Controlling rare events: optimizing disease extinction with limited vaccine

In rare events such as switching between stable states or disease extinction the system has to overcome an effective barrier. The barrier height can be changed by applying a control field. The change is determined by the effective work of the field along the most probable trajectory followed in a rare event. In turn, the barrier change results in an exponentially strong change of the event rate. We study the optimal temporal shape of the control field with a constraint that the time- average field value and the sign of the field are fixed. An example is vaccination with a limited vaccine production rate or control by light intensity with a limited laser power. For a comparatively weak field, for a broad class of rare events, optimal control is accomplished by periodically applying $\delta$-like pulses. We show that the barrier change may display resonant dependence on the pulse period and is linear in the pulse area. For a stronger field, the dependence of the barrier change on the field amplitude becomes system-dependent. The results are applied to simple models of population dynamics.