Evolving towards the optimal path to extinction in stochastic processes

A large, rare stochastic fluctuation can cause an epidemic or a species to become extinct. In large, finite populations, the extinction process follows an optimal path which maximizes the probability of extinction. We show theoretically that the optimal path also possesses a maximal sensitivity to initial conditions. As a result, the optimal path emerges naturally from the dynamics and may be characterized using the finite-time Lyapunov exponents. Our theory is general, and is demonstrated with several stochastic epidemiological models.