rate can allow one to tune the vaccination period in res- onance with the system dynamics. 

The analysis is based on the master equation for the population dynamics. We solve it in the eikonal approx- imation and reduce the problems of the tail of the dis- tribution and of the extinction to Hamiltonian dynamics of an auxiliary system. A general formulation of the cor- responding Hamiltonian problem is obtained for period- ically modulated systems. The optimal vaccination pro- tocol is found using this formulation with account taken of the constraint on the average vaccination rate. The feature of the problem that makes it different from other problems of optimal control of rare events is that the vac- 

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## Acknowledgments 

