Understanding variation in allele frequencies across populations is a central goal of population genetics. Classical models for the distribution of allele frequencies, using forward simulation, coalescent theory, or the diffusion approximation, have been applied extensively for demographic inference, medical study design, and evolutionary studies. Here we propose a tractable model of ordinary differential equations for the evolution of allele frequencies that is closely related to the diffusion approximation but avoids many of its limitations and approximations. We show that the approach is typically faster, more numerically stable, and more easily generalizable than the state-of-the-art software implementation of the diffusion approximation. We present a number of applications to human sequence data, including demographic inference with a five-population joint frequency spectrum and a discussion of the robustness of the out-of-Africa model inference to the choice of modern population.

Understanding variation in allele frequencies across populations is a central goal of population genetics. Classical models for the distribution of allele frequencies, using forward simulation, coalescent theory, or the diffusion approximation, have been applied extensively for demographic inference, medical study design, and evolutionary studies. Here we propose a tractable model of ordinary differential equations for the evolution of allele frequencies that is closely related to the diffusion approximation but avoids many of its limitations and approximations. We show that the approach is typically faster, more numerically stable, and more easily generalizable than the state-of-the-art software implementation of the diffusion approximation. We present a number of applications to human sequence data, including demographic inference with a five-population joint frequency spectrum and a discussion of the transferability of demographic histories across populations.