In the field of epidemiology, source attribution refers to a category of methods with the objective of reconstructing the transmission of an infectious disease from a specific source, such as a population, individual, or location. For example, source attribution methods may be used to trace the origin of a new pathogen that recently crossed from another host species into humans, or from one geographic region to another. It may be used to determine the common source of an outbreak of a foodborne infectious disease, such as a contaminated water supply. Finally, source attribution may be used to estimate the probability that an infection was transmitted from one specific individual to another, i.e., "who infected whom". Source attribution can play an important role in public health surveillance and management of infectious disease outbreaks. In practice, it tends to be a problem of statistical inference, because transmission events are seldom observed directly and may have occurred in the distant past. Thus, there is an unavoidable level of uncertainty when reconstructing transmission events from residual evidence, such as the spatial distribution of the disease. As a result, source attribution models often employ Bayesian methods that can accommodate substantial uncertainty in model parameters. Molecular source attribution is a subfield of source attribution that uses the molecular characteristics of the pathogen — most often its nucleic acid genome — to reconstruct transmission events. Many infectious diseases are routinely detected or characterized through genetic sequencing, which can be faster than culturing isolates in a reference laboratory and can identify specific strains of the pathogen at substantially higher precision than laboratory assays, such as antibody-based assays or drug susceptibility tests. On the other hand, analyzing the genetic (or whole genome) sequence data requires specialized computational methods to fit models of transmission. Consequently, molecular source attribution is a highly interdisciplinary area of molecular epidemiology that incorporates concepts and skills from mathematical statistics and modeling, microbiology, public health and computational biology. There are generally two ways that molecular data are used for source attribution. First, infections can be categorized into different "subtypes" that each corresponds to a unique molecular variety, or a cluster of similar varieties. Source attribution can then be inferred from the similarity of subtypes. Individual infections that belong to the same subtype are more likely to be related epidemiologically, including direct source-recipient transmission, because they have not substantially evolved away from their common ancestor. Similarly, we assume the true source population will have frequencies of subtypes that are more similar to the recipient population, relative to other potential sources. Second, molecular (genetic) sequences from different infections can be directly compared to reconstruct a phylogenetic tree, which represents how they are related by common ancestors. The resulting phylogeny can approximate the transmission history, and a variety of methods have been developed to adjust for confounding factors. Due to the associated stigma and the criminalization of transmission for specific infectious diseases, molecular source attribution at the level of individuals can be a controversial use of data that was originally collected in a healthcare setting, with potentially severe legal consequences for individuals who become identified as putative sources. In these contexts, the development and application of molecular source attribution techniques may involve trade-offs between public health responsibilities and individual rights to data privacy.
Microbial subtyping Microbial subtyping or strain typing is the use of laboratory methods to assign microbial samples to subtypes, which are predefined classifications based on distinct characteristics. The assignment of specimens to subtypes can provide a basis of source attribution, since we assume that a pathogen undergoes minimal change when transmitted to an uninfected host. Therefore, infections of the same subtype are implied to be epidemiologically related, i.e., linked by one or more recent transmission events. The assumption that the pathogen is unchanged when transmitted is generally reasonable if the rate of evolution for the pathogen is slower than the rate of transmission, such that few mutations are observed on an epidemiological time scale. For example, suppose host A is infected by a pathogen that we have categorized as subtype 1. They are more likely to have been infected by host B, who also carries the subtype 1 pathogen, than host C who carries the subtype 2 pathogen (Figure 1). In other words, transmission from host B is a more parsimonious explanation if there is a relatively small probability that the pathogen population in host C evolved from subtype 1 to subtype 2 after transmission to host A.
Today it is more common to use genetic sequencing to characterize the microbial sample at the level of its nucleotide sequence by sequencing the whole genome or proportions thereof. However, other molecular methods such as restriction length fragment polymorphism
have historically played an important role in microbial subtyping before genetic sequencing became an affordable and ubiquitous technology in reference laboratories. Sequence-based typing methods confer an advantage over other laboratory methods (such as serotyping or pulsed-field gel electrophoresis ) because there is an enormous number of potential subtypes that can be resolved at the level of the genetic sequence. Consider the above example again; however, this time host A carries the same infection subtype as many other hosts. In this case we would have no information to differentiate between these hosts as the potential source of host A's infection. Our ability to identify potential sources, therefore, depends on having a sufficient number of different subtypes. However, defining too many subtypes in the population makes it likely that every individual carries a unique subtype, especially for rapidly-evolving pathogens that can accumulate high levels of genetic diversity in a relatively short period of time. Hence, there exists an intermediate level of subtype resolution that confers the greatest amount of information for source attribution. When source attribution is considered for a pathogen with high diversity, such that most specimens have unique genetic sequences, it is useful to group multiple unique sequences with a clustering method.
Single and multi-locus typing Before whole-genome sequencing was cost-effective, targeting a specific part of the pathogen genome (a.k.a. single-locus typing) was an important step to facilitate microbial subtyping. For example, the ribosomal gene 16S is a standard target for identifying bacteria, in part because it is present across all known species and contains a mixture of conserved and variable regions. Within a pathogen species, sequencing targets tended to be selected on the basis of their length, ubiquity and exposure to diversifying selection, which may be dictated by the function of the gene product for expressed regions. For example, so-called "housekeeping" or core genes have indispensable biological functions, such as copying genetic material or building proteins. These genes are often preferred candidates for microbial subtyping because they are less likely to be absent from a given genome. Gene presence/absence is particularly relevant for bacteria where genetic material is frequently exchanged through horizontal gene transfer. Targeting multiple regions (loci) of the pathogen genome confers greater precision to distinguish between lineages, since the chance to observe informative genetic differences between infections is increased. This approach is referred to as multi-locus sequence typing (MLST). Similar to single-locus typing, MLST requires the selection of specific loci to target for sequencing. Moreover, for subtyping to be consistent across laboratories a reference database must be maintained that maps sequences from single or multiple loci to a fixed notation of allele numbers or designations.
Whole genome sequencing Although single- and multiple-locus subtyping is still predominantly used for molecular epidemiology, ongoing improvements in sequencing technologies and computing power continue to lower the barrier to whole-genome sequencing. Next-generation sequencing (NGS) technologies provide cost-effective methods to generate whole genome sequences from a given sample by individually amplifying and sequencing templates in parallel using customized technologies such as sequencing-by-synthesis. Shotgun sequencing applications of NGS generate full-length genome sequences by shearing the nucleic acid extracted from the sample into small fragments that are converted into a sequencing library, and then using a de novo sequence assembler program the genome sequence is reconstituted from the sequence fragments (short reads). Alternatively, short reads can be mapped to a reference genome sequence that has been converted into an index for efficient lookup of exact substring matches. This approach can be faster than de novo assembly, but relies on having a reference genome that is sufficiently similar to the genome sequence of the sample. While NGS makes it feasible to simultaneously generate full-length genome sequences from hundreds of pathogen samples in a single run, it introduces a number of other challenges. For instance, NGS platforms tend to have higher sequencing error rates than conventional sequencing, and regions of the genome with long stretches of repetitive sequence can be difficult to reassemble. Whole genome sequencing (WGS) can confer a significant advantage for source attribution over single- or multiple-locus subtyping. Sequencing the entire genome is the maximal extent of multi-locus typing, in that all possible loci are covered. Having whole genome sequences will tend to make one-to-one subtyping (Figure 1) less useful, since most genomes will be unique by at least one mutation for rapidly evolving pathogens. Consequently, applications of WGS for source attribution at a population level will likely have to cluster similar genomes together. The breadth of coverage offered by WGS is more advantageous for the epidemiology of bacterial pathogens than viruses. Bacterial genomes tend to be longer, ranging from about 106 to 107 base pairs, whereas virus genomes seldom exceed 106 base pairs. In addition, bacteria tend to evolve at a slower rate than viruses, so mutations tend to be distributed more sparsely throughout a bacterial genome. For example, WGS data revealed differences between isolates of Burkholderia pseudomallei from Australia and Cambodia that had otherwise appeared to be identical by multi-locus subtyping due to convergent evolution. WGS has also been utilized in several recent studies to resolve transmission networks of Mycobacterium tuberculosis in greater detail, because isolates with identical multi-locus subtypes (e.g., MIRU-VNTR profiles targeting 24 loci) were frequently separated by large numbers of nucleotide differences in the full genome sequence, comprising roughly 4.3 million nucleotides encodoing over 4,000 genes.
Genetic clustering When applied to genetic sequences, a clustering method is a set of rules for assigning the sequences to a smaller number of clusters such that members of the same cluster are more genetically similar to each other than sequences in other clusters. Put another way, a clustering method defines a partition on the set of genetic sequences using some similarity measure. Clustering is inherently subjective and there are usually no formal guidelines for setting the clustering criteria. Consequently, cluster definitions can vary substantially from one study to the next. In addition, clustering is an intuitive process that can be accomplished by a wide variety of approaches; because of this flexibility, numerous different methods of genetic clustering have been described in the literature. Genetic clustering provides a way of dealing with sequences from rapidly evolving pathogens, or whole genome sequences from pathogens with less divergence. In either case, there can be an enormous number of distinct genetic sequences in the data set. If each subtype must correspond to a unique sequence variant, then one could potentially have to track an unwieldy number of microbial subtypes for these pathogens when subtypes are defined on a one-to-one basis (Figure 1). The number of subtypes can be greatly reduced by expanding the definition of microbial subtypes from individually unique sequence variants to clusters of similar sequences. For example, pairwise distance clustering is a nonparametric approach in which clusters are assembled from pairs of sequences that fall within a threshold distance of each other. The distance between sequences is computed by a genetic distance measure (a mathematical formula that maps two sequences to a non-negative real number) that quantifies the evolutionary divergence between the sequences under some model of molecular evolution.
Frequency-based attribution When the potential sources are populations, not individuals, then we are comparing the frequencies of subtypes in the respective populations. The most likely source population should have a subtype frequency distribution that is the most similar to the reference population. Methods that employ this approach have been referred to as "frequency-based" or "frequency-matching" models. These subtypes are not necessarily derived from molecular data; for instance, these methods were originally applied to microbial strains defined by non-genetic antigenic or resistance profiling. For example, the "Dutch model"
was originally developed to estimate the most likely source of a number of foodborne illnesses due to Salmonella by comparing the relative frequencies of bacterial subtypes (based on phage typing) in different commercial livestock populations (including poultry, swine and cattle) through routine surveillance programs. For a given subtype, the expected number of human cases attributed to each source is proportional to the relative frequencies of that subtype among sources:
λ i j = p i j ∑ j p i j n i {\displaystyle \lambda _{ij}={\frac {p_{ij}}{\sum _{j}p_{ij}}}n_{i}}
where p i j {\displaystyle p_{ij}} is the proportion of (non-human) cases in the j {\displaystyle j} -th source population associated with subtype i {\displaystyle i} , and n i {\displaystyle n_{i}} is the number of cases of subtype i {\displaystyle i} in the recipient (human) population. For instance, if the frequencies of subtype X among three potential sources was 0.8, 0.5 and 0.1, respectively, then the expected number of cases (out of a total of 100) from the second source is 0.5 / ( 0.8 + 0.5 + 0.1 ) × 100 = 35.7 {\displaystyle 0.5/(0.8+0.5+0.1)\times 100=35.7} . This simple formula is a maximum likelihood estimator when the total force of infection from each source into the human population is uniform, e.g., the sources have equal population sizes.
Subsequently, this model was extended by Hald and colleagues
to account for variation among sources and subtypes using Bayesian inference methods. This extension, typically referred to as the Hald model, has become a standard model in source attribution for food-borne illnesses. The observed numbers of each subtype in the human population was assumed to be a Poisson distributed outcome with a mean λ i {\displaystyle \lambda _{i}} for the i-th subtype, after adjusting for cases related to travel and outbreaks:
λ i = ∑ i λ i j = ∑ i q i M j a j p i j {\displaystyle \lambda _{i}=\sum _{i}\lambda _{ij}=\sum _{i}q_{i}M_{j}a_{j}p_{ij}}
where q i {\displaystyle q_{i}} is the marginal effect of the i-th subtype (e.g., elevated infectiousness of a bacterial variant), M j {\displaystyle M_{j}} is the observed total amount (mass) of the j-th food source, a j {\displaystyle a_{j}} is the marginal effect of the j-th food source, and p i j {\displaystyle p_{ij}} is the same observed case proportion as the original "Dutch" model. This model is visualized in Figure 2.
Bayesian inference The addition of a large number of parameters to the "Dutch" model by Hald and colleagues yielded a more realistic model. However, it was too complex to solve for exact maximum likelihood estimates, in contrast to the original model. Many of the parameters could not be directly measured, such as the relative transmission risk associated with a specific food source. Consequently, Hald and colleagues adopted a Bayesian approach to estimate the model parameters. A similar approach has also been used to reconstruct the contribution of different environmental and livestock reservoirs of the bacteria Campylobacter jejuni to an outbreak of food poisoning in England, where the migration of different subtypes among reservoirs was jointly estimated by Bayesian methods. Although Bayesian inference is discussed extensively elsewhere, it plays an important role in computationally-intensive methods of source attribution, so we provide a brief description here. In the context of Bayesian inference every parameter is described by a probability distribution that represents our belief about its true value. Thus, the statistical principle that underlies Bayesian inference (i.e., Bayes' rule) can be expressed in terms of the model parameters ( θ {\displaystyle \theta } ) and the data ( D {\displaystyle D} ):
P ( θ ∣ D ) ∝ P ( D ∣ θ ) P ( θ ) {\displaystyle P(\theta \mid D)\propto P(D\mid \theta )P(\theta )}
where P ( θ ∣ D ) {\displaystyle P(\theta \mid D)} , P ( D ∣ θ ) {\displaystyle P(D\mid \theta )} and P ( θ ) {\displaystyle P(\theta )} are known as the posterior, sampling (likelihood), and prior distributions, respectively. A simple way to think about Bayesian inference is that our prior belief about the parameters is "updated" once we have seen the data. As a result, our posterior belief becomes a compromise between our prior belief and the data. To update our belief, we need to have a sampling distribution or model that describes the probability of different outcomes of an experiment. We also require a prior distribution that represents our belief in a statistical form. While modern computation allows almost any probability distribution to be used, the uniform distribution is commonly used because it assigns the same probability to every value within some range. After incorporating new information from the data, our updated belief about the model parameters is represented by the posterior distribution. This use of distributions to represent our belief distinguishes Bayesian inference from maximum likelihood, which results in a single combination of parameter values as a point estimate. Hald and colleagues used uniform prior distributions for many of their parameters to express the prior belief that the true value fell within a continuous range with specific upper and lower limits. They constrained some parameters to take the same numerical value as others. For example, the effects of domestic and imported supplies of the same food source were linked in this manner. This assumption expressed a strong belief that a given food source carried the same transmission risk irrespective of its origin, and simplified the model so that it was more feasible to fit the data. Other parameters were set to a fixed reference value to further simplify the model. Hald and colleagues employed a Poisson model to describe the probability of observing the number ( Y {\displaystyle Y} ) of rare transmission events that occur at a rate λ {\displaystyle \lambda } . As described above, the rate of cases due to a specific bacterial subtype was the sum of transmission rates across all potential sources. The Hald model was more realistic than the "Dutch" model because it allowed transmission rates to vary between subtypes and food sources. However, it was not feasible to directly measure these different rates — these parameters needed to be estimated from the data.
Comparative methods Instead of comparing the frequencies of subtypes to reconstruct the transmission of pathogens between populations, many source attribution methods compare the pathogen sequences at the level of individual hosts. One way of comparing sequences is to calculate some measure of genetic distance or similarity, a concept that we introduced earlier on the topic of pooling sequences into composite subtypes. For example, infections that are grouped into clusters are assumed to be related through one or more recent and rapid transmission events. Short genetic distances imply that limited time has passed for mutations to accumulate in lineages descending from their common ancestor. Consequently, these clusters are often referred to as "transmission clusters". Other studies have used genetic distances that exceed some threshold to rule out host individuals as potential sources of transmission. Although this application of clustering is related to source attribution, it is not possible to infer the direction of transmission solely from the genetic distance between infections. Furthermore, the genetic distance separating infections is not solely determined by the rate of transmission; for example, they are strongly influenced by how infections are sampled from the population. Sequences can also be compared in the context of their shared evolutionary history. A phylogenetic tree or phylogeny is a hypothesis about the common ancestry of species or populations. In the context of molecular epidemiology, phylogenies are used to relate infections in different hosts and are usually reconstructed from genetic sequences of each pathogen population. To reconstruct the phylogeny, the sequences must cover the same parts of the pathogen genome; for example, sequences that represent multiple copies of the same gene from different infections. It is this residual similarity (homology) between diverging populations that implies recent common ancestry. A molecular phylogeny comprises "tips" or "leaves" that represent different genetic sequences that are connected by branches to a series of common ancestors that eventually converge to a "root". The composition of the ancestral sequence at the root, the order of branching events, and the relative amount of change along each branch are all quantities that must be extrapolated from the observed sequences at the tips. There are multiple approaches to reconstruct a phylogenetic tree from genetic sequence variation. For example, distance-based methods use a hierarchical clustering method to build up a tree based on the observed genetic distances.
Phylogenetic uncertainty A common simplifying assumption in phylogenetic investigations is that the phylogenetic tree reconstructed from the data is the "true" tree — that is, an accurate representation of the common ancestry relating the sampled infections. For instance, a single tree is often used as the input for comparative methods to detect the signature of natural selection in protein-coding sequences. On the other hand, if the phylogeny is handled as an uncertain estimate derived from the data (including the sequence alignment), then the analysis becomes a hierarchical model in which the problem of phylogenetic reconstruction is nested within the problem of estimating the other model parameters that are conditional on the phylogeny (Figure 3). Sampling both the phylogeny and other model parameters from their joint posterior distribution using methods such as Markov chain Monte Carlo (MCMC) should confer more accurate parameter estimates. However, the greatly expanded model space also makes it more difficult for MCMC samples to converge to the posterior distribution. Such hierarchical methods are often implemented in the software package BEAST2
(Bayesian Evolutionary Analysis by Sampling Trees), which provides generic routines for MCMC sampling from tree space, and calculates the likelihood of a time-scaled phylogenetic tree given sequence data and sample collection dates. There are a number of sources of phylogenetic uncertainty. For instance, the common ancestry of lineages can be difficult to reconstruct if there has been little to no evolution along the respective branches. This can occur when the rate of evolution is substantially slower than the time scale of transmission, such that mutations are unlikely to accumulate between the start of one infection and its transmission to the next host (i.e., the generation time). It can also arise when existing divergence is not captured due to incomplete sequencing of the respective genomes. Furthermore, reconstructing the common ancestry of lineages is progressively more uncertain as we move deeper into the tree, forcing us to extrapolate the ancestral states at greater distances from the observed data.
Alignment Reconstructing phylogenies from molecular sequences generally requires a multiple sequence alignment, a table in which homologous residues in different sequences occupy the same position. Although alignments are often treated as observed data known without ambiguity, the process of aligning sequences is also uncertain and can become more difficult with the rapid accumulation of sequence insertions and deletions among diverging pathogen lineages. While there are Bayesian methods that address uncertainty in alignment by joint sampling of the alignment along with the phylogeny, this approach is computationally complex and is seldom used in the context of source attribution. Furthermore, sequences are themselves uncertain estimates of the genetic composition of individual pathogens or infecting populations, and next-generation sequencing technologies tend to have substantially higher error rates than conventional Sanger sequencing, and analysis pipelines must be carefully validated to reduce the effects of sample cross-contamination and adapter contamination.
Recombination Genetic recombination is the exchange of genetic material between individual genomes. For pathogens, recombination can occur when a cell is infected by multiple copies of the pathogen. If some hosts were infected multiple times by two or more divergent variants from different sources (i.e., superinfection), then recombination can produce mosaic genomes that complicate the reconstruction of an accurate phylogeny. In other words, different segments of a recombinant genome may be related to other genomes through discordant phylogenies in such a way that cannot be accurately represented by a single tree. In practice, it is common to screen for recombinant sequences and discard them before reconstructing a phylogeny from an alignment that is assumed to be free of recombination.
Inferring transmission history from the phylogeny The basic premise in applying phylogenetics to source attribution is that the shape of the phylogenetic tree approximates the transmission history, which can also be represented by a tree where each split into two branches represents the transmission of an infection from one host to another. In conjunction with reconstructing the transmission tree from other sources of information, such as contact tracing, reconstructing a phylogenetic tree can serve as a useful, additional information source especially when genetic sequences are already available. Because of the visual and conceptual similarity between phylogenetic and transmission trees, it is a common assumption that the branching points (splits) of the phylogeny represent transmission events. However, this assumption will often be inaccurate. A transmission event may have occurred at any point along the two branches that separate one sampled infection from the other in the virus phylogeny (Figure 3A). The transmission tree only constrains the shape of the phylogenetic tree. Thus, even if we can reconstruct the phylogenetic tree without error, there are several reasons why it will not be an accurate representation of the transmission tree, including incomplete sampling, pathogen evolution within hosts, and secondary infection of the same host.
Incomplete sampling Equating the phylogenetic tree with the transmission history implicitly assumes that genetic sequences have been obtained from every infected host in the epidemic. In practice, only a fraction of infected hosts are represented in the sequence data. The existence of an unknown and inevitably substantial number of unsampled infected hosts is a major challenge for source attribution. Even if the phylogenetic tree indicates that two infections are most closely related than any other sampled infection, one cannot rule out the existence of one or more unsampled hosts whom are intermediate links in the "transmission chain" separating the known hosts (Figure 3B). Similarly, an unsampled infection may have been the source population for both observed infections at the tips of the tree (Figure 3C). By itself, the phylogenetic tree does not explicitly discriminate among these alternative transmission scenarios.
Evolution within hosts The shape of the phylogenetic tree may diverge from the underlying transmission history because of the evolution of diverse populations of the pathogen within each host. Individual copies of the pathogen genome that are transmitted to the next host are, by definition, no longer in the source population. A split exists in the phylogenetic tree that represents the common ancestor between the transmitted lineages and the other lineages that have remained and persisted in the source population. If we follow both sets of lineages back in time, the time of the transmission event is the most recent possible time that they could converge to a common ancestor. Put another way, this event represents one extreme of a continuous range where the common ancestor is located further back in time. This process is often modelled by Kingman's coalescent, which describes the number of generations we expect to follow randomly selected lineages back in time until we encounter a common ancestor. The expected time until two lineages converge to a common ancestor, known as a coalescence event, is proportional to the effective population size, which determines the number of possible ancestors. Put another way, two randomly selected people in a large city are less likely to have a great-grandparent in common than two people in a small rural community. Longer coalescence times in a large, diverse within-host pathogen populations are a significant challenge for source attribution, because it uncouples the virus phylogeny from the transmission tree. For example, if a host has transmitted their infection to two others, then there can be as many as three sets of lineages whose ancestry can be traced in the source population in that host (Figure 3D). As a result, there is some chance that the branching order in the virus phylogeny implies a different order of transmission events if we interpret the phylogeny as equivalent to a transmission tree. For example, in Figure 3D hosts 1 and 3 are more closely related in the transmission history, but not in the phylogeny.
Clearance and secondary infection Many infections can be spontaneously cleared by the host's immune system. If a host that has cleared a previously diagnosed infection becomes re-infected from another source, then it is possible for the same host to be represented by different infections in the phylogenetic and transmission trees, respectively. In addition, some individuals may become infected from multiple different sources. For example, roughly one-third of infections by hepatitis C virus are spontaneously cleared within the first six months of infection. This previous exposure, however, does not confer immunity to re-infection by the same virus. In addition, co-infection by multiple strains of hepatitis C virus that persist simultaneously within the same host can occur relatively frequently in populations with a high rate of transmission, such as people who inject drugs using shared equipment (ranging from 14% to 39%). The persistence of strains from additional exposures may be missed by conventional genetic sequencing techniques if they are present at a low frequencies within the host, necessitating the use of "next-generation" sequencing technologies. For these reasons, the epidemiological linkage of hepatitis C virus infections through genetic similarity may be a transient phenomenon, leading some investigators to recommend using multiple virus sequences sampled from different time points of each infection for molecular epidemiology applications.
Ancestral host-state reconstruction Ancestral reconstruction is the application of a model of evolution to a phylogenetic tree to reconstruct character states, such as nucleotide sequences or phenotypes, at the different ancestral nodes of the tree down to the root. In the context of source attribution, ancestral reconstruction is frequently used to estimate the geographic location of pathogen lineages as they are carried from one region to another by their hosts. Drawing this analogy between character evolution and the spatial migration of individuals or populations is known as phylogeography, where the geographic location of an ancestral population is reconstructed from the current locations of its sampled descendants under some model of migration. Migration models generally fall into two categories of discrete-state and continuous-state models. Discrete-state or island migration models assume that a given lineage is in one of a finite number of locations, and that it changes location at a constant rate over time according to a continuous-time Markov process, analogous to the models used for molecular evolution. Ancestral reconstruction with a discrete-state migration model has also been utilized to reconstruct the early spread of HIV-1 in association with development of transport networks and increasing population density in central Africa. Discrete models can also be applied to the population-level source attribution of zoonotic transmissions by reconstructing different host species as ancestral character states. For example, a discrete trait model of evolution was used to reconstruct the ancestral host species in a phylogeny relating Staphylococcus aureus specimens from humans and domesticated animals. Similarly, Faria and colleagues
analyzed the cross-species transmission of rabies virus as a discrete diffusion process along the virus phylogeny, with rates influenced by the evolutionary relatedness and geographic range overlap of the respective host species. Continuous-state migration models are more similar to models of Brownian motion in that a lineage may occupy any point within a defined space. Although continuous models can be more realistic than discrete migration models, they may also be more challenging to fit to data. Taken literally, a continuous model requires precise geolocation data for every infection sampled from the population. In many applications, however, these metadata are not available; for example, some studies approximate the true spatial distribution of sampled infections by the centroids of their respective regions. This can become problematic if the regions vary substantially in area, and host populations are seldom uniformly distributed within regions.
Paraphyly Paraphyly is a term that originates from the study of cladistics, an evolutionary approach to systematics that groups organisms on the basis of their common ancestry. A group of infections is paraphyletic if the group includes the most recent common ancestor, but does not include all its descendants. In other words, one group is nested within an ancestral group. For example, birds are descended from a common ancestor that in turn shares a common ancestor with all reptiles; thus, birds are nested within the phylogeny of reptiles, making the latter a paraphyletic group. Thus, paraphyly is evidence of evolutionary precedence: the ancestor of all birds was a reptile. In the context of source attribution, paraphyly can be used as evidence that one infection preceded another. It does not provide evidence that the infection was directly transmitted from one individual to another, in part because of incomplete sampling. The application of paraphyly for source attribution requires that the phylogenetic tree relates multiple copies of the pathogen from both the putative source and recipient hosts. To elaborate, phylogenetic trees relating different infections are often reconstructed from population-based sequences (direct sequencing of the PCR amplification product), where each sequence represents the consensus of the individual pathogen genomes sampled from the infected host. If copies of the pathogen genome are sequenced individually by limiting dilution protocols or next-generation sequencing, then one can reconstruct a tree that represents the genealogy