An accurate approximation for the expected site frequency spectrum in a Galton-Watson process under an infinite sites mutation model

If viruses or other pathogens infect a single host, the outcome of infection often hinges on the fate of the initial invaders. The initial basic reproduction number R₀, the expected number of cells infected by a single infected cell, helps determine whether the initial viruses can establish a successful beachhead. To determine R₀, the Kingman coalescent or continuous-time birth-and-death process can be used to infer the rate of exponential growth in an historical population. Given M sequences sampled in the present, the two models can make the inference from the site frequency spectrum (SFS), the count of mutations that appear in exactly k sequences (k=1,2,…,M). In the case of viruses, however, if R₀ is large and an infected cell bursts while propagating virus, the two models are suspect, because they are Markovian with only binary branching. Accordingly, this article develops an approximation for the SFS of a discrete-time branching process with synchronous generations (i.e., a Galton-Watson process). When evaluated in simulations with an asynchronous, non-Markovian model (a Bellman-Harris process) with parameters intended to mimic the bursting viral reproduction of HIV, the approximation proved superior to approximations derived from the Kingman coalescent or continuous-time birth-and-death process. This article demonstrates that in analogy to methods in human genetics, the SFS of viral sequences sampled well after latent infection can remain informative about the initial R₀. Thus, it suggests the utility of analyzing the SFS of sequences derived from patient and animal trials of viral therapies, because in some cases, the initial R₀ may be able to indicate subtle therapeutic progress, even in the absence of statistically significant differences in the infection of treatment and control groups.