Analytical solution of the expiring fitness ODE system

Derive explicit time-dependent closed-form solutions for the expiring fitness model defined by the coupled differential equations dot{x} = s x (1 − x) and dot{s} = − x s, including exact expressions for x(t), s(t), and the sweep completion time (e.g., the time to reach a specified fraction r of the final frequency β), thereby eliminating reliance on logistic approximations used to estimate partial sweep dynamics and overlap probabilities.

Background

The paper introduces an effective model of expiring fitness to capture partial sweeps of immune-evasive variants, specified by the coupled ODEs dot{x} = s x (1 − x) and dot{s} = − x s. This model is used to paper eco-evolutionary dynamics when variant growth advantages dissipate due to host immunity.

To estimate sweep durations and the probability of overlapping sweeps, the authors resort to an approximate logistic expression for x(t), noting that they could not solve the coupled ODEs analytically. A closed-form solution for x(t) and s(t) would provide rigorous expressions for key quantities such as the time-to-threshold and overlap probabilities, strengthening the analytical foundations of the model and its predictions.