The impact of digital contact tracing on the SARS-CoV-2 pandemic—a comprehensive modelling study

Our deterministic models describe contacts between individuals in a population by the usual mass action law. The stochastic models allow for a more detailed investigation. Particularly, two different strategies are considered: For the first, in the following referred to as homogeneous population, we choose the contacts for each person randomly out of the entire population. The probability for a person to have n unique contacts close enough to transmit a respiratory virus on a given day is taken from the empirical distributions reported in [21]. These are well described by where \({n_{c}}\) is the mean number of contacts per day. We also consider a more realistic contact pattern, in the following referred to as social graph population. The population is described by a social graph, were each individual is represented by a node and contacts are represented by edges. Each individual is given a fixed set of contact persons for the entire simulation. We employ a modified version of the Lancichinetti–Fortunato–Radicchi benchmark graph (LFR) [22] as shown in [23] (therein referred to as LFR-BA). In this model, the population is divided into communities with sizes distributed according to a power-law distribution. Node edges are constructed according to the linear preferential attachment model [24] under the constraint that an average fraction of \((1-\mu )\) of the edges of each node connect nodes within the same community. A graph constructed in this manner results in a power-law probability distribution with index \(a=2\) for the node degree n [24]: where \(n_{\mathrm{min}}\) is the minimum node degree. Here, the mean number of contacts per day is given by \({n_{c}}= \frac{a\cdot n_{\mathrm{min}}}{a-1}\). We assume that each edge is active once per day, so that the node degree corresponds to the number of contacts per day.

Figure 1 schematically shows the probability distributions for symptom onset, transmission, and true positive test results. The parameters are explained in the following paragraphs.

Incubation period P(\({T_{\mathrm {Inc}}}\) ). The distribution of incubation periods is taken as (Fig. 1 upper curve) with shape parameters chosen to match the curve reported in [25], which has median and mean incubation periods of 7 days and 7.44 days, respectively, with a range from 0–23 days. The authors included a relatively large group of patients (\(n= 587\)) with a wide age (0–90 years) and symptom (asymptomatic to severe) range. Other studies, albeit with smaller number of patients and with a bias towards more severe symptoms, have reported lower medians [27, 28]. Therefore, in addition to the parameters matching [25] (printed in black in Table 1) we also model a curve with shorter median and mean incubation periods of 3.6 days and 4 days (printed in gray in Table 1).

Latent period \({T_{\mathrm {lat}}}\). The latent period for SARS-CoV-2 is shorter than the incubation period, meaning pre-symptomatic transmission can occur [1, 29]. The latent period is difficult to determine empirically, as it requires exact information about the time of exposure and contagiousness. As, to our knowledge, no reliable, large-scale studies have been published on the latent period of SARS-CoV-2 so far, we use the measured contagiousness relative to the incubation time as an auxiliary means to infer latency, where the value of 2.5 days comes from the transmission probability curve discussed in the next paragraph.

The transmission probability curve \(P_{\mathrm{trans}}({\tau })\). The transmission probability is the probability that a contagious person infects someone they have contact with. This probability is often given as an average “infectivity per day”, \({\beta _{i}}\), even though it changes significantly as a function of the time since infection τ. The infectivity was measured as a function of the time since onset of symptoms by He et al. [26]. They find that carriers become contagious approximately 2.5 days before the onset of symptoms, and that approximately 44% of transmissions occur during this pre-symptomatic phase.

We take as the contagious period \({T_{\mathrm {con}}}\) the time from the end of the latent period until 99% of the cumulative transmission probability is reached; a person is considered to be recovered afterwards. Therewith, the transmission probability as a function of time since infection is given by a scaled and truncated Gamma distribution (Fig. 1 lower curve). After infection, an individual has a (random) latent period \({T_{\mathrm {lat}}}\), during which the transmission probability is zero. Afterward, for \({\tau }={T_{\mathrm {lat}}}+t\) (and \(t\geq 0\)), we have where χ is 1 if the condition in the argument is met, and 0 otherwise.

The shape parameters shown in black in Table 1 are our defaults, taken to match the curve in [26]. Pre-symptomatic infectivity is strongly debated. Therefore, we model a second shape where only 18% of the transmission occurs during the pre-symptomatic phase (shape parameter values printed in gray).

The course of the disease for asymptomatic carriers is the same as that for symptomatic carriers as shown in Fig. 1, and the incubation time is the time when symptoms would have started.

Fraction of asymptomatic cases \(1-\alpha \). The proportion of asymptomatic carriers (\(1-{\alpha }\)) described in the literature ranges from ∼4% [30] to ∼40%[31] of all cases. Initial reports for (\(1-{\alpha }\)) derived from testing of specific cohorts (cruise ship, returning travellers) ranged from 17–31% [29, 32, 33], however larger studies suggest even higher numbers. Analysis of the mass screening of the full population of the municipality of Vo’, Italy [31] report that 41.1% of the confirmed SARS-CoV-2 infections were asymptomatic (as defined by the absence of fever and/or cough). Ferretti et al. [4] analyzed 40 selected transmission pairs and also derived a value of 40% for the proportion of asymptomatic infected individuals. We model several different values for α to cover the reported ranges.

Reduced asymptomatic transmission probability \({\eta _{as}}\). While initial studies assumed that asymptotic cases were less contagious [4, 34], newer reports indicate that the viral load of asymptomatic cases is similar to symptomatic cases, which suggests similar contagiousness [35‐40]. We nevertheless introduce the parameter \({\eta _{as}}\in [0,1]\), which scales the transmission probability for pre- and asymptomatic cases, and vary that parameter to explore its effects. We note that when we model with \({\eta _{as}}<1\), we apply the scaling to both pre- and asymptomatic phases; since there most likely is no difference in viral load, if asymptomatic transmission is suppressed, this is likely due to circumstantial factors, like lack of coughing, which apply to the pre-symptomatic phase as well.

The basic reproductive number \({R_{0}}\). If the contagiousness is independent of the symptom status, i.e. \({\eta _{as}}= 1\), the reproductive number is given by the number of contacts during the contagious period, \({n_{c}}{T_{\mathrm {con}}}\), times the average probability to transmit the infection in one contact, \({\beta _{i}}\). However, we need to distinguish between symptomatic, pre-symptomatic, and asymptomatic cases in order to allow for \({\eta _{as}}<1\). We find Note that—though \(P_{\mathrm{trans}}\) is a random function—the random part of the function is a pure translational offset (the latency period), s.t. the integral is deterministic.

\({R_{0}}\) is expected to be different in different populations because \({n_{c}}\) is different by factors of up to 4 just within the populations of different European countries. Values of \({R_{0}}\) ranging from 1.4 to 6.5 have been reported [45], though it is not always clear whether or not the reported \({R_{0}}\) is for the case where symptomatic individuals are quarantined. Furthermore, \({R_{0}}\) is usually not corrected for the contact rate in the population where it is studied. We consider \({R_{0}}\) the reproductive number without any form of interventions. We take \({R_{0}}\) of 3 (approximately the median reported in [45]) as our default, but also run the models for \({R_{0}}\) of 2 and 4.

Following [46], we introduce the random reproduction number for an individual \(R_{0,i}\). \({R_{0}}\) is the expectation value over the \(R_{0,i}\). For simplicity (and since two of our models are not based on continuous time but on discrete time/days), we state the time-discrete formula for \(P(R_{0,i}=n)\). Assume that an individual did infect n persons. These n persons can be arbitrarily distributed over the contagious period. In a slight abuse of notation, let \(T_{c}\) denote the number of days that a person is contagious for. Furthermore, let \(C_{n} = \{\vec{n}\in \mathbb {N} _{0}^{T_{c}}: \sum_{i=1}^{T_{c}} \vec{n}_{i} = n\}\) be all possible ways to distribute the n infectees to \(T_{c}\) days. Then, for a homogeneous population, where the convolution ⊛ is over the parameter m. This distribution is shown for some parameter combinations in Fig. 2.

The interventions considered here are (1) DCT, (2) quarantining, (3) testing, and (4) social distancing. Reported symptomatic cases are quarantined starting right at the beginning of the epidemic. The remaining interventions are turned on once a fraction \({f_{i}}\) of the population has become exposed.

(1) DCT. We assume that a fraction \({ p_{\mathrm {app}}}\) of the population uses the DCT system and that we can trace all contacts between users of this system with time delay \({T_{\mathrm {delay}}}\). In case that both infector and infectee have a DCT device, the probability for successful tracing is \({\eta _{\mathrm {DCT}}}\), while tracing always fails if either infector or infectee do not use the system. \({\eta _{\mathrm {DCT}}}\) accounts for situations where cell phones run out of battery, are not with the owner at all times, Bluetooth is turned off, or where an alert is ignored. The DCT system (DCTS) identifies contacts within the past \({\Delta T _{\mathrm {trace}}}\) days.

In the literature, the overall tracing probability across the population is often taken as \(p_{\mathrm{trace}} = p_{\mathrm{app}}^{2} {\eta _{\mathrm {DCT}}}\) (e.g. in [4]). We note that this formula is not correct but becomes approximately right if \({ p_{\mathrm {app}}}^{2} {\eta _{\mathrm {DCT}}}\) is small (see the Appendix).

To become an index case for tracing, a person must be reported. We assume that from the group of symptomatic carriers, a fraction \({f_{m}}\) sees a doctor to get tested with a reliable laboratory test and is then reported. A fraction (\(1-{\alpha }\)) of cases will go unreported because they do not exhibit symptoms, unless they get tested due to being traced. A fraction \({\alpha }(1-{f_{m}})\) of symptomatic cases will go unreported due to lack of access to medical tests. In the case where \({\eta _{as}}= 1\), α and \({f_{m}}\) are degenerate.

First order tracing refers to a protocol where contacts of an index case are traced. DCT also allows immediate tracing of contacts-of-contacts. We refer to this as second order tracing. If the \({\Delta T _{\mathrm {trace}}}\) is big enough, DCT will identify the infector. Second order tracing then can trace not just the people infected by an index case, but also the people who were infected by the same infector as the index case.

A DCT system will identify all contacts, regardless of their infection status. Since many models assume perfect accuracy in identifying only contacts that became infected (e.g. [4, 5, 47]), we run all parameters both with (closer to reality) and without (to be comparable to other models) tracing of the uninfected contacts.

All traced people immediately go into quarantine. This is necessary to suppress the pre- and asymptomatic transmission rates.

(2) Quarantining. Quarantining refers to any intervention that reduces the transmission probability significantly; this includes self-quarantine at home as well as being hospitalized. We assume that all reported symptomatic patients are immediately quarantined, regardless of any other interventions. This already reduces the reproductive number to

Figure 3 shows \(R_{e,Q}\) for combinations of α, \({f_{m}}\), and \({\eta _{as}}\). In Fig. 3 (top), \({\eta _{as}}= 1\), so only the product of α and \({f_{m}}\) is relevant, and results are shown for \({R_{0}}= 3\) and for \({R_{0}}= 2\). In Fig. 3 (bottom), \({\eta _{as}}<1\), so both \({R_{0}}\) and \(R_{e,Q}\) depend on α and on \({f_{m}}\). For each combination of α and \({f_{m}}\), the transmission probability \({\beta _{i}}\) was adjusted to obtain \({R_{0}}= 3\).

In response to being reported or being traced, people are quarantined by default for 14 days. Symptomatic cases may leave quarantine 8 days after the symptoms start. Uninfected contacts can leave quarantine early following a testing protocol.

(3) Testing. We consider two types of tests. A reliable laboratory test for symptomatic carriers seeing a medical professional, and a fast point of care (POC) test that can be performed at home or at mobile testing stations. In either case, carriers only test positive while they have a high enough viral load. We assume the tests have \(p^{\mathrm{true\ positive}} = 0.0\) while the carrier is in the latent period, that is up to approximately 2.5 days before symptom onset. The viral load rises quickly after the end of the latent period. We further assume that the laboratory test then has a true positive rate of 100% until the carrier has recovered. The POC test on the other hand has \(p^{\mathrm{max}}_{\mathrm{true\ positive}} = 0.9\) until approximately 5 days after symptom onset.² After this time, the true positive rate falls at the same rate as the transmission-probability curve; the true positive rate as a function of days since symptom onset is shown in Fig. 1 (bottom gray curve).

All people who are traced must be tested for two reasons: (a) A positive test result is the only way for asymptomatic individuals to become index cases for tracing, and index cases are needed for tracing to be effective, and (b) so uninfected traced people can be released from quarantine. Keeping all traced people in quarantine for the full quarantining duration means that a large fraction of the uninfected population may end up quarantined on any given day of the outbreak. We use the following release protocol: All traced people go into quarantine and get tested with a POC test. Regardless of the test result, everyone stays in quarantine, because the person may still be in the latent period. Those who tested negative on the first day are re-tested \({\delta T_{\mathrm {re\text{-}test}}}\) days later. If both tests were negative, the person may leave quarantine, but is tested again after another \({\delta T_{\mathrm {re\text{-}test}}}\) days in case they were still in the latent period when the second test was done.

In addition to testing in response to being traced, we simulate the option of randomly testing a fraction \({f_{\mathrm {RT}}}\) of the population each day. This is done with a testing protocol assumed to have a negligible number of false positives.

(4) Social Distancing. Social distancing includes both a reduction of the total number of contacts per day \({n_{c}}\) to \({n_{c}}\cdot {f_{\mathrm {SD}}}\), and limiting the maximum number of contacts per day. In the absence of second-order effects, and if the upper limit does not change the distribution mean significantly, this reduces the reproductive number to where R is the reproductive number without social distancing.

The early phase of an outbreak can be quantitatively described with compartmental models based on ordinary differential equations (ODE) [56] or with age-since-infection models [57].

The deterministic models used here bridge the different scales utilizing the mathematical analysis of the underlying, microscopic stochastic branching process with contact tracing. The effect of contact tracing on the removal rates is determined. These effective removal rates are then used in the deterministic models. Our first deterministic, compartmental model explicitly predicts the status (exposed/infectious) for a newly infected person, when he/she will eventually be traced. Eventually traced and never-traced individuals go to different compartments. In that, the (exponentially distributed) waiting times can be readily adapted. Particularly, the model is close to standard SEIR-models (see Fig. 4), and is feasible to analytical analysis (Appendix A.1). In contrast, the second model, based on age since infection, does not explicitly formulate an exposed and an infectious period. The basic assumption is that the state of an individual is a function of his/her age of infection, that is the time that has passed since he or she became infected. The structure is less pronounced, but it is possible to use transition rates that are more realistic (supplemental materials A.2). At the present time, the analytical treatment of the interdependence of contact tracing and correlations between infected individuals at the plateau phase of an epidemic is not well understood. Therefore, both models focus on the onset of the epidemics, where the reduction of the number of susceptible people by quarantine or recovery does not play a central role. The main outcomes of these two models are the doubling time \(T_{2}\) and effective reproductive number \(R_{\mathrm{eff}}\) for interventions starting on the first day of the epidemic, though both models are able to predict in a heuristic way the total course of the epidemic.

We developed two independent individual-based models (IBMs), which use the Monte Carlo (MC) technique to simulate social interactions, the progression of the viral disease, and interventions, at the level of individual people. The code for the models is available from [59] and [60]. The MC simulations proceed through the outbreak in steps of one day. Each day of the outbreak, every infected person not in quarantine has contact with a number of other people randomly drawn either from \(P_{\mathrm{social}}(n)\) (for a homogeneous population structure) or from the person’s social graph. The probability to infect each contact is given by \(P_{\mathrm{trans}}({\tau })\). When a contact becomes infected, the incubation time is drawn from \(P({T_{\mathrm {Inc}}})\). The intervention protocols are implemented as described in Sect. 2.

Figure 5 shows a chain of infections from one of the simulation runs. Each box represents a person, and arrows between boxes represent infections and tracing.

In this example, P936 is exposed to the virus on day 147 of the simulated epidemic and has a latent period of 5 days, but never develops symptoms (light blue background) or tests positive and is therefore never reported (R–). He or she infects three others—P576 on day 152, P747 on day 154, and P277 on day 155. All three infectees develop symptoms (purple background). P576 sees a doctor on day 155, tests positive, and is reported. This triggers tracing of his infector, P936, and of the person he or she infected, P392. Tracing to P392 fails because this person does not use the app. Since P392 also does not develop symptoms, he or she is never reported or quarantined and infects three others. The backward trace from P576 to P936 puts P936 in quarantine on day 155 and thus prevents him or her from infecting more people after this time. P936 does not test positive (dashed outline of the box indicates the person was traced but never reported), so never becomes an index case him- or herself. However, since second order tracing is active in this simulation, the ‘siblings’ of P576 are identified. P747 is put in quarantine before he or she can infect anyone else, and tests positive the same day. This makes him or her an index case, so that the common infector, P936, is traced again. P277 does not use the app, therefore the trace fails. However, P277 happens to not meet many people on the first two days of infectiousness, then develops symptoms, sees a doctor, and is quarantined.

In this example, the chain of infections was stopped at P747 through second order tracing, the chain was stopped at P277 due to luck, but the chain could not be interrupted at P576 because the person he or she infected did not use the tracing app.

For a given set of input parameters, that is for a specific scenario, each run of the MC simulation represents one possible course of the epidemic. To find the most likely outcome for a scenario, the simulation is run 50 to 1000 times and the outcomes are averaged. As an example, Figs. 6 shows the course of the epidemic for 50 MC runs. The stochastic nature of the processes involved creates a spread in outcomes. Especially near the beginning of the epidemic where only few people are infected, statistical fluctuations cause large differences in the outbreak dynamics.

The \({R_{e}}\) shown for each day is given as the average number of people infected by everyone who recovered on that day. After the interventions are turned on, \({R_{e}}\) begins to decrease and in the absence of non-linear effects reaches a plateau after approximately 10 to 14 days. In runs where more than a few percent of the population has been exposed at that time, \({R_{e}}\) declines naturally due to an increasing chance that contact persons are already infected or recovered, and therefore cannot be infected again. When reporting the \({R_{e}}\) for a simulation run, \({R_{e}}(t)\) is averaged in the time span of 10 days to 28 days after interventions start, or from 10 days to the day more than 50% of the population has been exposed, whichever period is shorter. This time window is a compromise between being far enough away from the start of interventions for the effect of the interventions to fully manifest, and not getting to close to the region where \({R_{e}}\) changes naturally. The \({R_{e}}\) reported for a scenario is the average \({R_{e}}\) over all the simulation runs for that scenario.

We consider the following outcomes:

Figure 7 and Fig. 8 show three outcomes each for the four simulated symptom/reporting fractions and for \({R_{0}}= 3\) (Fig. 7) and \({R_{0}}= 2\) (Fig. 8). Results are shown for the realistic case where tracing identifies contacts regardless of their infection status, and for the case where only infected contacts are traced. The latter is included so that results can be compared to other models, and because the difference in the number of quarantined people between the two cases indicates how many healthy people are quarantined when uninfected contacts are also traced. \({R_{0}}=2\) is likely too optimistic, the results, however, are also valid in the situation where R was lowered to \(R= 2\) by other interventions, such as mask wearing, before tracing and quarantining starts.

The \({R_{e}}\) shown in the top panels of Fig. 7 and Fig. 8 should be compared to Fig. 3. For example for \({R_{0}}=3\) and \({\alpha }\cdot {f_{m}}=0.6\), just quarantining reported symptomatic cases yields R\(_{e;q}=2.2\), so DCT only lowers \({R_{e}}\) by another 0.5 (if tracing is independent of infection status), or 0.3 (if tracing finds only infected contacts).

Assuming that α is about 60% in European populations and that not everyone who has symptoms sees a doctor or is tested, the region between \({\alpha }\cdot {f_{m}}=0.4\) to 0.6 is likely relevant for Europe. If the reports of higher α in Asian countries are due to true differences in symptom fraction rather than to differences in study methods, the higher \({\alpha }\cdot {f_{m}}\) values simulated should be more relevant to Asian countries.

We will use \({R_{0}}=3\), \({ p_{\mathrm {app}}}=0.6\), \({\alpha }\cdot {f_{m}}=0.6\), and \({\eta _{\mathrm {DCT}}}=1\) (points outlined in red in Fig. 7) as defaults.

In Fig. 9, the lightest-coloured points correspond to these defaults. The other colors indicate what happens when the tracing efficiency is reduced. For the lower app coverages, the results are barely sensitive to \({\eta _{\mathrm {DCT}}}\) because DCT is not very effective to begin with.

In the realistic case where traced uninfected contacts are quarantined until two test results are negative (see Sect. 2.3), as many as 15% of the population are in quarantine on each day of the simulated outbreak, most of them healthy. Without a POC test to release healthy contacts, this number rises to 25%. At the peak of the outbreak, approximately 30% of the population is quarantined and half of those quarantined are actually sick.

People are not available to be infected while in quarantine, so the mean number of contacts per day, and with it the effective reproductive number, goes down and fewer people become exposed. The number of people quarantined rises with higher app coverage (because more people are traced in that case) and with a higher number of people exposed (because there are more index cases). A higher app coverage eventually leads to fewer exposed people though. Hence for a given reported symptomatic fraction, the number of people quarantined rises until an app coverage of approximately 75% (for lower reported symptomatic fractions) or 90% (for higher reported symptomatic fractions) and then falls sharply.

Contact tracing cannot reduce R below 1 in any of the simulations presented here except for \({\alpha }\cdot {f_{m}}\geq 0.8\) and \({ p_{\mathrm {app}}}\geq 0.9\) (if \({R_{0}}=3\)) or \({ p_{\mathrm {app}}}\geq 0.7\) (if R when tracing and quarantining is started is (2), and perfect tracing probability.

We note that in some cases, the fraction of the population exposed after 1 year is higher than the herd immunity level. The herd immunity level is defined as the fraction of the population that must be immune for the increase in new infections to not be able to grow exponentially, that is for \({R_{e}}\) to become 1. In an ongoing epidemic, many people are infectious when this point is reached, and the number of exposed people continues to rise until enough people are immune for \({R_{e}}=0\), therefore the curve overshoots herd immunity level.

To control the epidemic, R must be reduced by additional measures. We simulated the effect of random testing (RT) and social distancing (SD). Figure 10 shows the outcomes for our standard scenario with the addition of RT of 5% and 20% of the population per day, and social distancing bringing \({n_{c}}\) to 0.8 and 0.6 of its original value. The reduction in contact rate is always connected to an upper limit in the number of contacts as shown in Table 1.

Random testing even at 20% of the population per day in combination with contact tracing can only achieve \({R_{e}}\leq 1\) for \({ p_{\mathrm {app}}}\geq 0.75\). It does however bring \({R_{e}}\) close enough to 1 to significantly reduce the fraction of the population that becomes exposed, even for lower app coverages.

Social distancing reliably reduces the reproductive number. Social distancing to just 80% of the contact rate does as well as randomly testing 20% of the population each day. Reducing the contact rate to 60% pushes \({R_{e}}\) below 1 for 60% app coverage.

As we saw in Fig. 3, in a situation where reported symptomatic cases are quarantined, down-scaling the contagiousness of asymptomatic carriers reduces R significantly. Figure 11 shows the outcomes when DCT is then applied.

In the case where \({\eta _{as}}=0.1\) and all symptomatic carriers are reported when symptoms start, the simulations generate no outbreaks (fewer than 400 people become infected in total). When 75% of symptomatic cases are reported, \({R_{e}}\) has large fluctuations between simulation runs, and outcomes are very sensitive to the fraction of symptomatic cases.

Figure 12 shows the outcomes as a function of tracing delay, that is the time in days between when an index case is identified and when his or her contacts are traced and quarantined. Outcomes are again grouped by whether or not uninfected contacts are identified by tracing. Results are also shown for both 1st order tracing and 2nd order tracing and for the two incubation time curves (the default one with mean incubation time (IT) of 7 days and the alternative one with the shorter IT = 4 days). The yellow marker uses the default incubation time curve with the alternate transmission probability curve (IC) where there is less pre-symptomatic transmission. In addition to the simulation results, the calculated \({R_{e}}\) from the age-of-infection model is shown for the settings with the default incubation time and transmission probability curves, and first order tracing. Approximations had to be made in the calculation, hence the absolute value is not expected to match the simulation results perfectly.

Results are shown for an app coverage of 75%. Some of the dynamics are quite sensitive to the app coverage (see supplemental materials), and at 75% trends are clearer than at 60% app coverage.

The difference between first and second order tracing is small in all three outcomes (this changes in some situations for higher app coverages). For the default incubation time curve with mean of 7 days, tracing delays of up to 6 days have only a small effect, increasing the number of people exposed after 1 year from approximately 72% to 82%. When the mean incubation time is only 4 days, \({R_{e}}\) is more sensitive to tracing delays, with a 7 day delay increasing the number of people exposed from approximately 60% to 81%. The infection probability curve with less pre-symptomatic transmission probability improves the outcomes only very slightly, though the effect becomes bigger with larger tracing delay.

Second order tracing can find the infector and through him or her, the ‘siblings’ of the index case. The chance that the infection took place within \({\Delta T _{\mathrm {trace}}}\) is higher with a longer \({\Delta T _{\mathrm {trace}}}\). However even for \({\Delta T _{\mathrm {trace}}}= 14\) days, the outcomes are not significantly different. The look-back time must be balanced against the number of healthy people quarantined. People typically become index cases before they have recovered, and thus would have had a chance to infect others in the approximately 7 days prior. Looking back longer than that means one has a bigger chance of finding the infector, but it also means tracing many uninfected contacts.

When considering the realistic case where uninfected contacts are traced, second order tracing with a 7 day look-back time sends about 1.3 times as many people into quarantine on average over 1 year as first order tracing.

The results discussed so far consider situations where an outbreak is ongoing and interventions are started at some point into the outbreak. But not all simulation runs result in an outbreak. The stochastic nature of the outbreak means that there are large statistical variations at the beginning of the chain. For example, if patient zero happens to not infect anyone, no outbreak happens.

The chance for an outbreak to occur increases with R. The more people a case typically infects, the less likely it is that cases at the beginning of the infection chain do not infect anyone. Therefore, keeping interventions in place even in populations without an ongoing outbreak can be useful to decrease the probability that an outbreak will occur when a case is introduced into the population, for example through travel.

Figure 13 shows the outbreak probability as a function of the reproductive number when an infected person enters a fully susceptible population.

The results presented so far assumed a homogeneous population of size \(1\times 10^{5}\) and a distribution of the number of contacts with infection potential per day from Eq. (3). We also ran some sets of parameters for different population sizes and for different contact structures. The results are shown in Fig. 14.

The introduction of a social graph introduces non-linear effects that change \({R_{e}}\) on timescales much longer than what is captured by our standard analysis. In some cases, this means that fewer people are exposed after one year, even though \({R_{e}}\) is higher (see Fig. 21 in Appendix A.8).

We assumed that all people, once they have recovered from the infection, are immune to a secondary infection. Whether and for how long a recovered person is immune remains to be answered. Studies show that neutralizing antibodies are produced during infection and to a higher degree in symptomatic carriers, but decline significantly 2–3 months after recovery [61, 62]. The minimal antibody titer to confer protection is, however, still unclear. Furthermore, memory T cells to SARS-CoV-2 have been found in patients including asymptomatic and mildly symptomatic ones, which likely contribute to protective immunity as well [63].

In our models, everyone adheres to quarantine protocols. That is, every time someone is alerted by the DCT system to having been in proximity of a contagious individual, this person must follow the quarantining and testing procedure. This is crucial to suppress pre- and asymptomatic transmission, but may be difficult to achieve in reality. We also assume that the fraction of symptomatic individuals who see a doctor/get tested do so the day they become symptomatic.

The transmission probability in our models changes with the time since infection, but not between individuals. Current research, however, suggests that COVID-19 is over-dispersed, meaning some individuals spread the virus to many others, in so-called “superspreading events”, while most do not transmit the virus at all or only to very few people [64]. Part of this over-dispersion is due to the random nature of the contact number—some people just meet more others, and is therefore included in our models (see Sect. 2.2 and Fig. 2).

We assume that no manual tracing is performed at all. Typically, the types of close contact persons to whom spreading the disease is most likely, that is friends and family, can be manually traced without much effort, hence the fraction of infected contacts traced could be larger in reality.