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We assume that vaccination is applied periodically in time. In this case, the optimal protocol is to apply the vaccine as a sequence of δ -like pulses. The disease ex- tinction rate can strongly depend on the period of this sequence. Furthermore, the extinction rate can display exponentially sharp peaks when the pulse sequence pe- riod is close to the characteristic period of oscillations of the system in the absence of fluctuations, or to its multiples. We illustrate this resonant phenomenon for the Susceptible-Vaccinated-Infected-Recovered (SVIR) model. 

Epidemics often display seasonal modulation [18]. It is natural to apply a vaccine with period equal to the modulation period. As we show, there is a qualitative difference between the effect of a periodic vaccination in this case and in the case where seasonal modulation is absent. For a system with seasonal modulation, an im- properly applied pulsed vaccination can actually reduce the disease extinction rate and therefore prolong the du- ration of the epidemic. The overall effect of the pulsed vaccination critically depends here on the phase at which the periodic pulses are applied. 

The analysis of periodic vaccination, with and with- out seasonal variations, necessitates a general formula- tion of the extinction problem in periodically modulated stochastic populations. We extend the previous results for single-population systems [19, 20] to provide a com- plete extinction theory for multi-population systems in the eikonal approximation, and emphasize the distinction from the well-understood problem of switching between metastable states in periodically modulated systems with noise [21]. 

Section II describes the class of epidemic models we consider in this work and develops an eikonal theory of disease extinction rate in periodically modulated sys- tems. Section III formulates the optimization problem for vaccination and presents its solution for a time-periodic vaccination in the limit of a small average vaccination rate. In Section IV we discuss the vaccination-induced reduction of the disease extinction barrier for two types of constraints on the vaccination period, a limited life- time of the vaccine and a limited vaccine accumulation. Section V illustrates, on the example of the stochastic SVIR model, the phenomenon of resonant response to vaccination. Section V contains concluding remarks. 

## II. THE DISEASE EXTINCTION RATE 

We consider stochastic disease dynamics in a well- mixed population which includes infected ( I ) and sus- ceptible ( S ) individuals and possibly other population groups such as recovered or vaccinated. The system state is described by a vector X = ( S, I, . . . ) with in- teger components equal to the sizes of different popula- tion groups. Along with X it is convenient to consider a 

## A. The model of population dynamics 

quasi-continuous vector x = X /N , where N is the char- acteristic total population size, N ≫ 1. We assume that the population dynamics is Markovian. It is quite gener- ally described by the master equation for the probability distribution P ( X , t ), 

Here W ( X , r , t ) is the rate of an elementary transition X → X + r in which the population size changes by r = ( r$_{1}$, r$_{2}$, . . . ). Examples of such transitions are infection of a susceptible individual as a result of contacting an infected individual, recovery of an infected individual or arrival of a susceptible individual. 

We assume that there is no influx of infected individ- uals into the population. Therefore, there are no tran- sitions from states where there are no infected to states where infected are present, 

where subscript E is used for the component of X which enumerates infected, X$_{E}$ ≡ I . 

In the neglect of demographic noise the population dy- namics can be described by the deterministic (mean-field) equation for the population size ¯ x , 

It immediately follows from Eq. (1) if the width of the probability distribution P ( X , t ) is set equal to zero. 

We start with the case where the transition rates W ( X , r ) are independent of time. An endemic state, where a finite fraction of population is infected for a long time, corresponds to an attracting fixed point x$_{A}$ of the dynamical system, Eq. (3). We will assume throughout this work that there is only one such point. We will also assume that Eq. (3) has one fixed point x$_{S}$ in the hy- perplane x$_{E}$ = 0. The state x$_{S}$ is stable with respect to all variables except x$_{E}$ . We call it the disease extinction state. If x$_{E}$ > 0 (there is a nonzero number of infected), the deterministic trajectory leaves the vicinity of x$_{S}$ and approaches the endemic state x$_{A}$ . 

Due to demographic noise the endemic state is ac- tually metastable . A rare large fluctuation ultimately drives the population into a disease-free state. The most probable fluctuation brings the system to the fixed point x$_{S}$ [13, 14]. The probability of such a fluctuation per unit time, i.e. , the disease extinction rate W$_{e}$ , is given by the probability current to x$_{S}$ , similarly to the prob- lem of escape from a metastable state [22]. For time- independent W ( X , r ) this current is quasistationary for 

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## 1. Stationary systems 

