The Barrier Method: A Technique for Calculating Very Long Transition Times

In many dynamical systems there is a large separation of time scales between typical events and "rare" events which can be the cases of interest. Rare-event rates are quite difficult to compute numerically, but they are of considerable practical importance in many fields: for example transition times in chemical physics and extinction times in epidemiology can be very long, but are quite important. We present a very fast numerical technique that can be used to find long transition times (very small rates) in low-dimensional systems, even if they lack detailed balance. We illustrate the method for a bistable non-equilibrium system introduced by Maier and Stein and a two-dimensional (in parameter space) epidemiology model.