Exponential Integral Solutions for Fixation Time in Wright-Fisher Model With Selection

TL;DR

This paper derives new analytic expressions for fixation time in the Wright-Fisher model with selection, using exponential integral functions and approximations, validated by extensive simulations.

Contribution

It introduces novel analytic solutions for fixation times incorporating selection effects, using a simplified differential equation approach.

Findings

Accurate approximation of fixation times for small populations and large selection.

Solutions involve exponential integral functions combined with elementary functions.

Validated results through extensive simulation studies.

Abstract

In this work we derive new analytic expressions for fixation time in Wright-Fisher model with selection. The three standard cases for fixation are considered: fixation to zero, to one or both. Second order differential equations for fixation time are obtained by a simplified approach using only the law of total probability and Taylor expansions. The obtained solutions are given by a combination of exponential integral functions with elementary functions. We then state approximate formulas involving only elementary functions valid for small selection effects. The quality of our results are explored throughout an extensive simulation study. We show that our results approximate the discrete problem very accurately even for small population size (a few hundreds) and large selection coefficients.