# An Approximate Markov Model for the Wright-Fisher Diffusion

## TL;DR

This paper introduces a new computational method to infer selection and population size from time series genetic data, improving accuracy in evolutionary studies.

## Contribution

A novel discrete approximation for diffusion processes called mean transition time approximation is introduced for joint inference of selection and demography.

## Key findings

- The approximation accurately describes allele trajectories through time with few states.
- The Bayesian inference approach accurately infers population size and selection coefficients.
- The method identifies selection targets and larger population sizes in Influenza virus evolution data.

## Abstract

The joint and accurate inference of selection and demography from genetic data is considered a particularly challenging question in population genetics, since both process may lead to very similar patterns of genetic diversity. However, additional information for disentangling these effects may be obtained by observing changes in allele frequencies over multiple time points. Such data is common in experimental evolution studies, as well as in the comparison of ancient and contemporary samples. Leveraging this information, however, has been computationally challenging, particularly when considering multi-locus data sets. To overcome these issues, we introduce a novel, discrete approximation for diffusion processes, termed mean transition time approximation, which preserves the long-term behavior of the underlying continuous diffusion process. We then derive this approximation for the particular case of inferring selection and demography from time series data under the classic Wright-Fisher model and demonstrate that our approximation is well suited to describe allele trajectories through time, even when only a few states are used. We then develop a Bayesian inference approach to jointly infer the population size and locus-specific selection coefficients with high accuracy, and further extend this model to also infer the rates of sequencing errors and mutations. We finally apply our approach to recent experimental data on the evolution of drug resistance in Influenza virus, identifying likely targets of selection and finding evidence for much larger viral population sizes than previously reported.