Birth-Death (BD) model with incomplete sampling.

Under the basic birth-death (BD) model [9] all the individuals are in the same state. They can transmit with a constant rate , get removed with a constant rate , and their pathogen can be sampled upon removal with a constant probability .

Under this model we can calculate the likelihood density of a tree under parameter values (see Mathematical formulation of BD and BD-CT(1) models). The likelihood density can then be used to estimate model parameters from a phylogenetic tree (e.g., by finding parameter values that maximize it, in the maximum-likelihood framework). In particular, it permits the estimation of the following epidemiological parameters:

• effective reproduction number , expected number of individuals directly infected by an infectious case;
• infectious time , expected time during which an infectious individual can further spread the epidemic.

Note that the BD model is asymptotically unidentifiable (see Remark 3.4 in [9]). To become identifiable it requires one of its parameters to be fixed. In practice, it is often the sampling probability , as it may be approximated from epidemiological data (e.g., the proportion of sampled cases among the declared ones) or the infectious time (estimated from observations of infected cases).

Multi-Type Birth-Death (MTBD) models.

Multi-Type Birth-Death (MTBD) models [12] add population structure to the BD model by allowing different individual states, transmissions between them and state changes. A general MTBD model with m individual states has parameters: An individual in state can be removed at a constant average rate (with pathogen sampling probability ), change their state to state l∈{1,...,m}at a constant average rate (), and transmit their pathogen to an individual in state r∈{1,...,m} at a constant average rate . The time between events is hence modeled with exponential distributions.

Contact Tracing (CT) extension.

We propose an extension of the MTBD models that adds contact tracing (CT). In the contact tracing process, an identified infected index patient notifies their potentially infected contacts (e.g., sexual or drug injection partners for HIV), either on their own (patient referral) or with the help of a health service provider (provider referral: a health advisor notifies the contacts without identifying the index patient; or contract referral: if the index patient fails to notify contacts within a specified time period, the health advisor proceeds to provider referral) [18]. The notified contacts are then provided with appropriate tests and treatment. Therefore, a successful detection of a new case B via contact tracing by an index patient A implies several conditions to be met: (i) A remembers the interaction with B and either notifies B on their own or gives the B’s details to a health advisor for assisted notification; (ii) B is successfully reached and notified; (iii) B agrees to take a test; (iv) B is indeed infected. For instance, according to the report from the Public Health England [19], in England in 2013 for HIV, reaching the stage (ii) had an estimated probability of about 50%; reaching the stage (iii) – of about 41% (as 83% of all the patients who attended a clinic through CT had a reported HIV test); and the stage (iv) – of less than 3%.

All these conditions are hardly identifiable separately from only sequence data, hence in our model we combine them into one, “an infected person B being successfully notified by A”, and model it with a constant notification probability which is applied upon sampling of A.

For simplicity, a positive draw with probability models a slightly modified scenario: (i) A remembers the interaction with B and either notifies B on their own or gives the B’s details to a health advisor for assisted notification; (ii’, similar to iv in the previous scenario) B is indeed infected, moreover, the infection of either A or B is caused by the pathogen transmission during their interaction; and either (iii’, same as ii) B is successfully reached and notified and (iv’, same as iii) agrees to take a test (which at the end of the considered time period might not yet be available in the database); or (iii”) by the time of notification, B is already removed by other means (already got sampled or stopped being infectious without sampling, e.g., died).

Note that in (ii’) in contrast to (iv) we exclude the possibility that B is indeed infected, however neither A infected B, nor B infected A (e.g., if they were both infected by a third person C). While in real life in such a situation B might still happen to get identified through tracing of A’s contacts, we consider such cases rare enough, and do not model them. From now on, we will use the term “contact” to denote a contact, during an interaction with whom a pathogen transmission happened.

Upon notification, if not yet removed, the contact’s pathogen is sampled almost instantaneously (modeled via a constant notified sampling rate ). Upon sampling, the contact might notify in their turn.

BD-CT hence has 3 classical BD and 2 additional parameters:

• – constant transmission rate: an infected individual transmits their pathogen to a newly infected individual with this rate. This leads to an internal node in the transmission tree;
• – constant removal rate: an infected individual stops being infectious at this rate. This leads to a tip in the transmission tree (observed or hidden);
• – constant sampling probability: upon removal the individual’s pathogen might get sampled with this probability. This leads to a tip in the transmission tree being observed;
• – constant contact tracing probability: upon sampling, a sampled individual’s contact might be notified with this probability. This leads to a removal rate change for the contact’s branch in the transmission tree;
• – constant notified sampling rate: a notified contact (if not yet removed via the standard procedure) gets removed and observed with this rate. This leads to an observed tip in the transmission tree.

We add an additional hyper-parameter , which defines how many contacts a sampled individual might notify. Under the BD-CT(1) model (i.e., ) each sampled individual can notify only their last contact. Under the BD-CT() model, each sampled individual can notify up to last contacts (their notification is independent, and depends for each of them on the probability ). Note that gives a standard BD model with no contact tracing.

This model is a simplification of the contact tracing process, in particular it makes 3 assumptions:

1. only observed individuals can notify (instead of any removed individual);
   This is a realistic hypothesis in which an individual is “observed” because he or she enters a diagnostic and follow-up process that includes contact tracing. Without entering this process, he or she is not observed and does not notify.
2. successfully notified contacts get sampled (i.e., their pathogens are always observed upon removal, unlike the standard case where the observation happens with a sampling probability );
   Realistically again, we assume here that the diagnostic and follow-up process works as expected, sampling notified contacts.
3. only the last contacts can get notified;
   In the case of sexually transmitted diseases, notifying only the last contact (k=1) could roughly correspond to notifying the “main” partner (e.g., spouse). Also note that the expected number of contacts per individual is R_e, so depending on the pathogen, a low value of might be enough (e.g., 1 or 2 for HIV).

Note that the “last contact” relationship is not necessarily symmetrical. The last contact j of individual i might have transmitted their pathogen further (to individual k) after the interaction with i. Hence, while j is the last contact of i, j’s last contact is k, and, once notified, j might notify k.

The -CT() extension can be generalized from BD to MTBD models: at the moment of sampling, an individual of any type can notify their last contacts (each independently, with the probability ). If notified, the contact’s removal rate gets replaced by a constant sampling rate . The -CT() extension therefore adds two parameters to an MTBD model: and υ.

Master equations and tree likelihood under BD model.

The BD model can be described with master equations representing the likelihood densities of evolving as on the observed transmission tree, with time t going forward from the root (t = 0) till the time of the last sampled tip (t = T). These likelihood densities depend on the probability U(t) of an unobserved transmission tree that started from one individual at time t and evolved till T, without this individual nor any of their induced cases being sampled:

(1)

With the Eq (1), we can write down the equation for the probability density of evolving as on an observed tree branch i till time t_i, starting on this branch at time . The master equations for the BD model were initially developed by Stadler et al. [9], however what we present here is their branch-specific formulation that we proposed in [29]: In our formulation, describes an evolution along a tree branch i, without taking into account the branch’s subtree. t_i corresponds to the time of the equation’s initial condition (typically at the end of the branch i).

(2)

Using this equation, we can calculate the likelihood density of a tree under parameter values as:

(3)

Master equations under BD-CT(1) model.

In the following we consider the simplest version of the BD-CT() model family, where and hence only the last contact can be notified.

Tree branches under BD vs BD-CT(1) models.

In the classical BD model with incomplete sampling, a branch of the tree corresponds to a virus evolution between the transmission corresponding to the node at the beginning of this branch and either a sampling event (if the branch is external) or a transmission corresponding to the node at the end of the branch (if it is internal). Note that since sampling is incomplete, hidden transmissions (leading to fully unsampled subtrees) could occur along this branch, where the unsampled subtree could correspond to either donor or recipient. Hence, we do not know whether the individual at the end of the branch (at time t_i) is the same as at any time t<t_i along the branch. These two individuals could be different if a hidden transmission where the donor subtree stayed unobserved, occurred between times t and t_i. in Eq 2 describes an evolution along such a tree branch, integrating over all the possibilities, starting from no hidden transmission along the branch and including any number of hidden transmissions with either donor or recipient tree staying unsampled (and hence potentially different individuals at times t and the branch end t_i).

In trees generated by the BD-CT(1) model, however, the individuals do not behave in the same way when they are notified or not and, hence, other types of branches are possible. As in a real tree we do not know who might have notified whom, we have to integrate over all the possibilities to calculate the tree likelihood. We calculate the likelihood density of a tree using a pruning algorithm [30], while considering all the combinatorial possibilities for each node. For instance, for each tree tip we consider four possibilities: whether the corresponding individual was notified or not by someone, and on top of that, whether the tip’s individual notified or not their last contact upon sampling. Before describing the combinations behind tree likelihood calculation, we will introduce additional tree branch types.

Additional master equations under BD-CT(1) model.

Let us consider a person A who got sampled at time t_A and successfully notified their contact B, who as a consequence got sampled at time . A and B hence correspond to tree tips, while their most recent common ancestor node AB corresponds to the transmission between A and B (see Fig 2A1).

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Fig 2. Examples of contact tracing in a transmission tree.

We show 12 scenarios (one per panel) where an index individual A (violet) tried to notify their contact B (orange). The violet tip at time t_A represents the sampling of A, the internal node AB corresponds to the transmission between A and B. In the first row ( A1-A3) B is observed by the end of the sampling period, in the other rows ( B1-B3 and C1-C3) B is hidden. In the last row ( C1-C3) also AB is hidden. In column 1 A was the first person to successfully notify B, in column 2 a third person C notified B before A did, in column 3 B was already removed via the standard procedure by the time A tried to notify them. The fact that A is the notifier of B implies that there is no hidden event along its branch AB-A (violet), and that in any transmission on the path AB-B (orange or dark-red) the donor is B. Moreover, between the moment of B’s first notification (t_A for column 1, t_C for column 2) and the moment of sampling of B (t_B) the removal rate of B becomes instead of (dark red). Finally, B might have been removed via a standard procedure before getting notified by their contact(s) (with rate and a probability , column 3). In the panels C1-C3 B’s subtree is fully hidden, and hence the transmission node AB is also unobserved. Therefore the A’s branch is composed of two parts, where the bottom part (violet) corresponds to A and contains no hidden transmissions, while the top part could correspond to A or not (e.g in C2 it corresponds to B) and might contain hidden transmissions. In panels B1, B2, C1 and C2 B is hidden as by the end of the sampling period (t = T) B is not yet sampled, while in panels B3 and C3 B is hidden as B got removed via the standard procedure without sampling before any notification. A’s branch is modeled with a notifier branch probability (4) in panels A1-A3 and B1-B3, and with a mixed branch probability (see S1 Appendix) where the bottom part corresponds to the notifier (A) in panels C1-C3. If B did not notify their last contact upon sampling in panels A1-A3, branches on the path AB-B are modeled with a combination of contact probability before notification (orange, (5)) and contact probability after notification (dark red, (6)). Otherwise, the part after the transmission from B to their last contact is modeled with either a both notified and notifier branch probability (7) (as in A1 and A2) or a notifier branch probability (4) (as in A3), depending on the B’s notified status at that time. In panels B1-B3, the AB-C branch is modeled with a mixed branch probability (see S2 Appendix), where the top part corresponds to the contact state (B).

https://doi.org/10.1371/journal.pcbi.1012461.g002

According to our model assumption 3, only the last contact can get notified. Hence, AB must be the parent node of A, and no hidden transmission is possible along the branch AB-A (as otherwise there would have been a contact of A who is more recent than B).

We will use a notifier branch probability for such cases (when no hidden transmission occurred along the branch A, colored violet in Fig 2):

(4)

As B corresponds to A’s contact, the individual on the path between the transmission node AB and the tip B cannot change (is always B) and only transmissions from B to someone else are possible along the path leading from AB to B. These transmissions can be either hidden (where the recipient tree stayed unobserved) or observed: in both cases we know which subtree corresponds to the donor (the one that contains B). However, depending on whether B already got notified ( in Fig 2A1) or not yet (t<t_A in Fig 2A1), B’s removal rate is different (is correspondingly or ). We will use a contact branch probability before notification for cases when the individual at the end of the branch (or considered time interval, here ) is the same as at time t, has not yet been notified, but might have transmitted to someone else (unobserved) along the way (colored orange in Fig 2):

(5)

We will use a contact branch probability after notification for cases when the individual at the end of the branch is the same as at time t, by time t has already been notified, and might have transmitted to someone else (unobserved) along the way (colored dark red in Fig 2):

(6)

Finally, once sampled, B could have notified their own last contact (represented by the most recent hidden subtree in Fig 2A1). This would imply that between the moment of transmission between B and their last contact (t_y) and the moment of B’s sampling (t_B) there was no event along the B’s branch. Hence, if B is a notifier, the part of the B’s branch after t_y should be modeled with a notifier probability instead (no event along the branch). Note that once B is notified, the removal rate becomes , and hence a both notified and notifier branch probability applies:

(7)

The Eqs 1, (2), (4)–(7) have the following closed form solutions:

(8)

Note that on top of the scenario we just described (shown in Fig 2A1), two other scenarios are possible for a person A who once sampled (at t_A) notified their last contact B, who got sampled at time t_B:

1. by the time t_A, B was already notified by someone else (e.g., at time in Fig 2A2).
   This would lead to a shorter contact-before-notification part on the B’s branch (finishing at t_C), and a longer notified-contact part (between t_C and t_B).
2. by the time t_A, B was already sampled via a standard procedure (at time , hence the branch B would finish before the branch A, as in Fig 2A3).
   This would lead to the contact-before-notification part on the B branch finishing at t_B and no notified-contact part. Note that the branches between AB and B stay oriented as B remains the last contact of A.

Note that if A is a notifier, A is always observed, as notification only happens upon sampling. However, B could have stay unsampled: either because once notified (by A or another contact, who notified them before A did) B has not yet got sampled (Fig 2B1, 2B2, 2C1 and 2C2); or because B got removed via the standard procedure without sampling before any notification (Fig 2B3 abd 2C3). Moreover, between the transmission AB and the moment of B’s notification/removal, B might have transmitted to someone else (e.g., C in Fig 2B, BC denoting the transmission). If that happened and someone in BC’s subtree got sampled, the branch AB-A would still correspond to the notifier A, while the other AB’s child branch would partially correspond to B (between AB and BC) and partially to C (starting at BC), as in Fig 2B. Otherwise (if there was no such transmission or all such subtrees stayed unobserved, as in Fig 2C), the node AB will not be observed either, and A’s branch will get extended with its parent’s branch. Hence only the bottom part (after AB) of the A’s branch would correspond to the notifier probability (see Fig 2C).

To model the cases when B is not sampled, we will need to describe the probability of a mixed branch for branches that include a hidden contact subtree (AB-C branches in Fig 2B and A’s tip branch in Fig 2C). s₁ here denotes the type of the top of the branch (e.g., between AB and BC in Fig 2B, contact) and s₂ denotes the type of the bottom of the branch (e.g., between AB and A in Fig 2C, notifier). Let us denote the first notification time of the hidden contact as t_r, and the time of the hidden contact subtree start as t_h, while as usual t_i stands for the time at the branch end. For instance, in Fig 2B1 , , ; in Fig 2B2 , ; in Fig 2C3 , .

combines three elements: the probability of the top part of the branch , which finishes with a transmission to or from the hidden contact, the probability of the bottom part of the branch (without taking into account the event at its end) and the probability of the hidden contact subtree . We describe the details behind calculation of in S2 Appendix.

In real trees we do not know sampled individual’s status (individual who did not notify, notifier, contact notified in time or not), and hence during the tree likelihood calculation, we will have to consider all these scenarios.

Tree likelihood under BD-CT(1) model.

Using the branch probability formulas described above, we can calculate the likelihood density of a tree using the pruning algorithm [30]. For each visited node i we calculate the likelihood density of its subtree including the branch ji connecting i to its parent node j for two cases depending on i’s receiving-a-notification status (shown with the first superscript, n^̄ or ⁿ). In the first case, , the node i corresponds to an unnotified individual. In the second case, l(ⁿ)(i,r), the branch ji and the node i correspond to an (eventually) notified contact (notified by the individual corresponding to a tip r at time t_r). If i is a tip, we additionally consider whether there is an observed contact in our tree, whom the individual i notified (shown with the second superscript, ⁿ or n^̄). For instance, represents the case where the tip i was not notified but notified their last contact (who is observed).

The tree likelihood density under BD-CT(1) model with parameters can be calculated as , as the root can only correspond to an unnotified individual (since there was no observed transmission before it).

We describe the subtree likelihood density calculation case by case in S3 Appendix.

Extension to forests.

As we previously described in [29], the likelihood calculation with MTBD models can be easily extended to forests. This extension applies also to the BD-CT models. Forests are useful for cases when a (sub-)epidemic started with several infected individuals (e.g., due to multiple pathogen introductions to a country of interest or due to a change of health policies leading to a change in parameter values). In this case the (sub-)epidemic leads to a forest of f observed trees: . The forest might also include a certain number u of unobserved trees, for which none of their tips got sampled. Forest likelihood formula hence combines the likelihoods of f observed and u hidden trees, and can be represented in a logarithmic form:

(9)

where and t_start is the (potentially averaged) start time of the hidden trees.

For given model parameter values we can estimate the number of hidden trees u from the number of observed trees f as:

(10)

The tree likelihood formula is a special case of Eq 9, where f = 1 and u = 0.

Code availability.

Our parameter estimators are implemented in Python 3. They use ETE 3 framework for tree manipulation [32] and NumPy package for array operations [33].

They are available as command-line programs (, ) and a Python 3 package via PyPi (bdct), and via Docker/Singularity (evolbioinfo/bdct). The source code, installation and usage documentation are available on GitHub at github.com/evolbioinfo/bdct.

The parameter estimator package also includes the cherry test (, see section “Testing for presence of contact tracing”).

CT test performance.

To illustrate the CT test performance, we applied it to the twelve 100-tree simulated datasets, corresponding to the three classical MTBD-family models: BD, BDEI (Birth-Death Exposed-Infectious [13]) and BDSS (Birth-Death with Super-Spreading [12]), and to their -CT versions with (possibility to notify the latest contact), (possibility to notify the last two contacts), and (possibility to notify all the contacts), see “MTBD-CT tree simulator and test datasets” for dataset details. The results are summarized in Table 1.

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Table 1.

CT test performance on trees generated under BD, BDEI, BDSS models and their -CT versions (100 trees per model, each having 500-1000 tips). We report the number of trees for which the CT-test p-value was <0.05, , the mean p-value and the mean number of cherries for each dataset.

https://doi.org/10.1371/journal.pcbi.1012461.t001

For 96/99/100 out of 100 trees in the BD/BDEI/BDSS dataset the CT test p-value was above 0.05 (i.e., 96%/99%/100% true negative and 4%/1%/0% false positive results).

For 97/94/91 out of 100 trees in the BD-CT(1)/BDEI-CT(1)/BDSS-CT(1) dataset the CT test p-value was below 0.05 (i.e., 97%/94%/91% true positive and 3%/6%/9% false negative results). It showed similar performance for higher values of .

Our test overall showed high specificity (98% on the combined 300-tree dataset without CT) and good sensitivity (95% on the combined 900-tree CT dataset).

Likelihood ratio test (LRT) performance.

For the BD and BD-CT datasets we performed an LRT with the parameters estimated by our bd and bdct estimators: (where 5.99 is the value of the chi-squared distribution with 2 degrees of freedom corresponding to a significance level of 0.95). The test selected the correct model (-CT vs no CT) for 97%, 99%, 100% and 99% of the corresponding BD, BD-CT(1), BD-CT(2) and BD-CT() trees.

These results and those obtained with the CT test show that the trees generated with the CT component are very different from those generated without it. It should also be noted that likelihood calculation and LRT performed using bdct seem to be robust with respect to the value of k(good results are obtained when k=2 and k=∞).

Point estimates.

The estimation results are shown in Fig 3. We report average relative errors and biases (, ) for the rate parameters () and average absolute errors and biases (, ) for the contact tracing probability , which can be very small.

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Fig 3. Comparison of inference accuracy of bd (blue) and bdct (yellow) estimators

on datasets of 100 trees of 500–1 000 tips generated under the BD (top left), BD-CT(1) (top right), BD-CT(2) (bottom left), and BD-CT() (bottom right) models, with fixed to its true value. We show the bar-plots (colored by estimator) representing mean errors for each parameter. The error values are displayed below the corresponding bar-plots. Below them the mean bias values are displayed. For the rate parameters relative errors and biases are shown (distance for errors, difference for biases, between the estimate and the true value divided by the true value), for the contact tracing probability absolute errors and biases are shown (as for the BD model the true value is zero). For only the cases with non-zero estimates of were considered.

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The estimators performed well on the models that correspond to them (BD for bd, BD or BD-CT(1) for bdct): the average relative errors for the BD-model parameters are within 10%, the average absolute error for (with prior within [0.01, 0.75]) was less than 0.04. However, the performance of our estimators got worse when the model was misspecified (presence of CT for bd, for bdct).

Interestingly, contact tracing had different effects on the estimation of different parameters. Not accounting for contact tracing at all led to a drastic overestimation of the standard removal rate (e.g., relative bias of 145% and 237% for bd on BD-CT(1) and BD-CT() trees, respectively). When accounting for less contacts than in the true model, the bias is less drastic but still significant (e.g., relative biases of 30% and 72%, for bdct on BD-CT(2) and BD-CT() trees). Not accounting for contact tracing also led to a slight overestimation of the transmission rate (e.g., relative bias of 24% for bd and of 5% for bdct on BD-CT() trees). Intuitively, in the absence (or decreased presence in case of wrong ) of a contact tracing mechanism, which allows for faster removal and sampling of a fraction of the infected population, the compensation mechanism is to increase the standard removal rate for the entire population. Both effects were much less present for the bdct estimator than for bd, i.e., when one contact was allowed to be traced. Combined with the performance of bdct (equivalent to bd’s) on the BD dataset, this advocates for using bdct in all the situations to at least reduce the estimation bias. Interestingly, the estimation of remained accurate (average absolute error <0.04) and unaffected by model misspecification.

The notified sampling rate was the hardest parameter to estimate in all the CT settings (average relative error of 18-42%, and a bias towards overestimation in all the settings). The difficulty in its estimation is not surprising as there is the least information on it in the trees. For instance, the contact tracing probability comes into play (positively or negatively) on every sampling event (i.e., every tree tip), while the notified sampling rate only participates in those of these cases where the index (i) decides to notify (i.e., the positive draw of ) and (ii) gets removed before their notified contact. Moreover, the bias in the estimation of this parameter could be due to the approximations in the BD-CT(1) likelihood calculation that we had to make in order to obtain a closed-form solution (e.g., setting hidden contact notification times to the middle of the corresponding intervals, see S2 Appendix).

Confidence Intervals (CIs).

The percentage of the trees for which true parameter values p_true were within the CI estimates and relative CI widths () are shown in Table 2. For the CIs as for the point estimates, bdct and bd estimators performed equally well on the BD trees (100% of estimates within the CIs), however the CIs estimated by bdct were slightly wider (15% and 49% for respectively and versus 12% and 39% for bd). On the BD-CT(1) trees the bd estimator performed significantly worse (only 51% and 7% of CIs included the true values of respectively and versus 100% and 98% for bdct). As increased, the performance of bd dropped (down to 22% for and 7% for on BD-CT() trees). The performance of bdct with respect to and also decreased with the increase of , but less drastically than for bd (reaching 84% and 36%, respectively, on BD-CT() trees). This finding is somewhat expected due to the bias in the point estimates for and especially . Interestingly, CI performance for and remained stable or even improved with the increase of (95% and 91%, respectively, on BD-CT() trees). As with the point estimates, was most affected by contact-tracing model misspecification, and much more so when contact tracing was not included at all (bd) than when it was partially included (bdct, ).

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Table 2. CI estimation performance.

Percentage of simulated BD, BD-CT(1), BD-CT(2) and BD-CT() trees for which the true parameter values were within the bd and bdct-estimated 95% CIs, and the median CI width: . (The sampling probability was fixed to its true value in all the settings.)

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Application 1: Men-having-sex-with-men HIV-1 B epidemic in Zurich.

We applied our estimators to assess the presence of contact tracing for HIV-1 subtype B epidemic among men-having-sex-with-men (MSM) in Zurich, using a 200-sample time-scaled phylogenetic tree from the study by Rasmussen et al. [20], and fixing the sampling probability to 25%: (see “Zurich men-having-sex-with-men HIV-1 B tree and sampling probability” for more details).

The CT test detected contact tracing in this tree (p-value of 0.014). The likelihood ratio test also suggested a BD-CT model: log-likelihood under the bdct-estimated parameters was vs. under the bd-estimated parameters, which gives an LRT statistic of . Our bdct parameter estimator inferred a contact tracing probability CI [0.001–0.314], R_e = 1.43 [1.17–1.76], infectious time [6–14] years, and 4.9 months [3 days–2.3 years] between contact notification and their sampling. These results globally agree (CIs intersect) with the R_e estimate for this data using a coalescent model from Rasmussen et al. [20] (between 1.0 and 2.5), and the estimates using the BDSS model from Voznica et al. [21]: R_e of 1.6 and 1.7 and an infectious period of 10.2 and 9.8 years (depending on the tree representation used for the deep learner).

Note that this tree covers a long non-homogeneous time interval (between 1974, the root of the tree, and 2012), including times with no or limited access to treatment, when the infectious time corresponded to progression to AIDS ( years), as well as later times, when antiretroviral treatment (ART) became available, and hence the infectious period would correspond to the time till ART start, as, when taken properly, it prevents transmission. Our estimate (8 years) represents a mix of these scenarios. This also holds for the other estimated parameters, whose values represent a somewhat averaged estimate over the time period. As this dataset was already small, we did not cut it to obtain a more homogeneous treatment policy (as we did in Application 2).

Moreover, as our results on the simulated data suggest, not accounting for (enough) contact tracing (i.e., estimating BD-CT(1) parameters on the data generated under ) leads to overestimation of and hence underestimation of the infectious time and potentially . Our estimate should therefore be treated as a lower bound (i.e., an infectious time of at least 8 years and a reproduction number of at least 1.43). In the same vein, parameter inference with the bd estimator gave a slightly lower estimate of R_e and a shorter infectious time, though with intersecting CIs (R_e = 1.32 [1.16–1.51], [6–9] years).

We could not find estimates of the percentage of HIV infections detected through contact tracing in Switzerland with which to compare our estimation of . An editorial in “Revue médicale Suisse” [41] stated in 2004 that there was no systematic assisted contact tracing for HIV in Switzerland. This situation might have improved by 2012. In addition, there might have been patient referral (patients notifying their contacts on their own). Overall, this supports a relatively low value for .

Application 2: HIV-1 B epidemic in the UK.

We also applied our estimator to asses the presence of contact tracing in HIV-1 subtype B infected individuals in the UK between 2012 and 2015. This time period was chosen to represent a homogeneous treatment policy (treatment start at CD4 count below 350 cells/mL [38]). We used 10 forests reconstructed from HIV-1 B sequences from the UK HIV Drug resistance database [36] (see “UK HIV-1 B forest reconstruction and sampling probability estimation” for details).

The CT-test p-values were below 0.05 for all the forests (between and ), strongly suggesting the presence of contact tracing. The likelihood ratio test also suggested a BD-CT model: For instance, on forest 1, log-likelihood under the bdct-estimated parameters was vs. under the bd-estimated parameters, which gives the LRT statistic of .

Using sampling probability (see “UK HIV-1 B forest reconstruction and sampling probability estimation” for its estimation), we estimated the contact tracing probability CI [0.002–0.015], R_e = 1.2 [1.20–1.29], infectious time [2.24–2.42] years, and 8 [2–17] days between contact tracing and their sampling. The results were consistent between 10 forests and are shown in Table 3.

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Table 3. BD-CT model parameters estimated for UK HIV-1 B epidemic between 2012 and 2015,

assuming that our data represents 58% of infected cases ().

https://doi.org/10.1371/journal.pcbi.1012461.t003

In the UK in 2015 the majority of infected population (72% [42]) was on antiretroviral treatment, which, when taken properly, prevents transmission. Hence, the average infectious time roughly represents the average time between infection and treatment start. Note that, according to our results on simulated data, this time might be underestimated if the true κ value (number of partners that can be notified) is greater than one. Hence, our result suggests that the average time between infection and treatment start is at least 2.3 years.

According to the HIV report from the Public Health England [19], in England in 2013, less than 3% of sampled infected individuals were detected through contact tracing. Our model considers only contacts that caused a virus transmission, while in real life partner notification may permit detection of seropositive individuals with whom there was no transmission. This explains why our estimate is even lower (1%). It should be interpreted as 1% of sampled infected individuals were detected through contact tracing between the transmitting and the acquiring partner.

For comparison, parameter estimation on the same forests with the BD model gave similar results (R_e = 1.2 [1.20–1.27], [2.25–2.39] years), however the infectious duration was slightly shorter, which is in line with overestimation of the removal rate when not accounting for contact tracing (which we have seen on simulated data).

As expected the estimates on the much smaller Zurich dataset show larger CIs than those on the larger UK one. The CIs estimated on the two datasets intersect for the effective reproduction number R_e, while the infectious time is larger for Zurich (CI of 6–14 years vs 2.2–2.4 years for the UK). This is however expected as the Zurich tree covers a longer, earlier and non-homogeneous time interval (between 1974 and 2012), including times with no or limited access to treatment, when the infectious time corresponded to progression to AIDS ( years). The CIs for contact tracing probability and contact sampling time also intersect on the two datasets, though they do not have to, as the contact tracing policy might differ between the two countries, especially during the long and non-homogeneous time interval represented by the Zurich tree. The point estimates suggest stronger contact tracing probability and longer notified contact sampling time in Zurich than in the UK. Longer delays in contact testing might be explained by the absence of assisted partner notification in Switzerland.