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 nonequilibrium system introduced by Maier and Stein and a two-dimensional (in parameter space) epidemiology model.