content_image/435828.jpg content_image/435811.jpg content_image/435815.jpg content_image/435820.jpg content_image/435825.jpg content_image/435829.jpg
in the absence of vaccination. For such systems 

<img src='content_image/435828.jpg'>

The quantity χ ( t ) is called logarithmic susceptibility [13, 20, 34, 35]; it gives the change of the logarithm of the extinction rate, which is linear in the vaccination rate for moderately low vaccination rate. 

The minimization over t$_{0}$ in Eq. (14) accounts for lift- ing the time-translational invariance of the optimal ex- tinction paths mentioned earlier. For ξ ( t ) ≡ 0, extinc- tion can occur at any time ( t$_{r}$ ≪ t ≪ W − $^{1}$) with rate W$_{e}$ . Periodic vaccination synchronizes extinction events; it periodically modulates the extinction rate, and the modulation is exponentially strong for N | s (1) $_{ext}$|  ≫ 1 (see below). Formally, in a modulated system there is only one optimal extinction path per period, as explained in Sec. II, which is here reflected in the minimization over t$_{0}$ . This optimal path minimizes the disease extinction barrier Q = N s$_{ext}$ [19, 20, 34, 35]. Equation (14) is closely related to the Mel’nikov theorem for dynamical systems [20, 36]. 

The constraint for minimizing the action over ξ ( t ) in Eq. (13) has a form of an integral over the vaccination period T . It is therefore convenient to write action s (1) ext also in the form of such an integral, 

The function χ$_{T}$ ( t ) is obtained by superimposing the parts of χ ( t ) which differ by T . By construction, χ$_{T}$ ( t ) is periodic in time t . 

To find the optimal shape of vaccination rate ξ ( t ) we first minimize the time integral in the variational prob- lem Eqs. (13) – (15) with respect to ξ ( t ) for a given t$_{0}$ . Since ξ ( t ) ≥ 0, it is convenient to perform the minimiza- tion with respect to ξ 1 / $^{2}$( t ) rather than ξ ( t ). The min- imization shows that ξ 1 / $^{2}$( t ) ̸ = 0 only for t = t$_{λ}$ , where t$_{λ}$ is given by equation χ$_{T}$ ( t$_{λ}$ − t$_{0}$ ) = − λ/T . From the constraint on the period-averaged ξ ( t ) we then have 

Substituting this expression into the functional ˜ s$_{ext}$ and minimizing with respect to t$_{0}$ , we obtain the action in a 

<img src='content_image/435811.jpg'>

## 2. Temporal shape of optimal vaccination 

<img src='content_image/435815.jpg'>

simple explicit form 

Alternatively, this expression can be rewritten in terms of the Fourier transform of the logarithmic susceptibility: 

where Ω = 2 π/T is the cyclic frequency of vaccination. 

We are interested in the solution for which s (1) ext is nega- tive, which requires min χ$_{T}$ ( t ) < 0. Only in this case will vaccination reduce the barrier for disease extinction and increase the disease extinction rate. The barrier reduc- tion due to vaccination, Q (1) = N s (1) ext ∝ N Ξ, becomes large for N ≫ 1 even if the average vaccination rate Ξ is small. Thus, for not too small vaccination rates, where the eikonal approximation is valid [20, 34], the effect of vaccination on the disease extinction rate is exponentially strong. (1) 

The expression for the action change s $_{ext}$, Eq. (17), can be also obtained in a more intuitive way. Indeed, since ξ ( t ) is non-negative, it follows from Eq. (15) that 

<img src='content_image/435820.jpg'>

The minimum is provided by ξ ( t ) = Ξ T ∑ n δ ( t − t$_{min}$ + nT ). Formally, t$_{min}$ is the instance of time where χ$_{T}$ ( t ) is minimal. In fact, it is the optimal path that adjusts to the vaccination pulses so as to increase the probability of disease extinction. This provides the mechanism of syn- chronization by vaccination. Equation (19) immediately leads to Eqs. (16) and (17) with t$_{λ}$ replaced by t$_{min}$ . 

In addition to the constraint on the average vacci- nation rate, there may be an upper limit on the in- stantaneous vaccination rate, which is imposed by con- dition w ( x , r ) = w $^{(0)}$( x , r$_{ξ}$ ) + ξ ( t ) w $^{(1)}$( x , r$_{ξ}$ ) ≥ 0. In the case where w $^{(1)}$( x , r$_{ξ}$ , t ) < 0, as for vaccination of newly arrived susceptibles, this condition is met provided ξ$_{0}$ ( t ) ≤ ξ$_{0}$$_{m}$ ≡ min { w $^{(0)}$( x , r$_{ξ}$ ) / ∣ w $^{(1)}$( x , r$_{ξ}$ ) ∣ } . In this case the optimal vaccination protocol changes. 

The new protocol can be found from the variational problem (13) by changing from ξ ( t ) ≡ ξ$_{0}$ ( t ) ∣ w $^{(1)}$( x$_{A}$ , r$_{ξ}$ ) ∣ to an auxiliary function η ( t ) such that ξ$_{0}$ ( t ) = ξ$_{0}$$_{m}$ [1 + η $^{2}$( t )] − $^{1}$, and then finding the minimum of ˜ s$_{ext}$ with respect to η ( t ). This choice satisfies the constraints 0 ≤ ξ$_{0}$ ( t ) ≤ ξ$_{0}$$_{m}$ . Variation with respect to η ( t ) shows that ˜ s$_{ext}$ has an extremum for η ( t ) = 0 or η ( t ) = ∞ for t ̸ = t$_{λ}$ , where t$_{λ}$ is given by equation χ$_{T}$ ( t$_{λ}$ − t$_{0}$ ) = − λ/T . The value of η ( t ) at the isolated instances t = t$_{λ}$ is arbi- trary. Only regions where η ( t ) = 0, so that ξ$_{0}$ ( t ) = ξ$_{0}$$_{m}$ , contribute to ˜ s$_{ext}$ . Obviously, ˜ s$_{ext}$ is minimal when ξ$_{0}$ ( t ) = ξ$_{0}$$_{m}$ for | t − t$_{min}$ | ≤ Δ t/ 2, where t$_{min}$ is the 

<img src='content_image/435825.jpg'>

<img src='content_image/435829.jpg'>

