are reported as 24 hour intervals ( days), which we used to derive a 0 , , ,
Gamma distribution considering the delay in each index case's awaiting a test result, sourcing of contacts and tracing of contacts as doubly-censored observations on the specified time intervals using the fitdistcens function from the fitdistrplus package in R (11) .
We consider th