1. Introduction

Rapid diagnosis and isolation of cases was critical in reducing further transmission of the SARS-CoV-2 virus throughout the pandemic. Since a large fraction of infections took place before infected individuals developed symptoms [1–3], isolation following the onset of symptoms is not a sufficient control mechanism to suppress the spread of infections in the pandemic [4,5]. To ensure rapid diagnosis and reduce transmission before symptom onset, the World Health Organization therefore recommended tracing, testing and isolating close contacts of COVID-19 positives [6].

The success of the test-trace-isolate strategy depends on many factors. In addition to social factors such as the population being informed and complying with guidelines, successful rapid diagnosis and isolation depend heavily on the following three factors. i. Test turnaround time (test TAT): How long it takes from when a person orders a test to the result of the test arrives and contact tracing begins. ii. Test sensitivity: The probability of a positive test result, conditioned on the individual truly being positive. iii. Tracing efficiency: The fraction of secondary cases that are successfully found through contact tracing of confirmed positives.

These three strategy parameters – test turnaround time, test sensitivity, and tracing efficiency – are not necessarily fixed. A surge in cases might cause an increase in test demands. Higher test demand could in turn cause longer test turnaround times as seen in some countries during the fall and winter of 2020 [7,8]. Longer turnaround times lead to slower tracing of secondary infections and thus, a vicious cycle begins. The 3 strategy parameters are also not independent. Faster test results often come at the cost of lower sensitivity. Polymerase Chain Reaction (PCR) tests are highly accurate, but take significant time to analyse. Faster results can be achieved by pooling tests [9–11], using saliva tests [12] or using rapid antigen tests [13].

One important direction of research is how details of the test-trace-isolate strategy influences the effectiveness of mitigation measures. In the beginning of the COVID-19 pandemic, several analyses concluded that test sensitivity is secondary to frequency and turnaround times in population-screening strategies [14,15]. Such a population screening strategy was implemented by Slovakia in November 2020 [16]. Another analysis by Wang et al. [17] concluded that contact tracing programs can effectively mitigate local COVID-19 risks if testing and contact follow-up are quick enough. Kretzschmar et al. [18] studied the interplay between tracing coverage and test turnaround time, focusing on the effect of test turnaround time delays on mitigation effectiveness of contact tracing strategies.

Here, we consider the interactions and tradeoffs between all three key parameters; test sensitivity, contact tracing efficiency, and the test turnaround time. Our work is motivated by the introduction of rapid antigen tests [13] during the COVID-19 pandemic. The rapid antigen test is a faster and more applicable test. However, the rapid antigen test has lower sensitivity compared to the slower PCR test. This raises the natural question: To what degree is it better to get a fast result than an accurate result? How does this change with the efficiency of contact tracing efforts, and with turnaround times of the slower, more accurate test? Here, we confront these questions with a modeling study.

2. Model description

To quantify the impact of test-trace-isolate strategies on growing epidemics, we simulate the branching structure of the chains of infections, also referred to as the “epidemic tree” [19,20]. Our model lets us keep track of who infected whom, which is essential when simulating contact tracing and quarantining infectious people. In the model, infected individuals cause new cases if they are not in isolation. An infected person isolates at symptom onset, or due to testing and forward contact tracing efforts. A visualisation of the branching structure can be found in Fig A in S1 Appendix. Throughout the paper, we will choose parameter settings that are based on and are qualitatively similar to the parameters estimated for SARS-CoV-2 during the pandemic. Our model only includes testing of infected people who either have symptoms or who have been traced as a potential infected through contact tracing. We do not consider the dynamics from screening or mass testing methods in our model, as we wish to focus on the direct interplay between testing and contact tracing.

We initiate a simulation with some number of newly infected people, . The simulation progresses in discrete timesteps, corresponding to days. A newly infected person becomes symptomatic after an assigned incubation time which is drawn from a Lognormal distribution similar to the one estimated by McAllon et al. [21] (cut off at 60 days for computational reasons).

Before symptom onset, the infected individual is presymptomatic and infectious [22]. Motivated by the nearly symmetrical infectiousness profile reported by He et al. and Ashcroft et al. [1,22], we make the simplifying assumption that each infected remains infectious and symptomatic for the same number of days as the length of the infected’s pre-symptomatic period. We denote the length of an infected person’s infectious period as t. t is a stochastic variable such that the incubation time follows a lognormal distribution [1,23]. We choose and as reported for SARS-CoV-2 by [1]. In Table A S1 Appendix, we give details on all distributions and parameter choices in this manuscript.

As for COVID-19 [24], some fraction of cases, , remain asymptomatic for the entire infectious period but can still be infectious; all other infected cases experience symptoms starting on the first day following the incubation period (as described above, the length of the incubation period is decided by a random number that we draw from the incubation period distribution). We note that with our model assumptions, the asymptotic infected can only be identified through contact tracing efforts.

In the absence of testing, tracing and isolation, an infectious person would give rise to k secondary cases in our model. We assume that k is proportional to the length of the infectious period of each infected, and therefore we draw k from a probability distribution p(k|t). For each of these k secondary cases, we determine the time of infection by drawing an integer from the probability distribution . represents the probability that an infectious person infects someone on day t of their own infection. takes positive values on the days where the infected is infectious and, mimicking SARS-CoV-2 [1,14], peaks around symptom onset. Concretely, for a person that is infectious for days, we take to be the most central values of the infectiousness profile reported by He et al. and Ashcroft et al. [1,22]. As an example, if the person is randomly assigned an infectious period , we take the person’s infectiousness profile to be described by He et al.’s and Ashcroft et al.’s estimate of infectiousness on the days [0,10]. In this example, day 6 marks the onset of symptoms. (Fig B in S1 Appendix plots the infectiousness as a function of time.) We normalize the infectiousness profile such that the infectiousness values (10 values in total in this example) sum to 1. The k secondary cases distributed throughout the infectious period will take place unless the infected is in isolation at the time the infection would occur.

In our model, infectious people isolate when they wait for a test to be taken, when they wait to receive a test result, and after a positive test result. An infectious person orders a test if either of two things happens: 1) The person is traced; 2) The person develops symptoms. In either case, the person orders a test immediately and then waits days for the result. In our work, we refer to as the test turnaround time (TAT). This TAT is divided into days waiting for the test to be taken followed by days to receive the result. The test correctly identifies the case with probability equal to its sensitivity, Se. This sensitivity can either be identical throughout the course of disease, or vary during the course of disease; in this work, we mainly focus on the assumption that the test sensitivity is a constant parameter, however, in Section 3.1, we assume that the test sensitivity can vary through the course of the infected’s disease period, by modelling the test sensitivity as a function of the infectiousness profile. If the test comes back positive, each of the person’s secondary cases is traced with independent probability . The results are indistinguishable when simulating the same with varying values of and . Thus, the key parameters of the model are the test turnaround time (TAT) , test sensitivity Se, and tracing efficiency . Finally, we note that due to our model construction, it is possible for an infected person to test twice. This can happen if the person gets traced, gets a false negative test result, and then develops symptoms, which makes the person test again. Furthermore, our model does not account for the depletion of susceptibles, and therefore does not incorporate herd immunity effects, as we wish to only study the effect of test and tracing efforts at a given instance of the pandemic.

After starting the simulation with infected people, the simulation can continue for several generations. When the first people to become infected have recovered, we end the simulation. We then count the number of people that they infected, . The output of the simulation is the effective reproduction number of the simulated disease: .

3. Results

We use the model to examine the trade-off between test turnaround time and test accuracy depending on the tracing efficiency. In our simulations, we therefore vary the parameters , Se, and and fix all other parameters.

As we wish to take the effect of superspreading individuals [26–28] into account, we choose P(k) as a negative binomial distribution with dispersion parameter r = 0.1 [28] and mean R₀ = 2, which is slightly lower than estimated in early stages of the pandemic [29,30]. In Figs E and F in S4 Appendix, we also run the simulation for R₀ = 3 and R₀ = 10 to demonstrate robustness to this assumption. Unless otherwise stated, we assume that the test sensitivity is identical on all days throughout the course of disease. Finally, we choose [24], and .

To develop some intuition, let us first introduce the results we obtain when fixing the tracing efficiency . This constraint leaves two free parameters: The test sensitivity, Se, and the test delay, . We now compare the effective reproduction number obtained by using a slow, but accurate test (parameters: and some ) to the reproduction number, , obtained with a less accurate, but rapid test (parameters: some and ).

To evaluate whether speed or accuracy is to be preferred, we compute the difference in obtained effective reproduction numbers of the virus under the different choices of tests, . Let us choose some test delay, e.g., (perhaps corresponding to a PCR test with a 1-day waiting time to get tested and a subsequent 1-day waiting time to get the result). In this case, how will depend on the risk of getting a false negative test result from the rapid test? For very high sensitivity (high ), this rapid test will be almost as accurate as the slower test it is being compared to. For this reason, the rapid test will be preferable to the slower one (Fig 1A top colourbar). If we now imagine slowly decreasing the test sensitivity, will gradually increase until it reaches a breaking point where the slow and rapid tests reduce the effective reproduction number equally well: . Decreasing the sensitivity even further makes positive, meaning that for these high probabilities of false negatives, the accurate test is to be preferred.

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Fig 1. A Comparison of effective reproduction number, , when using a slow but accurate test with sensitivity 1 (no false negatives), and a test TAT of days and using a rapid but less accurate test with a given sensitivity and test TAT of days.

In this inset, we assume that 80% of secondary cases are successfully traced following a positive test. The colourbars show results obtained for different choices of (top colourbar: days, middle: days, bottom: days), and the rapid-test sensitivity (horizontal axis in each colourbar). In each colourbar a breaking-point separates sensitivity values favouring the slower, accurate test and values favouring the rapid test. The values of the colour bar corresponds to those of the colourbar in panel C. B Comparison of effective reproduction numbers when the result of the slower test arrives after days, as a function of the sensitivity and tracing efficiency. Horizontal dashed line marks tracing efficiency 0.80, the value chosen for the colourbars in A. C Same as in B but with days. D Same as in B, C but with the slower result arriving after days. E Heatmap of the effective reproduction number obtained using the rapid, less accurate test, . The breaking-point lines of days are plotted in black, showing the exact points where . The blue dotted lines display the 95% CI for the test sensitivity for the COVID-19 rapid antigen test as reported by Brümmer et al. [25].

https://doi.org/10.1371/journal.pcsy.0000134.g001

If we now make the same plot but with a higher test turnaround time to receive the slow test, the breaking-point moves to a lower test sensitivity. The second colourbar in Fig 1A shows the result for . The breaking-point moved from sensitivity to . Increasing the waiting time once more, setting , moves the breaking-point further to the left. In this latter case, a sensitivity down to 0.76 is better than a 6 day wait for accuracy.

Having established some intuition for the simulations, we proceed to varying the third parameter: the tracing efficiency. By varying the tracing efficiency, for each choice of we get 2-dimensional heatmaps instead of the one-dimensional colourbars in Fig 1B–1D that we presented in the previous paragraphs. In these heatmaps, the breaking-points become white curves. Every point to the right of the breaking-point curve is a parameter combination where a faster test result is preferable. Every point to the left of the breaking-point curve is a parameter combination that favours an accurate test. Notice how all the breaking-point curves start in the lower-right corner (where tests are completely accurate but no contact tracing is done) and then slowly move to the left with increasing tracing efficiency. Fig 1E plots the obtained . For each simulated choice of parameters, each computed is averaged over 500 simulations (the standard deviation on is barely noticeable, as seen in Fig A in S5 Appendix). On top of the heatmap in Fig 1E, we plot the breaking-point curves that we obtain for different choices of .

We note that flipping the axes in Fig 1E can yield new insights from the simulations. In Fig B in S5 Appendix, for example, we visualize the breaking points with the test sensitivity on the horizontal axis, the test TAT on the vertical axis, and the contact tracing efficiency as the colour bar. This visualizes the minimum sensitivity a rapid test requires to outperform a slower, more precise alternative, under a given contact tracing efficiency.

The black breaking-point lines in 1E look similar when changing the shape of the offspring distribution p(k|t) and when changing the fraction of individuals who stay asymptomatic throughout their infectious period; the lines move slightly left – indicating that faster test results become more favourable – when increasing the mean R₀ (the basic reproduction number) of the offspring distribution p(k|t) (see Figs A- F in S4 Appendix).

When changing the infectiousness profile, , the breaking-point lines shift even more to the left. In Fig 2, we show an example of such a shift. Here, we simulate a disease that is twice as infectious during the pre-symptomatic part of the infectious period as it is during the last half. All breaking-point lines move left, indicating that fast turnaround times become more important when more infections happen early in the course of disease. This makes intuitive sense: When most infections happen early (and in asymptomatic patients), early identification of infected is crucial. Delays in test turnaround times are inconvenient in this case and a faster turnaround time can more easily compensate for lower sensitivity of rapid tests.

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Fig 2. Heatmap of the effective reproduction number obtained using a rapid, less accurate test, for a disease that is twice as infectious before infected develop symptoms.

The infectious profile is visualised in Fig B in S1 Appendix. Black lines show effective reproduction numbers where obtained when using tests with sensitivity 1.0 for which results arrive after 2, 4, 6, 8 and 10 days respectively. For simplicity we assume that the infectiousness profile is the same for asymptotic infected who never develop any symptoms. The blue dotted lines display the 95% CI for the test sensitivity for the COVID-19 rapid antigen test as reported by Brümmer et al. [25].

https://doi.org/10.1371/journal.pcsy.0000134.g002

We note that the breaking-point lines can take shapes that look qualitatively different from those shown in Figs 1E and 2. In the next section, we also see a qualitative change in the shape of the breaking-point lines when we introduce test sensitivity as a function of the viral load of the infected. We will examine this result in detail in Section 3.1.

For clarity, all heatmaps have been smoothed with a Gaussian filter with a kernel bandwidth of 0.75. This is done to make the visualisations more readable, and it does not change the contour positions from our simulations qualitatively. In Figs D and E in S5 Appendix we present Figs 1E and 3 without any smoothing.

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Fig 3. Heatmap of the effective reproduction number obtained using a rapid, less accurate test, when test sensitivity is modelled as function of the viral load of the infected throughout the infectious period.

Black lines show effective reproduction numbers obtained when using tests with sensitivity 1.0 and TATs of 2, 4, 6, 8 and 10 days, respectively. The x-axis indicates the initial given parameter, however the test sensitivity is modelled entirely different in the simulation as described in Section 3.1, and the black breaking-point lines are not directly comparable to the other figures. The blue dotted lines display the 95% CI for the test sensitivity for the COVID-19 rapid antigen test as reported by Brümmer et al. [25]. In this plot, negative binomial offspring distribution gives rise to significant noise on the breaking point lines. To smooth the curves, we simulated 4000 experiments for each parameter configuration.

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3.1 The effect of correlations between test sensitivity and viral load

So far, we have assumed that test sensitivity for the rapid test is unchanged throughout the course of disease. This has been a simplifying assumption that is not entirely accurate: It is well known that the viral load of an infected patient influences the test sensitivity [25,31–35]. Brümer et al. [25] reported test sensitivity across several studies, examining different subgroups of rapid antigen tests for SARS-CoV-2. When manufacturers’ instructions were followed and sensitivity was evaluated at any given cycle threshold (Ct) value among infected individuals, the average test sensitivity was 76.3% (95% confidence interval [CI]: 73.7–78.7). However, in cases with high viral loads (Ct < 20), test sensitivity increased to over 90%, indicating that the rapid antigen tests correctly identified nearly all infected individuals under these conditions.

We now relax our simplifying assumption of fixed sensitivity and study the effect of viral-load-dependent sensitivity on the tradeoff between rapid tests and accurate tests. We introduce a weighting of the initial test sensitivity parameter, , which depends on the infectiousness profile , leaving a higher sensitivity on days where is large, and a lower test sensitivity on days where is low. At the same time, we make sure that the test sensitivity is equal to when we compute its average over the course of disease. We also make sure that no test sensitivities are less than zero or higher than 1. This weighting algorithm is described in detail in S2 Appendix. We aim to match the qualitative trajectory of the test sensitivity such that when the initial is 0.76, then the average Se is above 0.9 when viral loads are high (i.e., when the infected is most infectious) for every individual in our simulation. This mimics the results from the survey study by Brümer et al.[25]. The trajectory for different lengths of infectious period and a weighted average is displayed in Fig 4.

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Fig 4. The theoretically sampled test sensitivity curve, averaged across all different lengths of infectious periods and normalized to the length of the infection period.

As for the plot above, we choose an average test sensitivity of as reported in [25]. We have normalized the length of the infectious periods and plot an average across the normalized x-axis using interpolation.

https://doi.org/10.1371/journal.pcsy.0000134.g004

Since the assumption Se = 1 causes the weighting scheme to leave the test sensitivity of the accurate test unchanged, the correlation between viral load and test sensitivity only impacts the rapid test. We still keep the conservative assumption that the slow test has sensitivity equal to one throughout the course of infection.

Fig 3 plots the results we obtain when we let the sensitivity of rapid tests depend on viral load. The vertical dashed lines indicate the 95% confidence interval for the empirical rapid test sensitivity as reported by Brümmer et al. [25]. For these realistic values of test sensitivity and almost all levels of contact tracing, the rapid test is preferable to the PCR test when TAT is more than 4 days.

In Fig 3, we observe two notable differences compared with Fig 1. First, the rapid test performs better overall when its sensitivity depends on viral load. Second, increasing contact tracing from 0% to 20% substantially improves the performance of the rapid test. For larger values of , however, both the rapid and accurate tests benefit similarly, resulting in almost vertical breaking-point lines. This suggests that for each there is a threshold value of Se and beyond which additional contact tracing results in diminishing marginal advantages for the rapid test.

This pattern can be explained by the time dependent profile of the test sensitivity of the rapid test. Early in the infectious period, sensitivity is low, so rapid identification provides limited benefit. Around peak viral load, however, sensitivity approaches 100%, making the rapid test nearly comparable to the accurate test while retaining a substantially shorter TAT.

If we zoom in on the middle part of the normalised infectious period in Fig 4 (where infectiousness is highest), we see that for people who have a rather short infectious period (2, 4 or 6 days), the Se profile will always be lower compared to people who have a rather long infectious period. This implies that those with the highest probability of testing positive at the peak of their viral load, are also the most infectious, due to our assumption that the offspring distribution depends directly on the length of the infectious period.

3.2 Analytical results

We now proceed to study our branching process model analytically. In this section we assume for simplicity that Se is fixed and not time-dependent.

As a start, we study the simple scenario where no contact tracing is being enforced: the limit . In this case, we can describe as

(1)

where p(k,t)=p(k|t)p(t), , and p(k|t) is the offspring distribution conditioned on the length of the infectious period, t, where and p(t) is the distribution of the length of infectious periods (measured in days, an even number (see Model Description)). The derivations and a more thorough explanation of the equation can be found in S2 Appendix in equation (B).

Equation (1) yields an intuitive interpretation of the model. We are given an initial R₀. Any enforced mitigation policy is now represented by the following subtraction term. By adjusting the model parameters , Se and , we can directly influence the resulting effective reproduction number and quantify how much the initial R₀ is reduced.

Say, if all infected are asymptotic (), no individuals are getting tested. Consequently, the mitigation efforts will yield no reduction in the effective reproduction number: .

If , all symptomatic infected are guaranteed to go into isolation when the symptoms start at (first integral). However, the symptomatic infected only stay in isolation if they obtain a positive test result, which depends on the test sensitivity, Se (second integral). In Fig 5, we plot given different test sensitivities and test turnaround times. We also plot our analytical results from Equation (1) and our simulated results next to each other and see that the curves are indistinguishable.

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Fig 5. The empirical (full lines) plotted against the theoretical (dashed lines) for different values of and zero contract tracing.

We observe an excellent agreement between the empirical estimates and the theoretical predictions across the different TAT regimes. This plot underlines that the contribution coming from changing the Se parameter is indeed linear as we have derived in equation (1).

https://doi.org/10.1371/journal.pcsy.0000134.g005

In Fig 5 we can see that all the lines are either monotonically decreasing when test sensitivity is growing or constant due to a sufficiently high TAT. This gives rise to an interesting result, which we state through the following theorem.

Theorem 1. In the absence of contact tracing, a slow test with perfect accuracy always reduces infections better than rapid tests with lower accuracy. That is, for all values of Se, , and .

Proof: By inserting the same parameter values used in the simulations in Fig 1, we can use Equation (1) to compare the slow and the rapid tests’ ability to mitigate infections in the absence of contact tracing.

For the rapid test, we have , and the equation becomes

(2)

For the accurate test with higher TAT we get

(3)

Deriving this latter equation from Equation (1), the disappears from the integral as we can collapse the two integrals in Equation 1 since Se = 1. In the case where no contact tracing is enforced, we subtract these from each other to get the following formal expression of :

(4)

Equation (4) is a product of only non-negative terms, and therefore we conclude that when no tracing efforts are present, the rapid test will always perform worse in reducing infections than the slow and precise test as for any value of Se, , and .

This analytical result is analogous to our observation in the simulations that all the breaking-point lines are always intersecting in the bottom right corner in Fig 5: the point and Se = 1. So, in order for rapid tests to be preferred, we need to support the mitigation effort with contact tracing.

Introducing contact tracing to the analytical formulation introduces some complexity. This is due to the fact that the probability that a node gets traced is not simply the probability that the node’s parent traces you. The probability of getting traced through a parent depends on whether the parent tests positive; the parent only receives a positive test if i) its own parent gets tested after either being traced themselves or if symptoms appear, or ii) if the parent then receives a positive test. The dependence of parents being tested positive then introduces a recursiveness in the problem: For the parents to get traced, the parents of the parents must test positive, etc. Finally, tracing is only effective if the traced individual has not already entered isolation and is still actively infectious at the time of tracing. We denote the probability that a node is both still infectious and not already isolated at the time it is traced by . We refer to this as a valid tracing event. See equation (I) in the Supplementary Information for an analytical formulation of .

From the recursive dependence described above, it is clear that the probability that an infected node happens to get traced through a parent is non-trivial to estimate analytically. We call the probability that an infected individual eventually takes a test due to being traced for the effective probability of being traced through a parent, . is dependent on all the parameters – Se, , , – and can be expressed as a recursive formula and reduced to the following expression,

(5)

We provide the detailed derivation of this expression in S3 Appendix.

Now that we have a symbolic representation of , we can express the effective reproduction number under our mitigation efforts, , when contact tracing efforts are present,

(6)

where is used to ease the notation and denotes the time at which a traced node was traced. See S3 Appendix for more details.

To evaluate the theoretical expressions, we need to know the distributions of the variables , , k, and t. Although we started out by defining distributions for the latter two variables, we defined the distributions for the population of infected population as a whole. When conditioning on whether or not a node gets traced (as we do in Equation (6)) the distributions of k and t change. This is due to variable interdependencies: t affects how many people an infectious person infects, and and t together influence how likely it is that an infected node gets traced. Therefore, in order to evaluate our theoretical expressions, we infer the conditioned distributions of the variables t, and k from the simulations: We simply run the simulation and store sample triplets of t, , and k for traced nodes and pairs of t and k for untraced nodes, in order to calculate the Monte Carlo estimates of the integrals. Due to the intractability of , we also estimate directly from the simulations. We do this by storing the number of traced individuals from each simulation such that we can estimate by . In Fig J in S5 Appendix, we report the standard deviation of the estimated for each parameter configuration, and we observe that the standard deviation is small and increases slightly when the simulations are near or below the epidemic threshold . For robustness, we also calculate the difference between the we get when we insert the estimated into Equation (6) and the we get when we instead insert into Equation (6). The resulting differences are shown in Fig K in S5 Appendix, and we observe that both differences are negligibly small.

With these approximations, we can evaluate the theoretical expression in Equation (6) numerically as,

(7)

In this equation, denotes the time at which a given node i gets traced, M denotes the number of sample triplets (, , ) from the distribution of infected who get traced and N denotes the number of pairs that are tested but not traced for the Monte Carlo estimate.

With the distributions properly sampled, we evaluate our analytical expression for for the full range of Se and values. In Fig 6, we compare the analytical and simulated results and see very good agreement: In most of the parameter space, we have that . However, the analytical expression and the simulated results start to diverge when we are nearing the epidemic threshold where . In the subcritical regime, the agreement between simulations and theory is even worse. This is because of the uncertainty that comes from the simulations dying out very quickly. In the expanded Figs A and B in S3 Appendix we see that the simulations end very early in this part of parameter space, often at less than 1000 infected. This means that the distributions we sample of the parameters , , , and are not accurate, leading to poor analytical approximations of the dynamics.

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Fig 6. The difference between obtained through simulations and the analytical expression in Equation (6).

In most of the parameter space, we have that . When the values for tracing efficiency and test sensitivity are both high, we see some disagreement between the simulation and the analytical expression. We see that the instabilities only appears when is close to the epidemic threshold 1, and when . The reason for this is that the epidemic being simulated declines and dies out completely. In fact, in the expanded Figs A and B from S3 Appendix, we see that the simulations ends at less than 1000 infected. This means that the distributions we sample of the parameters , , and are not accurate, leading to poor analytical approximations of the dynamics.

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We have used simulation outputs in both the evaluation and the validation of the theoretical results. Although such a procedure could cause circularity, the simulation outputs here serve two fundamentally different purposes when estimating and , respectively. When we estimate , we simulate the discrete-time branching process and compute the mean from the generated realizations. For , we use the triplet samples of , , for traced and untraced infected individuals in our simulations, to estimate the three terms in the theoretical expression , , and . We then input these estimates into Equation (7) to estimate . Although the parameter estimates are derived from the simulations, the accuracy of the closed-form solution is not guaranteed, which justifies a meaningful comparison of the two results even though they rely on the same simulation outputs.

Our derived analytical expression for the effective reproduction number can be used to study the separate contributions of mitigation effort to lowering infections. For example, Fig C in S5 Appendix examines the isolated impact of perfect contact tracing conditional on different rapid test sensitivities. The figure displays the difference between a 0% and a 100% contact tracing scenario, and demonstrates that the isolated impact of contact tracing increases super-linearly with increasing rapid test sensitivity.

In the analytical expressions described in this section we have considered the test sensitivity as a constant parameter throughout the length of the infected individuals. However, if we were to allow for the test sensitivity to depend on viral load in the analytical expression, we would replace Se by the weights, w, described in S2 Appendix. Depending on the equation, we would evaluate this new function at different times: We would replace Se with for the untraced term and replace Se with in the traced term of Equation 6.

Importantly, the analytical expression is agnostic to the specific epidemiological kernels. Any infectiousness profile , offspring distribution for secondary cases, or generation-time distribution can be substituted directly into the integrals.

4. Discussion

Testing, tracing and isolating positive cases often constitute the backbone of disease mitigation measures, and these efforts were central in many countries’ strategy to fight the COVID-19 pandemic [36–38]. We have demonstrated that there is a trade-off between test sensitivity and test turnaround times in such strategies, and that it can be beneficial to prioritize test speed over test sensitivity if waiting time is long for accurate tests. This indicates that it can make sense to use lower-sensitivity tests in order to avoid such additional waiting time. We show how the efficiency of the contact tracing combines with test sensitivity and the test turn around time to yield effective mitigation measures. Moreover, we find that the infectiousness profile greatly influences whether rapid or accurate tests are preferable; if many infections happen early in the course of disease, fast turnaround is more important.

Depending on the sensitivity of the rapid tests available, our results indicate when rapid tests are preferable to long-wait accurate tests. It is to be expected that testing systems will occasionally get under stress during a pandemic, and having a way to avoid build-up of waiting times in this scenario is crucial. Designing such stress-relieve strategies presents an interesting direction for future research.

An interesting observation can be made from Theorem 1. Recall that the theorem states that, in the absence of contact tracing, a slow, perfectly accurate test will always outperform a faster, inaccurate test. This interesting result is visible in Fig 5, our plot of mitigation effectiveness in strategies without contact tracing. For all TAT values, a test sensitivity of 1.0 yields the same reduction in reproduction number. For lower test sensitivity, however, strategies with longer TAT yield better mitigation. This is due to our assumption that people isolate until they receive a test answer. Because of this perfect self-isolation, false negative results release infected people early and result in more infections. Curiously, this means that if no contact tracing is enforced, a mitigation strategy that relies only on self-isolation upon symptoms for an appropriate number of days (4–5 days in this case) is as effective as a mitigation strategy that relies on tests with sensitivity 1.0 and TAT 0. In practice, one downside to relying on such self-isolation could be a large economic cost associated with keeping people who were not infected by SARS-CoV-2 virus, but displayed similar symptoms for other reasons in isolation. However, during the Covid-19 pandemic a massive reduction in other respiratory diseases was observed [39–41], meaning that people who experienced symptoms would be likely to be infected with Covid-19. So, given a situation where i) people adhere strictly to self-isolation and where ii) other common diseases with similar symptoms are being suppressed due to mitigation efforts, our model predicts that self-isolation is at least as effective as a test-isolate effort with no contact tracing. This is true even for a test with perfect test sensitivity and zero days test TAT.

Some of the model assumptions we have made can be questioned. Four such assumptions are: That isolation hinders any transmission; that symptomatic individuals do not isolate after testing (false) negative; that we have R₀ = 2 being slightly less than estimated in the early stages of the pandemic; and that the slow test has a perfect test sensitivity. We note that these assumptions will favour reducing transmission by slower, perfectly accurate tests: That people do not break isolation benefits the tests with long waiting times; and false negative results leading to completely normal behaviour damages only the tests that allow for false negative results. In Figs F - I in S5 Appendix, we run the simulations from Figs 1E and 3, with a lower test sensitivity for the tests with long TAT. We see that lowering the test sensitivity for a slow test indeed favours the rapid test; especially when the TAT is sufficiently low. When the TAT is high (6–10 days), the efficiency of a slow test where Se < 1 is comparable to the efficiency that we obtain for a slow test with perfect sensitivity. This is because the long delay makes testing and tracing so ineffective, that the main reduction in comes from self-isolation before testing and while waiting for the test result. That choosing a slightly low R₀ disfavours the rapid test, can be seen in Figs E and F in S4 Appendix, where we run the simulation with R₀ = 3 and R₀ = 10. Here we see that the breaking-point lines have moved slightly to the left as compared to the lines plotted in Fig 1E. Hence, a higher basic reproduction number makes faster test results even more important.

That we have chosen our assumptions as to disfavour rapid low-sensitivity tests means that our results can be interpreted as conservative estimates of the benefits of reducing test waiting time with less sensitive tests.

Our choices of the probability distributions p(k) and can also be questioned. As it is difficult to empirically identify the chains of infection accurately [42], we do not know what the actual p(k) would be for COVID-19. The distribution would presumably also change with interventions such as banning of large social gatherings. That our results are robust when changing p(k) from a negative binomial to a geometric or a Poisson distribution indicates that the uncertainty on p(k), including its overdispersion, is unimportant for the obtained results. Further work in this direction, including rigorous theoretical explorations of different choices of p(k) would be interesting going forward.

For , we have chosen an empirical estimate from He et al. and Ashcroft et al. [1,22]. Other empirical estimates of the infectiousness profile have been reported [43–46] and in future work, it would be interesting to see whether results change qualitatively with some of these other infectiousness profiles.

Our implementation of the effect of viral load on test sensitivity is designed such that the curve of the test sensitivity directly reflects the infectious profile given by He et al. and Ashcroft et al. [1,22]. This implies an underlying assumption that the viral load and the infectious profile for any infected individual are directly correlated with each other. We believe that the implementation of the viral load-dependent test sensitivity is a step closer to a realistic depiction of modelling the dynamics of contact tracing, test sensitivity, and test turnaround time rather than assuming the test sensitivity to be constant throughout the infectious period. For pathogens such as COVID-19, which is characterized by a substantial proportion of asymptomatic and presymptomatic infections, our model indicates that it has an influence on the outcome of different test-trace-isolate strategies whether you consider viral load-dependent or constant test sensitivity. Our assumption that the test sensitivity curve (or, equivalently, the viral load curve) is the same for any infected person could be too clean compared to reality. In future work, it would be interesting to introduce heterogeneity as described by Jones [47] et al. and Smith-Jeffcoat et al. [34], rather than assuming one viral load curve for all infected. We leave the discussion on heterogeneity on test sensitivity across the population for future work.

Three other limitations arise not from parameter choices, but the model itself. First, our model does not take depletion of susceptibles into account. Such effects are unimportant to what we have been interested in: the effect of testing and tracing on the effective reproduction number at a given instance of the pandemic. Secondly, our implementation of contact tracing allows only for descendants of the confirmed positive to be traced, meaning that we only consider forward contact testing as a strategy. Not including two-way contact tracing or the possibility of identifying individuals through non-parents could cause us to underestimate the effect of contact tracing [48–56]. This again biases our results against the rapid tests that so rely on tracing efforts. We also do not consider higher-frequency testing as a screening-based mitigation strategy. We acknowledge that there may be important interdependencies between high-frequency testing and contact tracing dynamics. For example, superspreading individuals may be more likely to be identified through backward contact tracing under mass-screening programs, as individuals will undergo testing independently of whether their infector has been detected. The interplay between mass testing, backward and forward contact tracing, test sensitivity, and turnaround time (TAT) remains to be investigated. Thirdly, we do not model explicit contact-tracing delays, even though such delays can become substantial when the tracing system is under stress. If contact tracing delays were applied equally across both test types in our study, this would, in our model, favour the slow, precise test. The reduction in achieved by the fast test relies precisely on the enhanced mitigation effect from contact tracing, meaning that any delay in contact tracing would undermine it more substantially. In contrast, the slow, precise test would be less affected since the impact from contact tracing is already diminished due to test delays. If we consider a real-world scenario with heavy stress on the test capacity, an additional line of investigation could be to consider the correlation between contact tracing delay and test TAT, as these might rise simultaneously and possibly non-linear.

Lastly, in this work, we describe the recursive nature of the contact tracing and how it is difficult to measure the distinct impact on the reduction of cases from the tracing compared to the other key parameters Se and . In future investigations, it would be valuable to examine the selections of and for respectively traced and untraced individuals. A first question would be whether certain infected individuals are more likely to get traced than others. Does the chance of getting traced depend on the length of the infectious period or on the number of secondary infections the infected has caused? Knowing such conditional distributions would be informative for future mitigation efforts.

Supporting information

Acknowledgments

We thank SSI for contributing to funding this research. JLJ thanks Steven H. Strogatz for helpful discussions and Gorm G. Jensen for comments on an early version of the manuscript. MRB and JLJ thank Jacob L. Curran-Sebastian and Lasse E. Christiansen for helpful discussions on the modelling components of this research.