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Vaccination increases the number of individuals who are at least temporarily immune to the disease. It thus reduces the pool of susceptible individuals and ultimately leads to a reduction of the number of infected. When the available amount of vaccine is small, so that the disease extinction still requires a large fluctuation, the goal of vaccination is to reduce the disease extinction barrier Q . 

An outcome of vaccination is often modeled as the cre- ation of a sub-population of vaccinated individuals out of susceptibles. The corresponding elementary transi- tion rate is W ( X , r ) = ξ$_{0}$ ( t ) X$_{S}$ for r$_{S}$ = − 1, r$_{V}$ = 1 and r$_{i}$$_{̸}$$_{=}$$_{S,V}$ = 0, where subscripts V and S refer to vaccinated and susceptible individuals, respectively, and ξ$_{0}$ ( t ) is the control field that characterizes the vaccina- tion (subscript S should not be confused with subscript S used to indicate the extinction state). Another model is vaccination of newly arrived susceptibles, which leads to an effective reduction of the arrival rate µN . In this model, the elementary transition rate for the arrival is W ( X , r ) = N [ µ − ξ$_{0}$ ( t )] for r$_{S}$ = 1 and r$_{i}$$_{̸}$$_{=}$$_{S}$ = 0, with ξ$_{0}$ ( t ) N being the change in the arrival rate due to vacci- nation. 

We will consider a general model where vaccination modifies the rate of an elementary transition of a certain type; the change of the population in the corresponding transition is r$_{ξ}$ . The field ξ$_{0}$ ( t ) characterizing the vacci- nation is assumed to be weak. The affected rate has the form W ( X , r$_{ξ}$ , t ) = W $^{(0)}$( X , r$_{ξ}$ , t ) + ξ$_{0}$ ( t ) W $^{(1)}$( X , r$_{ξ}$ , t ), with W (0) being the rate without vaccination. The vac- cination either increases or decreases the rate, as for tran- sitions from susceptibles to vaccinated or for vaccination of newly arrived susceptibles, respectively. Therefore, we will assume without loss of generality that ξ$_{0}$ ( t ) ≥ 0 and that W $^{(1)}$( X , r$_{ξ}$ , t ) is either positive or negative. We con- sider models in which the number of susceptibles changes by 1 in an elementary transition associated with vacci- nation, ( r$_{ξ}$ )$_{S}$ = ± 1. We note that the analysis can be immediately extended to describe other processes, like modification of the infection rates [3] or recovery accel- eration by administrating medicine. 

It should be noted that the vaccination model adopted in this work is probabilistic by nature. An alternative is where vaccination is done in a pre-determined fashion, when a certain number of individuals are vaccinated per unit time at a given time. The analysis of such determin- istic vaccination lies beyond the scope of this paper. 

We will assume that vaccination is periodic, ξ$_{0}$ ( t ) = ξ$_{0}$ ( t + T ), and that the amount of vaccine available per period T is limited. We model this limitation as a con- straint on the ensemble-averaged number of individuals vaccinated per period T . The constraint can be written 

## III. OPTIMAL VACCINATION 

## A. Constraint on the vaccination protocol 

as 

Here, Ξ is the average vaccination rate rescaled by the characteristic population size N . The constraint is well- defined for t$_{r}$ ≪ t ≪ W − 1 e , where P ( X , r , t + T ) ≈ P ( X , r , t ). Since for N ≫ 1 the population distribution sharply peaks at the endemic state X$_{A}$ ( t ), the sum over X in Eq. (11) can be replaced by ∣ W $^{(1)}$( X$_{A}$ ( t ) , r$_{ξ}$ , t ) ∣ , in the leading order in 1 /N . 

In the presence of vaccination, one can still seek a so- lution of the master equation in the eikonal form. The exponent Q in the extinction rate is again given by the action of an auxiliary Hamiltonian system, Eq. (8). The Hamiltonian now has the form 

Our goal is to find the optimal form of ξ$_{0}$ ( t ) which would minimize the disease extinction barrier Q subject to con- straint (11). Since w $^{(1)}$( x$_{A}$ ( t ) , r$_{ξ}$ , t ) is a known periodic function of time, we can equivalently search for the op- timal vaccination rate ξ ( t ) ≡ ξ$_{0}$ ( t ) ∣ w $^{(1)}$( x$_{A}$ ( t ) , r$_{ξ}$ , t ) ∣ . It minimizes the functional 

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where λ is the Lagrange multiplier. The functional s$_{ext}$ [ ξ ] is given by Eq. (8), where the Lagrangian corresponds to the Hamiltonian (12) and depends on the vaccination rate ξ ( t ). 

## B. Vaccination protocol for stationary systems 

For a low average vaccination rate Ξ it suffices to keep in the action s$_{ext}$ [ ξ ( t )] only the leading-order term in ξ ( t ). Since Hamiltonian (12) is linear in ξ , this term is linear in ξ , too. In the spirit of the standard perturbation the- ory for Hamiltonian systems [33], the change in the ac- tion, caused by the small perturbation, can be calculated along the unperturbed trajectory ( x (0) $_{opt}$( t ) , p (0) $_{opt}$( t )) of the Hamiltonian H $^{(0)}$, which describes the optimal path of disease extinction in the absence of vaccination. 

We start with the case of systems, which are stationary 

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## 1. Double optimization problem 

