Demographic inference of pathogen-infected populations from partially observed transmission forests

Abstract

Genomic epidemiology has several methods for identifying which sampled individuals are linked by close proximity in the transmission chain, whether by grouping them into clusters under a genetic distance threshold or by identifying probable direct transmission pairs. Individuals found to have no sampled neighbours---singletons---are usually set aside. We argue that they are informative. Whether any two sampled individuals prove to be linked depends on the size of the sampling frame and the number of independent lineage introductions into it, and thus the balance of linked and unlinked individuals is itself data about these quantities. We formalise this by treating the intersection of a transmission tree with a sampling frame as a labelled rooted forest, from which a fixed number of nodes are sampled uniformly at random. Using the all-minors matrix-tree theorem, we derive a closed-form expression for the number of labelled rooted $k$-forests on $N$ nodes in which a specified set of nodes is independent. From this we obtain, again in closed form, the probability that a sample of a given size contains no linked individuals at all, and the likelihood of an arbitrary observed configuration of clusters, both when the transmission structure within each is known, and when only the cluster sizes are. The approach is closer to a survey, or to mark-recapture, than to conventional phylodynamic model fitting: the information comes from the linkage structure of a single sample alone with no assumptions regarding pathogen dynamics. We outline three applications: power calculations for prospective studies, a test of the uniform sampling assumption that underlies the reading of clusters as transmission hotspots, and demographic inference itself. We finally set out the assumptions that a more flexible implementation would need to relax.