key: cord-0838433-wr82hjcq
authors: Ashcroft, P.; Lehtinen, S.; Bonhoeffer, S.
title: Quantifying the impact of test-trace-isolate-quarantine (TTIQ) strategies on COVID-19 transmission
date: 2020-12-07
journal: nan
DOI: 10.1101/2020.12.04.20244004
sha: a725885b9a5b0f186dd07b4a23a540c617147673
doc_id: 838433
cord_uid: wr82hjcq

The test-trace-isolate-quarantine (TTIQ) strategy is used to break chains of transmission during a disease outbreak. Confirmed-positive pathogen carriers are isolated from the community to prevent onward transmission and their recent close contacts are identified and pre-emptively quarantined. TTIQ, along with mask wearing and social distancing, make up the non-pharmaceutical interventions that are utilised to suppress the ongoing SARS-CoV-2 pandemic. The efficacy of the TTIQ strategy depends on the probability of isolating a case, the fraction of contacts quarantined, and the delays in these processes. Here we use empirical distributions of the timing of SARS-CoV-2 transmission to quantify how these parameters individually contribute to the reduction of onwards infection. We show that finding and isolating index cases, and doing so with minimal delay after symptom onset, have the largest effects on case reduction, and that contact tracing can make up for deficiencies in testing coverage and delays. These results can be used to assess how TTIQ can be improved and optimised. We provide an online application to assess the efficacy as a function of these parameters.

tive test result, such that their onward transmission is reduced. This is exemplified In a previous study of TTIQ efficacy, Ferretti et al. (2020b) used an approach 62 based on the empirically-observed timing of transmission events -but with sub-63 stantial approximations around the TTIQ process -to get to an analytically tractable 64 prediction of the impact of TTIQ on SARS-CoV-2 transmission. They concluded lighted the optimal use of test-and-release strategies (Ashcroft et al., 2020b) . Briefly, 79 we use the empirically-observed distributions of transmission timing [ Fig. 2 ; Fer-80 retti et al. (2020a)] to determine when infections occur (Fig. 1) . We then introduce 81 five parameters to describe the TTIQ process: i) f , the probability that an index case 82 is isolated from the population and is interviewed by contact tracers; ii) ∆ 1 , the time 83 delay between symptom onset and isolation of the index case; iii) τ, the duration 84 prior to symptom onset in which contacts are identifiable; iv) g, the fraction of iden-85 tifiable contacts that are quarantined; and v) ∆ 2 , the delay between isolation of the 86 index case and the start of quarantine for the contacts. We compute the expected 87 number of tertiary cases per index case under the TTIQ interventions, with the aim 88 being to reduce this number below one to suppress the growth of the epidemic (see 89 Methods for details). We systematically explore this parameter space, first for the 90 "testing & isolation" intervention in the absence of contact tracing (Fig. 1A) , and 91 then with additional "tracing & quarantine" (Fig. 1B) . cases are identified and isolated from the population after a delay ∆ 1 after they develop symptoms (at time t S 1 ). This curtails their duration of infectiousness and reduces the number of secondary cases. B) Under tracing & quarantine, the contacts of an index case are identified and quarantined after an additional delay ∆ 2 . This reduces the onward transmission from the secondary cases. Only contacts that occur during a contact tracing window can be identified. This window extends from t S 1 − τ (i.e. τ days before the index case developed symptoms) to t S 1 + ∆ 1 (i.e. when the index case was isolated). Shown distributions are schematic representations of those shown in Fig. 2. 3 . CC-BY-NC 4.0 International license It is made available under a perpetuity.

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The copyright holder for this this version posted December 7, 2020. ; https://doi.org/10.1101/2020.12.04.20244004 doi: medRxiv preprint 2 Methods

Our primary goal is to quantify the reduction of transmission by isolating indi-94 viduals who test positive for SARS-CoV-2 and by quarantining their recent close 95 contacts with an increased risk of infection. We refer to the initial confirmed case 96 as the index case, and the infected contacts as secondary cases. We know that the 97 index case developed symptoms at time t S 1 , but the time at which they were in-98 fected, t 1 , is generally unknown. Secondary cases will be infected by the index case 99 at some time t 2 (t 2 > t 1 ), and develop symptoms at time t S 2 ( Fig. 2A) . The relationships between the times t 1 , t S 1 , t 2 , t S 2 are determined by: the generation 102 time distribution, q(t 2 − t 1 |θ q ), describing the time interval between the infection 103 of an index case and secondary case (Fig. 2B) ; the infectivity profile, p(t 2 − t S 1 |θ p ), 

where p(t|θ p ) is the infectivity profile and P(t|θ p ) = t −∞ dt p(t |θ p ) is the cumula- is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted December 7, 2020. incubation period (days) probability density incubation period D Fig. 2 A) The timeline of infection for an infector-infectee transmission pair. The infector (index case) is initially infected at time t 1 , and after a period of incubation develops symptoms at time t S 1 . The infectee (secondary case) is infected by the infector at time t 2 , which can be before (presymptomatic) or after (symptomatic) t S 1 . The infectee then develops symptoms at time t S 2 . The generation time is then defined as t 2 − t 1 (the time between infections), while the serial interval is defined as t S 2 − t S 1 (the time between symptom onsets). B) The generation time distribution [q(t|θ q ) = q(t 2 − t 1 |θ q )] follows a Weibull distribution (Ferretti et al., 2020a) . C) The infectivity profile [p(t|θ p ) = p(t 2 − t S 1 |θ p )] follows a shifted Student's t-distribution (Ferretti et al., 2020a) . D) The distribution of incubation times [g(t) = g(t S 1 − t 1 )] follows a meta-distribution constructed from the mean of seven is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted December 7, 2020. ; https://doi.org/10.1101/2020.12.04.20244004 doi: medRxiv preprint are not isolated (T → ∞). We can compute the number of secondary infections, n 2 , as a function of testing coverage f and delay ∆ 1 , as shown in Fig. S1 . For a given 131 symptom onset time t S 1 and degree k 1 of the index case, we have

(2)

Averaging over k 1 , which is distributed as p k 1 , and keeping t S 1 fixed as the reference 133 time point, we arrive at

where R e = k 1 is the mean of p k 1 , i.e. the average number of secondary infections 135 in the absence of testing & isolation ( f = 0).

Each secondary case has some potential to cause further infections, which will be 138 the tertiary cases of the index case. The number of tertiary infections caused by a 139 secondary case who is infected at t 2 and isolated at time T, will be

where k 2 is the number of contacts of the secondary case, t 3 is the infection time antine individuals who have recently been exposed to the confirmed case. Quar-149 antining these individuals prevents the onward infection of tertiary cases (Fig. 1B) . 150 We introduce three further parameters to quantify contact tracing: i) τ > 0, the 151 duration of lookback prior to symptom onset of the index case; ii) 0 ≤ g ≤ 1, the 152 probability to identify and quarantine a secondary contact that was infected within 153 the contact tracing window; and iii) ∆ 2 > 0, the delay between isolating the index 154 case and quarantining the identified secondary contacts.

There are many permutations of events that contribute to the number of tertiary 156 cases under TTIQ, as shown in Fig. S2 . The index case may not be detected (1 − f ), the non-traced contacts may themselves be tested and become index cases that are isolated at time t S 2 + ∆ 1 , where t S 2 is the symptom onset time of the secondary case.

By considering these different scenarios, we arrive at an expression for the number 165 of tertiary cases per index case under TTIQ,

We now have to average Eq. (5) over t S 2 , k 1 , and k 2 to obtain the expected number 167 of tertiary cases per index case under TTIQ. We first note that t S 2 = t 2 + γ for

where g(γ) is the incubation period distribution. Note that we have assumed the 170 independence between symptom onset and infectivity, which may lead to an over-171 estimation of the fraction of tertiary cases prevented. Keeping t S 1 fixed as the refer- 

where we have substituted t = t 2 − t S 1 such that

Eq. (7) can be further simplified to

7 . CC-BY-NC 4.0 International license It is made available under a perpetuity.

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The copyright holder for this this version posted December 7, 2020. ; https://doi.org/10.1101/2020.12.04.20244004 doi: medRxiv preprint Finally, in the absence of contact tracing (g = 0), the number of tertiary cases under testing & isolation only is given by

The primary sources of uncertainty in the outcomes of this model come from the 

(2020a), we use a likelihood ratio test to extract sample parameter sets for each 183 distribution that lie within the 95% confidence interval.

Concretely, we first identify the maximum likelihood parameter setsθ p andθ q 185 for the infectivity profile and generation time distribution, respectively. We then 186 randomly sample the parameter space of each distribution, and keep 1,000 param-187 eter sets whose likelihood satisfies ln L(θ) > ln L(θ) − λ n /2, where λ n is the 95% 188 quantile of a χ 2 distribution with n degrees of freedom. The infectivity profile is 189 described a shifted Student's t-distribution, which has n = 3 parameters, while the 190 generation time is described by a Weibull distribution with n = 2 parameters. 191 We then use these sampled parameter sets to generate the number of secondary is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted December 7, 2020. here focus on the number of tertiary cases, but results for the number of secondary 218 cases are qualitatively equivalent (Fig. S3) .

The region of ( f , ∆ 1 ) parameter space in which the number of tertiary cases 220 is less than one, i.e. the region in which the epidemic is controlled by testing & 221 isolating, is shrinking for higher R e epidemics (Fig. 3A) is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted December 7, 2020. The impact that contact tracing has on epidemic control can be seen by varying 249 the parameter g. For g = 0, no contacts are traced & quarantined, and hence we 250 return to the testing & isolation strategy (Fig. 3) . By increasing g, we expand the 251 parameter space in which n 3 < 1 (Fig. 4) , i.e. contract tracing allows an epidemic 252 to be controlled for lower fractions of index cases found ( f ) and/or longer delays 253 to isolating the index case after they develop symptoms (∆ 1 ). Fig. 4 The impact of tracing & quarantine on the number of tertiary cases per index case, n 3 , as a function of the testing coverage f (x-axis) and delay to isolation after symptom onset ∆ 1 (y-axis), for different contact tracing success probabilities g (colour) across different R e values (columns) [Eq. (7)]. We fix ∆ 2 = 2 days and τ = 2 days. The contours divide the regions where n 3 > 1 (the epidemic is growing) and n 3 < 1 (the epidemic is suppressed). The contours for g = 0 are equivalent to the contours in Fig. 3 . We do not show confidence intervals for clarity of presentation.

To visualise the impact of each parameter on the number of tertiary cases, we 255 consider focal parameter sets for the five TTIQ parameters, ( f , g, ∆ 1 , ∆ 2 , τ). We 256 then calculate the expected number of tertiary cases when we perturb each single 257 parameter, keeping the remaining four parameters fixed (Fig. 5) . is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted December 7, 2020. ; https://doi.org/10.1101/2020.12.04.20244004 doi: medRxiv preprint We set R e = 1.5 throughout, which is the intensity of the epidemic in the absence of TTIQ. We consider four focal TTIQ parameter combinations, with f ∈ {0.3, 0.7}, ∆ 1 ∈ {0, 2} days, g = 0.5, ∆ 2 = 1 day, and τ = 2 days. The number of tertiary cases for the focal parameter sets are shown as thin black lines. With f = 0 (no TTIQ) we expect R 2 e tertiary cases (upper grey line). We then vary each TTIQ parameter individually, keeping the remaining four parameters fixed at the focal values. The upper panel shows the probability parameters f and g, while the lower panel shows the parameters which carry units of time (days). The critical threshold for controlling an epidemic is one tertiary case per index case (lower grey line).

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Modifying the fraction of index cases that are identified and isolated ( f ) has the largest effect of all parameter changes. By identifying more index cases (increasing 260 f ), we not only prevent the onward transmission to new secondary cases through 261 isolation, but we also allow infected contacts to be traced and quarantined.

Increasing the fraction of secondary cases that are quarantined (g) has a smaller 263 return than increasing f . If only 30% of index cases are identified, then increasing 264 g results in a small reduction of the number of tertiary cases and for R e = 1.5 the 265 epidemic cannot be controlled even if all secondary cases (g = 1) of known index 266 cases are quarantined (Figs. 5A & B) . However, if a large fraction of index cases are 267 identified ( f = 0.7), then increasing g can control an epidemic that would be out of has on the number of tertiary cases (Fig. 6) . We find that f is the dominant param-282 eter to determine the number of tertiary cases, followed by ∆ 1 , g, ∆ 2 , and finally τ 283 has the smallest impact (Fig. 6B) . is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted December 7, 2020. Fig. 6 A) Linear discriminant analysis (LDA) of the impact of TTIQ strategies on the number of tertiary cases. We fix R e = 1.5 and then uniformly sample parameter combinations from f ∈ [0, 1], g ∈ [0, 1], ∆ 1 ∈ [0, 5] days, ∆ 2 ∈ [0, 5] days, and τ ∈ [0, 5] days. The number of tertiary cases is calculated [Eq. (7)] for each parameter combination, and the output (n 3 ) is categorised into bins of width 0.2 (colour). We then use LDA to construct a linear combination (LD1) of the five (normalised) TTIQ parameters which maximally separates the output categories. We then predict the LD1 values for each paramter combination, and construct a histogram of these values for each category. B) The components of the LD1 vector. By multiplying the (normalised) TTIQ parameters by the corresponding vector component, we arrive at the LD1 prediction which corresponds with the number of tertiary cases under that TTIQ strategy. Longer arrows (larger magnitude components) correspond to a parameter having a larger effect on the output.

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The copyright holder for this this version posted December 7, 2020. ; https://doi.org/10.1101/2020.12.04.20244004 doi: medRxiv preprint transmission prevented by quarantining these cases. We do account for uncertainty 326 in the infection time distributions, and this uncertainty is carried through into our 327 analysis and is captured by the confidence intervals shown in the figures and re-328 ported in the text. 329 In terms of modelling the TTIQ process, we have assumed that identified in-330 dex cases are isolated and have their contracts traced. If the index case fails to 331 adhere to the isolation protocol, then we will overestimate the amount of transmis-332 sion prevented by isolation. However, uncertainty in whether contacts adhere to 333 quarantine protocols, or whether contact tracers actually identify contacts, is con-334 tained in the parameter g. Lower adherence to quarantine or missed cases due to 335 overwhelmed contact tracers is captured by lowering g.

Here we have shown through systematic analysis that TTIQ processes can be is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

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A) The impact of testing & isolation on the number of secondary cases per index case, n 2 , as a function of the testing coverage f (x-axis) and delay to isolation after symptom onset ∆ 1 (y-axis) for different R e values (columns) [Eq. (3)]. The black line shows n 2 = 1. Above this line (red zone) we have n 2 > 1 and the epidemic is growing. Below this line we have n 2 < 1 and the epidemic is suppressed. Dashed lines are the 95% confidence interval for this threshold. B) Lines correspond to slices of panel A at fixed delay ∆ 1 = 0, 2, or 4 days (colour). Shaded regions are 95% confidence intervals for the number of secondary cases per index case. Horizontal grey line is the threshold for epidemic control (n 2 = 1). is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

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