Bayesian Reconstruction of Disease Outbreaks by Combining Epidemiologic and Genomic Data

1This is a PLOS Computational Biology Methods article.

2Statistical methods for analyzing detailed epidemiological data collected during infectious disease outbreaks have seen rapid development in recent years,,,,,,,. These methods probabilistically reconstruct likely transmission links between cases using data on the timing of symptoms and, where available, contact tracing data or other proximity information. The resulting transmission trees allow estimation of the number of secondary infections generated by each case, and thus of the transmission intensity (characterized by the reproduction number, R) over time. Pathogen genetic sequence data provides valuable additional information on potential transmission links between cases in a disease outbreak, particularly when reliable contact tracing data is not available. Indeed, using sequence data alone to estimate transmission rates during epidemics is increasingly frequent,,,,,,,,,,. As genetic sequences can now be obtained nearly in real-time,, this new source of information opens up exciting perspectives not only for understanding past outbreaks, but also for unraveling the transmission routes of ongoing outbreaks and subsequently adapting public health responses.

3Integrated analysis of both epidemiological and sequence data clearly would maximize our ability to reconstruct transmission trees, but there are methodological and computational challenges. These challenges center on constructing and evaluating a unified likelihood for both the genetic and epidemiological data. One of the first attempts at integrated analysis used phylogenetic trees to constrain the set of transmission trees then explored by an epidemiological transmission tree inference algorithm. An alternative approach highlighted limitations of phylogenetic methods for reconstructing densely sampled outbreaks, and proposed an alternative graph theoretic approach for reconstructing ‘genetically parsimonious’ transmission trees, i.e. trees implying the smallest number of genetic changes amongst the sampled isolates. While simple and fast, this method also has a number of limitations: dates of infection are not inferred, the probability of a given transmission event cannot be assessed, and unobserved cases or multiple introductions of the disease cannot be detected. Substantive methodological developments have been made by Ypma et al. and subsequently by Morelli et al., both of which proposed unified likelihoods for genetic and epidemiological data to analyze livestock disease outbreaks (avian influenza H7N7 and foot-and-mouth disease ). However, those methods require that the outbreak has a single introduction event and that all cases are observed, which limits their applicability to restricted epidemic contexts.

4Here we introduce a novel and generic framework for the reconstruction of disease outbreaks based on pathogen genetic sequences and collection dates. We use the distribution of the generation time (i.e. time interval between a primary and a secondary infection), to define the epidemiological likelihood of a given transmission tree. This is coupled with a simple model of sequence evolution defining the probability of the genetic changes observed between the pathogen genomes along a chain of transmission. Our model is embedded within a Bayesian framework allowing estimation of dates of infections, mutation rates, separate introductions of the pathogen, the presence of unobserved cases, and the transmission tree. Estimate of the effective reproduction number over time, R(t), can also be obtained. As an improvement over previous approaches,, our method does not require all cases to be observed or there to be a single introduction event which triggers an outbreak. After evaluating the performance of our method using simulated outbreaks, we illustrate our approach by analyzing the 2003 Severe Acute Respiratory Syndrome (SARS) outbreak in Singapore,,. Our method is implemented in the package ‘outbreaker’ for the R software and represents the first widely available tool for the reconstruction and analysis of disease outbreaks from genomic data.

5We analysed simulated outbreaks to assess the performance of our method under a variety of conditions, including different basic reproduction numbers (R0), sampling coverage, rates of evolution, and generation time distributions, with our base scenario resembling an influenza-like illness (Table 1). The outbreak size varied from 10 to nearly 200 infections in a fixed population of 200 susceptible hosts (plus imported cases), with a median sample size of 110 (quartile range: [66–132], Fig. S1). Wherever applicable, reported results refer to the marginal distributions.

6Transmission trees were overall very well reconstructed, with 70% to 90% of true ancestries being recovered in most simulation settings (Fig. 1 and Table S1 in Text S1). Better results were achieved when the sampling coverage was high (compare settings ‘Base’ to 75%, 50% and 25% of missing cases). In the absence of genetic information, the transmission tree was very difficult to infer (setting ‘No mutation’). Differences in basic reproduction numbers (settings ‘Low R’ and ‘High R’) and in the shape of the generation time distribution (settings ‘Short generation’ and ‘Long generation’) induced some variation in the proportions of successfully recovered ancestries, although these remained satisfying in every case (Fig. 1 and Table S1 in Text S1). Dates of infections were inferred with accuracy in most settings (Fig. S2 and Table S1 in Text S1). However, this result was mostly driven by the shape of the generation time distribution, with broader distributions leading to greater uncertainty in the dates of infection (Fig. S2). While perfectly inferred in fully sampled outbreaks, the number of generations between ancestor and descendents became ambiguous as the proportion of missing cases increases (Table S1 in Text S1). Mutation rates were also mostly well estimated (Table S1 in Text S1, Fig. S3), albeit with a tendency to over-estimation. This bias was stronger when sampling grew sparser (settings with 75% and 50% missing cases), and to a lesser extent when the number of imported cases grew large (setting ‘Many imports’). Detailed investigation of individual simulations suggested that misdetection of imported cases and increased numbers of erroneous ancestries may be responsible for over-estimating the mutation rates in these settings. The inference of sampling coverage varied largely amongst different simulation settings (Table S1 in Text S1, Fig. S4): well recovered in fully sampled outbreaks, it was largely overestimated in sparse samples (settings with 75%, 50% and 25% missing cases), and slightly underestimated with longer generation time.

7The detection of imported cases showed excellent specificity and good sensitivity pooling results across the simulated datasets examined, with a majority of simulations exhibiting perfect results (Fig. 2). However, substantial variations were observed between simulation settings (Fig. S5, Table S1 in Text S1). Unsurprisingly, detection of imported cases was more difficult when imported cases were more frequent and when a higher fraction of cases was unobserved. With longer generation times, the larger numbers of mutations accumulated between ancestors and descendents made the detection of genetic outliers, and thus of imported cases, nearly impossible (Fig. S2).

8While our model does not explicitly estimate the effective reproduction number ‘R’ (i.e., the number of secondary cases per infected individual), this quantity can easily be computed from the posterior trees. Our ‘base’ simulations show that reliable estimates of R at an individual level can be obtained when genetic information is available (Fig. 3, left). In contrast, such inference was impossible in the absence of genetic data (Fig. 3, right).

9To gain a better understanding of disease outbreak dynamics, identifying systematic heterogeneity in R across cases is also essential. To assess whether our approach could detect such heterogeneity, we implemented two types of simulations in which there were systematic differences in infectivity between groups of hosts. In a first set of simulations, the host population was divided into two groups of equal sizes (e.g. adults and children) with low and high infectivity (infectivity in one group was twice that of the other group, with equal susceptibility). In the second setting, we included rare (5%) super-spreaders, who had the same susceptibility to infections as non super-spreaders, but were 13-fold more infectious. In both sets of simulations, infectivity was fixed for each individual at the beginning of the simulations. The classification of individuals into super-spreaders and regular spreaders was considered as known when comparing estimated reproduction numbers.

10Results showed that our method was able to recover contrasted infectivity between different groups (Fig. 4, S6, 7, 8, 9). In the simulations with equally-sized groups, the overall distributions of R for each group were almost perfectly recovered (Fig. 4, top panel), while values of R at an individual level were also well estimated (Fig. S6). Importantly, when ignoring the genetic information, differences between groups were barely detectable (Fig. 4 and S7). Similar results were observed in simulations including super-spreaders (Fig. 4, bottom panel), in which estimates of R values at an individual level were excellent when using genetic information (Fig. S8), and very poor without it (Fig. S9). The reconstruction of average R values over time was not improved by the inclusion of genetic information (Fig. S10, S11), which is unsurprising as this mainly depends on correctly inferring the dates of infections, which was unaffected by the absence of genetic data (Fig. S1, S2).

11We analyzed data collected during the beginning of a SARS outbreak which took place in Singapore in 2003,. Previous studies proposed different reconstructions of this outbreak based on indirect contact tracing information and genetic data, and while all agreed on the necessity to combine these two streams of information, a clear consensus on the initial transmission tree has not been reached,,. Here, we aimed to reconstruct the early stage of this outbreak using 13 full SARS genomes collected from the putative index patient and primary and secondary cases, and previously published estimates of the generation time distribution (Fig. S12).

12The genetic diversity amongst isolates was limited, with less than 15 mutations separating any pair of genomes (Fig. S13). For most cases, transmission events could not be readily inferred from the phylogenetic tree (Fig. S14). According to previous estimates of the mutation rate, we expect that most direct transmissions (>99%) will exhibit between 0 and 5 mutations. Using this result, we performed a simple graph analysis to derive possible clusters of direct transmissions, which suggested the existence of one main cluster of cases that may be linked directly, the remaining 4 isolates falling into three groups (Fig. S15). However, this crude analysis only relied on genetic diversity, and did not take into account information on the collection dates of the isolates or on the duration of the infectious period.

13We used outbreaker to exploit all these data simultaneously. Results of the inferred likely scenarios (Fig. 5 and 6) show that for half of the cases, a well-supported ancestor can be identified from the data (see also Fig. S16). These correspond to all of the first and second generations of infections (Sin2677, Sin2679, Sin2748, Sin2774) and to the last sampled case (Sin850). Ancestries of most cases were compatible with a single generation, although one or two unobserved infections may have taken place between Sin849 and Sin850 (Fig. S17). We found no evidence for separate index cases after Sin2500, in agreement with contact tracing information,,. However, the small number of cases may impair the detection of outliers and thus the identification of imported cases, so that multiple introductions of the pathogen cannot be ruled out.

14The most recent investigation of this outbreak suggested a dual introduction of the pathogen, with a separate index case (Sin2679) nearly 20 days after the initial index case Sin2500,,. This may be deemed surprising as this case is genetically close to some preceding cases (Fig. S14, S15). Here, our results suggest that Sin2679 would in fact be part of the second generation of infection, and was infected by Sin2748 (Fig. 5 and 6). Indeed, while the collection dates of Sin2748 and Sin2679 are relatively close, the generation time of SARS (Fig. S12) may have allowed for this transmission to occur. Closer examination of the patterns of mutations between Sin2500, Sin2748 and Sin2679 bring further support to this scenario (Fig. 6, Data S3). Indeed, the four mutations separating Sin2500 from Sin2679 are the simple addition of the mutations accumulated on the chain of transmission, from Sin2500 to Sin2748 (position 26,430: a→g), and from Sin2748 to Sin2679 (18,284: c→a; 19,086: t→c; 23,176: c→t).

15Building on past work,, we have presented a flexible analytical framework for the reconstruction of densely sampled outbreaks from epidemiological and sequence data. We extended previous work by accounting for unobserved cases and proposing a new approach for identifying multiple introductions of the pathogens based on the detection of genetic outliers. Our method is also the first tool for outbreak reconstruction widely available as a free software (the R package ‘outbreaker’) and able to run on standard desktop computers. The analysis of simulated data suggests that our approach will be applicable to a wide range of pathogens with various basic reproduction numbers, generation time distributions, and genetic diversity. We have shown how our approach can be used to infer effective reproduction numbers at an individual level. Importantly, this allows for detecting differences in infectivity of different groups of cases, and for the identification of super-spreaders. Our results suggest that while epidemiological data may suffice for the estimation of mean aggregated quantities such as the mean effective reproduction number, R, genetic data are useful to tease individual heterogeneities apart.

16As in other tree reconstruction methods,,,, we did not explicitly model the population of susceptible individuals. This is because information on individuals who were not infected during the outbreak (the “denominator” data) is quite often unavailable. Compared with case-only analyses, availability of denominator data also makes it possible to estimate the force of infection and risk factors for infection. We note that our framework could easily be extended to model the uninfected population. This could be done by modifying our likelihood so that the probability of the time of infection of a case would be based on an explicit model of the force of infection; individuals not infected during the outbreak would also contribute to the epidemiological likelihood as is standard in such situations. Integrating and validating these additional features in our approach will be the subject of future research.

17Our method relies on several assumptions which can be used to define the scope of its possible applications. The most important element in this respect is the proportion of cases represented in the sampled data, and thus often the scale of the epidemics considered. Our approach aims to reconstruct ancestries in closely related cases. As such, it should be most useful for detailed outbreak investigations. While the reconstruction of transmission tree seems relatively robust to large proportions of unobserved cases (up to 75% of missing cases, Fig. 1), our method is clearly tailored to densely sampled outbreaks, and not meant for the analysis of large-scale, more sparsely sampled epidemics. In such cases, phylogenetic methods are preferred as they explicitly reconstruct unobserved common ancestors of the sampled pathogen genomes, and can be used to infer, if not the transmission tree, the past dynamics of the disease,,.

18One of the novelties of our approach is the detection of imported cases, which are identified as genetic outliers. While this method should be useful to detect separate introductions of different pathogenic lineages in an epidemic, it may be sensitive to other events prone to creating genetic outliers, such as sequencing errors or recombination. Care should therefore be devoted to ensuring data quality and filtering out polymorphism due to recombination. Moreover, the assumption that imported cases are genetically distinguishable from other cases may not always be true, especially when multiple introductions take place from a closely related lineage. Such cases cannot be detected by genetic data only, and would require other sources of information (e.g. contact tracing) to be considered. In this respect, an interesting feature of outbreaker is the ability to fix known imported cases (as well as any other known transmissions) before reconstructing the transmission tree.

19Another important point is that following a previous, widely-used approach for the analysis of outbreaks, we assume the distributions of the generation time and of the time from infection to sample collection to be known. In some situations such as outbreaks of new emerging pathogens, accurate estimates of the generation time may not be readily available. In this case, a conservative approach should allow for a wide range of possible times to infection, at the expense of increased uncertainty in the inferred ancestries. As our method is numerically efficient for the analysis of small outbreaks, we suggest testing different generation time distributions to assess the robustness of the results. As a longer-term alternative, our approach could be extended to include an explicit parameterization and estimation of the generation time distribution.

20More fundamentally, the use of a generation time distribution also implies that our method is less appropriate for diseases in which long periods of asymptomatic carriage are frequent. For instance, bacteria such as Staphylococcus aureus can cause infections after months of asymptomatic colonization of the host, but may equally cause outbreaks of cases linked by only a few days,. In such cases, the collection dates of isolates effectively carry less information about possible transmissions, which would hamper our current approach. However, our model could be adapted to the analysis of carried pathogens by incorporating specific data on known exposures (e.g. shared occupancy on a hospital ward),,.

21Moreover, carried pathogens are also more likely to cause multiple colonizations of the host, resulting in several lineages coexisting within the same patient. Our model assumes that a single pathogen genome exists within each host, and is therefore not designed to account for multiple infections. A simple workaround would consist in duplicating cases of multiple infections into single infections, assuming that multiple infections are made of independent, single colonization events. However, this would not allow for disentangling multiple infections from mere within-host evolution of a single lineage. A more satisfying approach would consist in modeling explicitly the evolution of isolates within host, but this will likely result in a much more complex model and is beyond the remit of our current approach.

22A major simplification made in our model, that could be relaxed in future work, is that we do not consider within host diversity of pathogens. Within-host diversity is particularly prominent in pathogens that infect a host for a long time relative to their within-host replication cycle (e.g. HIV or Hepatitis C Virus), pathogens that can be carried for a long time (e.g. Staphylococcus aureus), pathogens where the infectious inoculum is large (e.g. blood-transmitted HIV), or super-infection is frequent (e.g. Streptococcus pneumoniae in hyper-endemic settings). Limited host diversity leads us to assume that genomes sampled from infectors are effectively ancestral to genomes sampled from secondary cases, allowing us to equate phylogenetic and transmission trees. This substantially reduces the complexity of the inferential problem, and reduces by orders of magnitude the dimensionality of the space of linked augmented variables to be explored. The assumption of no within-host diversity will likely be appropriate for acute infectious pathogens in outbreaks, but will also be relatively appropriate for situations where there is a strong bottleneck on diversity upon transmission and limited opportunities for superinfection, such as sexually transmitted HIV. Inclusion of within-host diversity in the model inference is an important but likely complex task, though efficient approximations may be possible. A related development will be the inclusion of multiple samples per individual, used to sample cross-sectional and longitudinal genetic diversity within infected hosts. Another somewhat simpler extension would be the inclusion of a ‘relaxed’ molecular clock, which would allow accounting for heterogeneities in mutation rates amongst different pathogen lineages.

23Finally, we wish to emphasize the importance of including all available prior information in the analysis. Because the estimates of parameters governing an outbreak are often correlated, accurate knowledge of one can be used to refine the estimation of the others. For instance, specifying known transmission chains or imported cases will improve the estimation of the mutation rates, as well as the overall reconstruction of the transmission tree. Conversely, fixing the mutation rate to its ‘true’ value (or a good estimate thereof) is likely to improve the detection of imported cases. As currently implemented, our method allows for fixing any parameter as well as individual ancestries, which are used in the likelihood computations but not changed during the MCMC. This feature should be especially useful for incorporating known transmission events or introductions of the pathogen into the population, based for instance on clinical investigations and contact tracing information. However, results of contact tracing studies should always be considered cautiously, and could be contradicted by the analysis of corresponding sequences, as illustrated by the SARS outbreak in Singapore.

24There are other promising avenues for incorporating various streams of information into our approach. The likelihood of our model allows for additional ‘plug-in’ terms for individual transmissions, which could be used to model spatial dispersion processes as well as movement over a contact network. Therefore, we hope that the present method will not only be applied widely, but also motivate further developments for the investigation of infectious disease outbreaks.

25We developed a discrete-time stochastic model for reconstructing likely transmission trees of an outbreak based on pathogen genetic sequences and their collection dates (see notations summary in Table 2 and Figure S18). Our model considers a single pathogen genome for each case. We note the genetic sequence of case, sampled at time. The function computes the number of mutations between and, while computes the number of nucleotide positions which can be compared between the two sequences. is the distribution of the generation time, defined as the time interval between the infection of an individual and his seeding of new secondary cases. is the distribution of the time interval between infection and collection of an isolate. Both and are assumed to be known, and are not part of the estimated parameters. Augmented data are used to model the transmission process, which is not observed directly,. We denote the index of the most recent sampled ancestor of case, and the number of generations separating cases and. For imported cases, is fixed to 0. The date of infection for case is denoted. We use the simplest model of sequence evolution considering one single mutation rate, measured per site and per generation of infection. Unlike approaches based on strict molecular clocks (e.g. ), a generational clock models the accumulation of genetic diversity with new infections while overlooking within-host evolution. Lastly, the parameter is the proportion of cases of the outbreak that have been sampled over the time span of the dataset, assuming a constant reporting rate over time.

26Our model is embedded within a Bayesian framework. We denote Y the observed data, A the augmented data, and the model parameters. The joint posterior distribution of parameters and augmented data is defined as:which is proportional to:where the first term is the likelihood of the data and augmented data, and the second, the joint prior distribution. Likelihood computations are described below. Priors and estimation procedures are described in Supporting Methods.

27The likelihood is computed as a product of case-specific terms, in which we assume that all cases are independent conditional on their ancestries:where is the probability of the first collection date given the first infection date, and is a constant. In the case of partially sampled transmission chains, several cases could share some common (unsampled) ancestry, and would thereby no longer be independent conditional on their most recent sampled ancestor. It follows that in the general case, Eq. 2 is not a true likelihood but a composite likelihood used to approximate the likelihood.

28The general term of the pseudo-likelihood for case is:which can be decomposed into:where is a constant. We refer to as the genetic pseudo-likelihood and to as the epidemiological pseudo-likelihood.

29As in, mutations are modeled as features of the transmission events. This is a direct corollary of the assumption of no within host diversity. This approach has the advantage of being computationally very efficient, as only the genetic distances between isolates need to be known to compute the pseudo-likelihood of a transmission event, and all transmission events are independent. The genetic pseudo-likelihood of case is defined as the probability of observing the genetic differences between the sequence and the ancestral sequence with and being separated by generations. In practice, if case has not been sequenced, we look for another ancestral sequence by moving up the transmission chain, replacing by the number of generations between the two compared sequences. Given the short timescale considered between pairs of sequences, reverse mutations are considered negligible. Accordingly, sites under strong selection such as immune epitopes or drug-resistance associated SNPs should be removed from the analyzed sequences. Assuming that all sites mutate independently and in the absence of reverse mutations, the genetic pseudo-likelihood is given by:

30The epidemiological pseudo-likelihood is computed as:The first term corresponds to the pseudo-likelihood of the collection date. The second term is the probability of the infection date for generations between the infection dates considered., where is the convolution operator. The last term is the probability of unobserved intermediate cases, modeled with a negative binomial distribution (equivalent to a geometric distribution with parameter p), indicating the probability of obtaining one ‘success’ (here, sampling a case) after ‘failures’ (unobserved cases) with a probability of success.

31Imported cases are not explicitly included in the model, but detected using a preliminary run of the model, during which genetic outliers are identified and the corresponding cases classified as imported. The ancestry of these cases is fixed as ‘unknown’ in the second and final run. We use a leave-one-out procedure for detecting cases with outlying genetic log-likelihood which has been used previously in a similar context. This approach defines the global influence of case (considering genetic data only) as:where denotes the expectation of the corresponding terms, approximated by the average over a number of samples (50 by default) from the MCMC of the preliminary run. Large values of reflect unlikely numbers of mutations, and therefore a probable genetic outlier. By default, cases with a global influence greater than 5 times the average global influence are classified as outliers. While this threshold is arbitrary, it was determined empirically to have excellent specificity and appreciable sensitivity on a range of simulation settings (see Fig. 2).

32Because our model uses a mutation rate expressed per generation of infection, estimated values cannot be readily compared to classical rates of evolution, typically expressed per unit of time. As a workaround, we can re-estimate a classical mutation rate from the distribution of posterior trees. The mutation rate can be inferred from one transmission event as the ratio of the number of mutations from ancestor to descendent and the amount of time separating the infection dates of these cases. For each tree, we compute the average mutation rate across all ancestries, which provides one estimate of the mutation rate for each posterior sample. This procedure is implemented in the function get.mu in outbreaker.

33Our approach is implemented in the R package outbreaker (version 1.1-0), freely available at: http://cran.r-project.org/web/packages/outbreaker/index.html.

34Outbreaks were simulated using the function simOutbreak in the package outbreaker. Each simulation starts with a single infection in a population of susceptible hosts. For simplicity, the same function was used for and. is the fixed basic reproduction number, and the number of susceptible hosts at time. The probability for a susceptible individual to become infected on day t is:At each time step, the number of new cases is drawn from a binomial distribution with draws and a probability. Infectors of a case infected at time ti are sampled from a multinomial distribution with probabilities:In addition to endogenous cases, external cases are imported at a constant rate.

35Mutations are simulated using a single mutation rate, all sites mutating independently. Pathogens of separate introductions of the disease (including the index case) are assumed to all coalesce to the same common ancestor ten generations ago.

36We evaluated the overall performance of the method using a basic scenario, and assessed the impact of different factors on the results by changing one aspect of the simulation at a time. These factors included the shape of the generation time distribution (from peaked to flat), the basic reproduction number (from 1.1 to 4), the mutation rates (from 0 to 2 mutations on average per generation and genome), the proportion of cases observed (from 0.25 to 1), the rate at which external cases are imported (from 0 to 0.2), and the proportion of sampled cases with DNA sequences (from 0.25 to 1). The different values for each element are summarized in Table 2. For each setting, 50 epidemics were simulated with 200 susceptible hosts and a minimum of 10 cases, and analyzed using outbreaker with the default settings, described in supporting information.

37In addition, two other types of simulation were used to test our approach's ability to detect heterogeneous infectivity amongst cases. First, we generated outbreaks where the host population was divided into two groups of equal sizes, one being twice as infectious (equivalent R0 = 3) as the other (equivalent R0 = 1.5). Second, we simulated outbreaks with super-spreader dynamics, were 5% cases were super-spreaders, with an equivalent R0 of 20, while the rest of the population had an equivalent R0 of 1.5. In both cases, 50 outbreaks with minimum sizes of 10 cases were simulated using a single pathogen introduction and 100 susceptible hosts, and fully sampled outbreaks were analysed using outbreaker, fixing values to 1 generation and using defaults otherwise. For the super-spreader simulations, super-spreaders were identified first from the data and their reproduction number compared to that of the non super-spreaders. For such comparisons, effective reproduction numbers of the different groups were calculated based on cases during the whole outbreak.

38Thirteen previously published full SARS genomes, (Data S1) were obtained from Genbank and aligned using MUSCLE. The resulting alignment contained 29,731 columns, 39 of which were polymorphic (Data S2). We used a generation time distribution modeled as a discretized gamma distribution with a mean of 8.4 days and a standard deviation of 3.8 days, using the function DiscrSI from the R package EpiEstim. The same distribution was used for the the time to collection. Details of the parameters used to run outbreaker are provided in Supporting Methods. The statistical confidence in determining the ancestry of a given case was quantified using the entropy of the frequencies of the posterior ancestors. With different ancestors of posterior frequencies, the entropy is defined as:The entropy is 0 if one of the, is 1, indicating high confidence in allocation of an ancestry, while larger values of the entropy indicate poorer confidence.