The earliest use of phylogenetics in infectious disease epidemiology was to confirm or rule out a suspected source of the human immunodeficiency virus ( HIV ) ., Phylogenetic analyses were used to confirm that five HIV patients were infected at a dental practice in Florida between 1987 and 1989 19 and to rule out infection of a Baltimore patient in 1985 by an HIV-positive surgeon 20 ., A more ambitious use of phylogenetics is to reconstruct a transmission tree , which is a directed graph with an edge from node i to node j if person i infected person j ., An analysis by Leitner et al . 21 , 22 of an HIV-1 transmission cluster in Sweden from the early 1980s compared reconstructed phylogenies based on HIV genetic sequences to a true phylogeny based on the known transmission tree , times of transmission , and times of sequence sampling ., The reconstructed phylogenies accurately reflected the topology of the true phylogeny , and the accuracy increased when sequences from different regions of the HIV genome were combined ., The increasing availability of whole-genome sequence data has renewed interest in combining pathogen genetic sequence data with epidemiologic data to reconstruct transmission trees ., One approach to this problem is to reconstruct the transmission tree using genetic distances ., Spada et al . 23 reconstructed the transmission tree linking five children infected with hepatitis C virus ( HCV ) by finding the spanning tree linking the HCV genetic sequences that minimized the sum of the genetic distances across its edges , excluding edges inconsistent with the epidemiologic data ., The SeqTrack algorithm of Jombart et al . 24 generalizes this approach ., It constructs a transmission tree by finding the spanning tree linking the sampled sequences that minimizes ( or maximizes ) a set of edge weights ., Snitkin et al . 25 used this algorithm to investigate a 2011 outbreak of carbapenem-resistant Klebsiella pneumoniae in the NIH Clinical Center , penalizing edges with large genetic distances , between patients who did not overlap in the same ward , or that required a long silent colonization ., Wertheim et al . 26 constructed a network among HIV patients in San Diego by linking individuals whose sequences were <1% distant ., This was used to estimate community-level effects of HIV prevention and treatment ., A second approach to transmission tree reconstruction is to weight possible infector-infectee links using a pseudolikelihood based on genetic and epidemiologic data ., Ypma et al . 27 analyzed a 2003 influenza A ( H7N7 ) outbreak among poultry farms in the Netherlands by combining data on the times of infection and culling at each farm , the distances between the farms , and RNA consensus sequences ., The weight of each possible transmission link was the product of components based on temporal , geographic , and genetic data ., The weight of a complete transmission tree was the product of the edge weights ., The R package outbreaker implements an extension of this approach that allows multiple introductions of infection and unobserved cases 28 ., Like the spanning tree methods above , these methods model pathogen evolution as a process that occurs at the moment of transmission ., Morelli et al . 29 proposed a variation of these methods that allows within-host pathogen evolution by incorporating the times of infection and observation into the likelihood component for the genetic sequence data ., A third approach is to reconstruct the transmission tree by combining a phylogeny with epidemiologic data , which was first done by Cottam et al . 30 , 31 in an investigation of a 2001 foot-and-mouth disease virus ( FMDV ) outbreak among farms in the United Kingdom ( UK ) ., The phylogeny and the transmission tree were linked by considering possible locations of the most recent common ancestors ( MRCAs ) of viruses sampled from the farms ., The probability pij that farm i infected farm j was calculated using epidemiologic data on the oldest detected FMDV lesion and the dates of sampling and culling on each farm ., The weight of each possible transmission network was proportional to the product of the pij for all edges i → j ., Similar methods were used to track farm-to-farm spread of a 2007 FMDV outbreak 32 ., Gardy et al . 33 combined social network analysis with a phylogeny based on whole-genome sequences to construct a transmission tree for a tuberculosis outbreak in British Columbia ., Didelot et al . used the time of the most recent common ancestor ( TMRCA ) to identify possible person-to-person transmission events in studies of Clostridium difficile transmission in the UK 34 and Helicobacter pylori transmission in South Africa 35 ., In a study of Mycobacterium tuberculosis transmission in the Netherlands , Bryant et al . 36 ruled out transmission between individuals whose samples did not share a parent in the phylogeny ., Recent research has identified problems with using genetic sequence data to reconstruct transmission trees ., Simulations by Worby et al . 37 , 38 found that pairwise genetic distances cannot reliably identify sources of infection ., Methods based on phylogenies often underestimate the complexity of the relationship between the phylogenetic and transmission trees ., Branching events in a phylogeny do not necessarily correspond to transmissions , and the topology of the phylogenetic tree need not be the same as the topology of the transmission tree 39–41 ., These differences are especially important for diseases with significant within-host pathogen diversity and long latent or infectious periods 41 , 42 ., Ypma et al . 40 and Didelot et al . 42 have developed Bayesian methods that enforce consistency between phylogenetic and transmission trees in Markov chain Monte Carlo ( MCMC ) iterations ., More recently , Lau et al . 43 have outlined a Bayesian integration of epidemiologic and genetic sequence data that uses likelihoods based on survival analysis , but their approach does not use pathogen phylogenies directly , assuming that a single dominant lineage within each host can be transmitted ., Here , we build a systematic understanding of the relationship between pathogen phylogenies and transmission trees under much weaker assumptions about within-host evolution , allowing the incorporation of genetic sequence data into frequentist and Bayesian survival analysis of infectious disease transmission data ., Reconstruction of transmission trees is most useful to public health if it leads to generalizable scientific insights about disease transmission ., The transmission tree from one outbreak does not generalize to future outbreaks , but a phylogeny provides partial information about who-infected-whom ., Survival analysis provides a rigorous but flexible statistical framework for infectious disease transmission data that explicitly links parameter estimation to the set of possible transmission trees 44–46 ., In this framework , estimates of transmission parameters such as covariate effects on infectiousness and susceptibility and evolution of infectiousness over time in infectious individuals are based on sums or averages over all possible transmission trees ., Since a phylogeny linking pathogen samples from infected individuals constrains the set of possible transmission trees , pathogen genetic sequence data can be combined with epidemiologic data to obtain more efficient estimates of transmission parameters ., At any time , each individual i ∈ {1 , … , n} is in one of four states: susceptible ( S ) , exposed ( E ) , infectious ( I ) , or removed ( R ) ., Person i moves from S to E at his or her infection time ti , with ti = ∞ if i is never infected ., After infection , i has a latent period of length εi during which he or she is infected but not infectious ., At time ti + εi , i moves from E to I , beginning an infectious period of length ιi ., At time ti + εi + ιi , i moves from I to R , where he or she can no longer infect others or be infected ., The latent period εi is a nonnegative random variable , the infectious period ιi is a strictly positive random variable , and both have finite mean and variance ., If person i is infected , the time elapsed since the onset of infectiousness at time ti + εi is the infectious age of, i . After becoming infectious at time ti + εi , person i makes infectious contact with j ≠ i at time t i j = t i + ε i + τ i j * ., We define infectious contact to be sufficient to cause infection in a susceptible person , so tj ≤ tij ., The infectious contact interval τ i j * is a strictly positive random variable with τ i j * = ∞ if infectious contact never occurs ., Since infectious contact must occur while i is infectious or never , τ i j * ∈ ( 0 , ι i  or τ i j * = ∞ ., For each ordered pair ij , let Cij = 1 if infectious contact from i to j is possible and Cij = 0 otherwise ., For example , the Cij could be the entries in the adjacency matrix for a contact network ., However , we do not require that Cij = Cji ., We assume the infectious contact interval τ i j * is generated in the following way: A contact interval τij is drawn from a distribution with hazard function hij ( τ ) ., If τij ≤ ιi and Cij = 1 , then τ i j * = τ i, j . Otherwise , τ i j * = ∞ ., Survival analysis of infectious disease transmission data can be viewed as a generalization of discrete-time chain binomial models 47 to continuous time , and it includes parametric methods 44 , nonparametric methods 45 , and semiparametric relative-risk regression models 46 ., For simplicity , we use parametric methods and assume that exogenous infections are known ., Let the hazard of infectious contact from i to j at time τ after the onset of infectiousness in i be, h i j ( τ ) = exp β 0 ⊤ X i j ( τ ) h 0 ( τ ) , ( 1 ), where β0 is an unknown coefficient vector , Xij ( τ ) is a covariate vector , and h0 ( τ ) is a baseline hazard function ., The vector Xij ( τ ) can include individual-level covariates affecting the infectiousness of i or the susceptibility of j as well as pairwise covariates ( e . g . , membership in the same household ) ., The coefficient vector β0 captures covariate effects on the hazard of transmission , and the baseline hazard function h0 ( τ ) captures the evolution of infectiousness over time in infectious individuals ., We assume that τij can be observed only if j is infected by i at time ti + εi + τij ., The contact interval τij will be unobserved if i recovers from infectiousness before making infectious contact with j , if j is infected by a someone other than i , or if observation of j has stopped ., Let Ii ( τ ) = 1τ ∈ ( 0 , ιi be a left-continuous process indicating whether i remains infectious at infectious age τ ., Let Sij ( τ ) = 1ti+εi+τ ≤ tj be a left-continuous process indicating whether j remains susceptible when i reaches infectious age τ ., Assume that the population is under observation until a stopping time T and let Oij ( τ ) = 1ti+εi+τ ≤ T be a left-continuous process indicating whether j is under observation when i reaches infectious age τ ., Then, Y i j ( τ ) = C i j I i ( τ ) S i j ( τ ) O i j ( τ ) ( 2 ), is a left-continuous process indicating whether infectious contact from i to j can be observed at infectious age τ of, i . The assumptions above ensure that censoring of τij is independent for all ij , and they can be relaxed if independent censoring is preserved ., Let θ be a parameter vector for a family of hazard functions h ( τ , θ ) such that h0 ( τ ) = h ( τ , θ0 ) for an unknown θ0 ., To allow maximum likelihood estimation , we assume that h ( τ , θ ) has continuous second derivatives with respect to θ ., Let, h i j ( τ , β , θ ) = exp β ⊤ X i j ( τ ) h ( τ , θ ) ., ( 3 ) Let W j = { i : t i + ε i < t j and C i j = 1 } denote the set of all infectious individuals to whom j was exposed while susceptible , which we call the exposure set of, j . When we observe who-infected-whom ( i . e . , v is known ) , the likelihood is, L v ( β , θ ) = ∏ j = 1 n h v j j ( t j - t v j - ε v j , β , θ ) 1 v j ∉ { 0 , ∞ } ∏ i ∈ W j e - ∫ 0 ι i h i j ( τ , β , θ ) Y i j ( τ ) d τ ., ( 4 ) The hazard terms depend on v , but the survival terms do not 44 ., When we do not observe who-infected-whom , the likelihood is a sum over all possible transmission trees: L ( β , θ ) = ∑ v ∈ V L v ( β , θ ) 44 ., Each v ∈ V can be generated by choosing a v j ∈ V j for each endogenous infection, j . Given the epidemiologic data , each vj can be chosen independently 48 ., This leads to the sum-product factorization, L ( β , θ ) = ∏ j = 1 n ∑ i ∈ V j h i j ( t j - t i - ε i , β , θ ) 1 v j ∉ { 0 , ∞ } ∏ i ∈ W j e - ∫ 0 ι i h i j ( τ , β , θ ) Y i j ( τ ) d τ ., ( 5 ) The probability of a particular transmission tree v is, Pr ( v | β , θ ) = L v ( β , θ ) L ( β , θ ) = ∏ j : v j ∉ { 0 , ∞ } h v j j ( t j - t v j - ε v j , β , θ ) ∑ i ∈ V j h i j ( t j - t i - ε i , β , θ ) , ( 6 ), and Lv ( β , θ ) = Pr ( v|β , θ ) L ( β , θ ) ., In this framework , estimation of ( β , θ ) , and the probabilities of possible transmission trees is simultaneous ., An interesting special case is when hij ( τ , β , θ ) = λ for all ij ., Then Pr ( v|β , θ ) does not depend on λ , so the transmission tree is an ancillary statistic 44 ., The relationship between phylogenies and transmission trees we develop here is similar to the approach taken by Cottam et al . 31 who linked phylogenetic and transmission trees via the locations of common ancestors ., It is logically equivalent to the approaches of Ypma et al . 40 who joined the within-host phylogenies of infectors and infectees into a single phylogeny , Didelot et al . 42 who colored lineages in the phylogeny with a unique color for each individual , and Hall and Rambaut 49 who represented transmission trees as partitions of phylogenies ., We begin with these assumptions: The first two assumptions concern the biology of disease ., The last three assumptions concern the resolution of the epidemiologic data , which can be controlled through study design ., Initially , we use only the topology of the pathogen phylogeny to infer the set of possible transmission trees ., Later , we consider how branching times at interior nodes further restrict the set of possible transmission trees ., To study the impact of a phylogeny on the efficiency of transmission parameter estimates , we conducted a series of 1 , 000 simulations ., In each simulation , there were 100 independent households of size 6 ., Each household had an index case with an infection time chosen from an exponential distribution with mean one ., Each individual i had a binary covariate Xi that could affect infectiousness and susceptibility ., Given a parameter vector β = ( βinf , βsus ) , the hazard of infectious contact from i to j at infectious age τ of i is, h i j ( τ , β ) = exp ( β inf X i + β sus X j ) λ 0 ., ( 16 ), In each simulation , βinf and βsus were independently chosen from a uniform distribution on ( −1 , 1 ) ., In all simulations , the baseline hazard was λ0 = 1 and the infectious periods were independent exponential random variables with mean one ., In each simulation , we analyzed data from the first 200 infections in three ways: using only epidemiologic data via the likelihood in Eq ( 5 ) , using epidemiologic data with who-infected-whom via the likelihood in Eq ( 4 ) , and using epidemiologic data with a phylogeny via the likelihood in Eq ( 7 ) ., In the phylogenetic analysis , we assumed a single pathogen sample from each infected individual ., The within-host phylogeny for each individual who infected m > 0 individuals was chosen uniformly at random from all rooted , bifurcating phylogenies with m + 1 tips ., Within-individual phylogenies were chosen independently and combined into a single phylogeny as in Ypma et al . 40 ., Thus , the conditional probability Pr ( Φ|v , Epi ) for a phylogeny Φ given a transmission tree v in which each individual j infected mj ≥ 0 other individuals was proportional to, ∏ j : m j > 0 2 m j - 1 ( m j - 1 ) !, ( 2 m j - 1 ) !, ., ( 17 ), The set of transmission trees consistent with the phylogeny was determined using Algorithms 1 and 2 ., We calculated the mean error , mean squared error , 95% confidence interval coverage probability , and relative efficiency of βinf , βsus , and ln λ0 estimates in all three analyses ., The simulations were conducted in Python 2 . 7 ( www . python . org ) and analysis was conducted in R 3 . 2 ( cran . r-project . org ) via RPy2 2 . 7 ( rpy . sourceforge . net ) ., The Python code is in S1 Text ., Parameters , point estimates , and 95% confidence limits are in S1 Data ., R code for the simulation data analysis is in S1 Text ., To illustrate an application of these algorithms and likelihoods , we use them to analyze farm-to-farm transmission trees of foot and mouth disease virus ( FMDV ) in a cluster of 12 epidemiologically linked farms in Durham , UK in 2001 ., The genetic and epidemiologic data are publicly available as Data S3 and Data S4 in Morelli et al . 29 ., These data were previously analyzed by Cottam et al . 31 , 32 , Morelli et al . 29 , Ypma et al . 40 , and Lau et al . 43 ., FMDV is a picornavirus that causes a highly contagious disease in cattle , pigs , sheep , and goats 51 ., Upon infection , there is an incubation period of approximately 1–12 days in sheep , 2–14 days in cattle , and two or more days in pigs ., The incubation period is followed by an acute febrile illness with painful blisters on the feet , the mouth , and the mammary glands ., It is transmitted through secretions from infected animals , fomites , virus carried on skin or clothing , and aerosolized virus ., Outbreaks of foot-and-mouth disease are difficult to control and can devastate livestock ., During the FMDV outbreak , teams from the UK Department for Environment , Food , and Rural Affairs ( DEFRA ) visited each infected farm 30 , 31 ., They recorded the number and types of susceptible and infected animals , examined infected animals to determine the age of the oldest lesions , and collected epithelial samples ., Finally , they recorded the date of the cull ., We assume that infectiousness begins on the day that the first lesions appeared and ends with the cull , and we assumed a latent period ( between infection and the onset of infectiousness ) of 2–16 days ., Fig 2 shows the relative locations of the farms , and Fig 3 shows the timeline of the latent and infectious periods ., Analysis was conducted in R 3 . 2 ( cran . r-project . org ) , and the code is available in S3 Text ., Table 1 shows the mean error , mean squared error , 95% confidence interval coverage probability , and relative efficiency of βinf , βsus , and ln λ0 estimators in the simulations ., In all cases , the point estimates were nearly unbiased ( indicated by the mean error squared being much smaller than the mean squared error ) and the 95% confidence interval coverage probabilities were near 0 . 95 ., Fig 4 shows that estimates of βinf using a phylogeny were more efficient than estimates using epidemiologic data only and less efficient than estimates using who-infected-whom ., By mean squared error , the phylogenetic estimates had a relative efficiency of 1 . 39 compared to estimates using only epidemiologic data and 0 . 80 compared to estimates using who-infected-whom ., Because knowledge of who-infected-whom does not add to our knowledge of who was infected , all three analyses were equally efficient for βsus ( similar results were obtained for estimates with and without who-infected-whom in Ref 46 ) ., Fig 5 shows that estimates of ln λ0 using a phylogeny were more efficient than those using epidemiologic data only and less efficient than those using who-infected whom ., By mean squared error , the phylogenetic estimates had a relative efficiency of 1 . 17 compared to estimates using only epidemiologic data and 0 . 90 compared to estimates using who-infected-whom ., Table 2 shows the mean error , mean squared error , 95% confidence interval coverage probability , and relative efficiency of βinf , βsus , and ln λ0 estimators that excluded data on uninfected household members ., The mean squared errors were much higher than the corresponding estimators in Table 1 , so their relative efficiency was very low ., In all cases , the efficiency loss from excluding data on individuals who escape infection was much larger than the efficiency gain from incorporating a phylogeny or from knowing exactly who infected whom ., For estimators of βinf and βsus , the square of the mean error was much smaller than the mean squared error , indicating little bias ., Estimates of ln λ0 were biased upward , resulting in extremely low relative efficiencies and coverage probabilities ., In 44 , similar results were seen for estimates of the basic reproduction number ( R0 ) when approximate likelihoods for mass-action models , which do not require data on uninfected individuals , were used to analyze data from network-based epidemics ., Table 3 shows results the mean error , mean squared error , 95% confidence interval coverage probability , and relative efficiency of βinf , βsus , ln λ0 , and ln γ estimators from models with Weibull contact interval distributions with rate parameter λ0 = 2 and shape parameter γ = . 5 ., All estimators are unbiased with 95% confidence interval coverage probabilities near 0 . 95 ., The relative efficiencies are similar to those in Table 1 , showing that the gains in efficiency for estimates of infectiousness hazard ratios and baseline hazards occur under weak assumptions about the baseline hazard ., With no phylogeny , there are 19 , 440 possible transmission trees linking the 12 farms in the Durham cluster ., A phylogeny was constructed in SeaView 52 using consensus RNA sequences from 15 farms , including three farms not epidemiologically linked to the cluster 29 ., We used a generalized time reversible ( GTR ) nucleotide substitution model with four rate classes on 8 , 196 sites ., Fig 6 shows the rooted phylogeny for the 12 farms in the cluster with branch tips scaled to reflect the time of infectiousness onset at each farm ( interior branch lengths do not indicate branching times ) ., The order of infectiousness onsets is known , so first ( x ) is the host with the earliest onset of infectiousness in clade Cx ., Fig 7 shows the postorder host set Dx for each node x in the phylogeny , and Fig 8 shows the host sets ., The host is uniquely determined by the phylogeny for all interior nodes except three ., Fig 9 shows the six possible interior node host assignments and the corresponding transmission trees ., The simulations suggest that a phylogeny can recover much of the information that would be obtained by observing who-infected-whom ., Incorporating a phylogeny generated more precise estimates of βinf and ln λ0 ., This increase in efficiency remained when infectiousness varied over the course of the infectious period , as in the Weibull models ., The simulations used only phylogenetic topologies and assumed that all within-host topologies were equally likely , limiting the ability of the phylogeny to constrain the set of possible transmission trees ., Using branching times and more realistic models of within-host pathogen evolution would allow greater information about who-infected-whom to be extracted from a phylogeny ., The simulations that excluded data on household members who escape infection showed that this information is critical to estimating βinf , βsus , and ln λ0 accurately ., These individuals do not appear anywhere on the pathogen phylogeny , so this point has escaped the attention of many researchers working on incorporating phylogenetics into the analysis of infectious disease transmission data ., Any analysis that excludes this data should have an explicit justification based on a complete-data model—for example , the initial spread of a mass-action epidemic can be analyzed without data on escapees 44 ., In general , epidemiologic studies of emerging infections should be designed to capture information on individuals who were exposed to infection but not infected , which might justify greater emphasis on detailed studies of households , schools , or other settings with rapid transmission and a clearly defined population at risk ., The data analysis showed that the increased precision found in the simulations can be obtained in practice ., The incorporation of phylogenies allowed more precise estimates of the hazard of FMDV transmission from infected to susceptible farms ., For simplicity , our analysis assumed that the times of infectiousness onset were accurately estimated ., A data-augmented MCMC 53 could be used to account for uncertainty in the onset of infectiousness , showing the importance of extending our methods to account for missing data ., A more important limitation of this analysis was the lack of data on uninfected farms ., The hazard function estimates were highly sensitive to the number of uninfected farms in the area where the cluster occurred ., These data often go uncollected in outbreaks because their importance is unrecognized ., This insight has important implications for the theory and practice of molecular infectious disease epidemiology .
Recent work has attempted to use whole-genome sequence data from pathogens to reconstruct the transmission trees linking infectors and infectees in outbreaks ., However , transmission trees from one outbreak do not generalize to future outbreaks ., Reconstruction of transmission trees is most useful to public health if it leads to generalizable scientific insights about disease transmission ., In a survival analysis framework , estimation of transmission parameters is based on sums or averages over the possible transmission trees ., A phylogeny can increase the precision of these estimates by providing partial information about who infected whom ., The leaves of the phylogeny represent sampled pathogens , which have known hosts ., The interior nodes represent common ancestors of sampled pathogens , which have unknown hosts ., Starting from assumptions about disease biology and epidemiologic study design , we prove that there is a one-to-one correspondence between the possible assignments of interior node hosts and the transmission trees simultaneously consistent with the phylogeny and the epidemiologic data on person , place , and time ., We develop algorithms to enumerate these transmission trees and show these can be used to calculate likelihoods that incorporate both epidemiologic data and a phylogeny ., A simulation study confirms that this leads to more efficient estimates of hazard ratios for infectiousness and baseline hazards of infectious contact , and we use these methods to analyze data from a foot-and-mouth disease virus outbreak in the United Kingdom in 2001 ., These results demonstrate the importance of data on individuals who escape infection , which is often overlooked ., The combination of survival analysis and algorithms linking phylogenies to transmission trees is a rigorous but flexible statistical foundation for molecular infectious disease epidemiology .
Recent work has attempted to use whole-genome sequence data from pathogens to reconstruct the transmission trees linking infectors and infectees in outbreaks ., However , transmission trees from one outbreak do not generalize to future outbreaks ., Reconstruction of transmission trees is most useful to public health if it leads to generalizable scientific insights about disease transmission ., Accurate estimates of transmission parameters can help identify risk factors for transmission and aid the design and evaluation of public health interventions for emerging infections ., Using statistical methods for time-to-event data ( survival analysis ) , estimation of transmission parameters is based on sums or averages over the possible transmission trees ., By providing partial information about who infected whom , a pathogen phylogeny can reduce the set of possible transmission trees and increase the precision of transmission parameter estimates ., We derive algorithms that enumerate the transmission trees consistent with a pathogen phylogeny and epidemiologic data , show how to calculate likelihoods for transmission data with a phylogeny , and apply these methods to a foot and mouth disease outbreak in the United Kingdom in 2001 ., These methods will allow pathogen genetic sequences to be incorporated into the analysis of outbreak investigations , vaccine trials , and other studies of infectious disease transmission .
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