8 ✁ ✂ ✄ ☎ GLYPH<a0> ✆ ✝ ✞ ✟ ✠ ✡ ☛ ☞ ✌ ✍ ✎ ✏ ✑ ✒ ✓ ✔ d dT min 0 <t<T χ$_{T}$ ( t ) = ∂ ∂T ∑ n χ ( t$_{∗}$ − a$_{∗}$T + nT ) = ∑ n ( dχ ( t − a$_{∗}$T + nT ) dt )$_{t}$$_{=}$$_{t}$$_{∗}$ ( n − a$_{∗}$ ) > 0 , (23) H $^{(1)}$( x , p ) = x$_{1}$ ( e p 4 − p 1 − 1 ) . (24) The behavior of s (1) ext with varying vaccination period T depends on the form of the logarithmic susceptibil- ity χ ( t ). Let us first consider the case where χ ( t ) has a single local minimum (at t = t$_{∗}$ ), and | χ ( t ) | mono- tonically decays to zero with increasing | t − t$_{∗}$ | . Here, once the vaccine accumulation has reached saturation with increasing T (which happens for Ξ T = M ), func- tion | s (1) $_{ext}$| = M | min χ$_{T}$ ( t ) | monotonically decreases with further increase in T . Indeed, FIG. 1: The SVIR epidemic model with susceptible, vacci- nated, infected and recovered sub-populations. The arrows indicate processes leading to changes of sub-population sizes; the corresponding rates are scaled per individual. where a$_{∗}$ gives the position of the minimum of χ$_{T}$ ( t ) over t and is given by equation ∂χ$_{T}$ ( t$_{∗}$ − a$_{∗}$T ) /∂a$_{∗}$ = 0; we choose 0 < a$_{∗}$ < 1 and take into account that min$_{0}$$_{<t<T}$ χ$_{T}$ ( t ) < 0. In Eq. (23) we have used that, if χ ( t ) is minimal for t = t$_{∗}$ , then dχ/dt > 0 for t > t$_{∗}$ and dχ/dt < 0 for t < t$_{∗}$ . It follows from Eq. (23) that, once the vaccine accumulation has reached saturation, further increase in T will only reduce the effect of the vaccine. This result is understandable because, if T in- creases beyond M/ Ξ, the actual average vaccination rate Ξ$_{a}$ decreases. For β > Γ ≡ γ + µ the SIR model possesses a sin- gle endemic state. This state corresponds, in the mean- field theory, to an attracting fixed point on the two- dimensional phase plane of susceptibles and infected. At µ < 4 ( β − Γ)(Γ /β ) 2 this attracting point is a focus. The populations of susceptibles, infected and recovered exhibit decaying oscillations in time as the system ap- proaches the endemic state. It was found in Ref. [14] that, in this parameter range, the populations oscillate also on the optimal disease extinction path. These oscil- lations are illustrated in Fig. 2. We will now incorporate vaccination and introduce a sub-population of vaccinated X$_{4}$ = V . The vaccination is described by the transition rate W ( X , r ) = ξ$_{0}$ ( t ) x$_{1}$ for r$_{1}$ = − 1 , r$_{4}$ = 1 , r$_{i}$$_{̸}$$_{=1}$$_{,}$$_{4}$ = 0. The corresponding term in the Hamiltonian Eq. (12) has the form ξ$_{0}$ ( t ) H (1) with A counterintuitive situation may occur if χ ( t ) is os- cillating. Here the inequality (23) may be violated. As a result, the dependence of the effect of the vaccine on T and, consequently, on the actual vaccination rate Ξ$_{a}$, may be nonmonotonic. An example of this behavior is discussed in the next section. Vaccinated individuals leave at the same rate µ as in- dividuals in other populations. For simplicity, we as- sume that the immunity from the vaccination is never lost. In this case fluctuations of the vaccinated pop- ulation do not affect fluctuations of other populations, and p$_{4}$ ≡ 0 along the optimal extinction path. Then from Eq. (14), the logarithmic susceptibility is χ ( t ) = x (0) $_{1opt}$( t ) x − 1 1 A ( 1 − exp[ − p (0) $_{1opt}$( t )] ) , where x (0) $_{1opt}$( t ), p (0) $_{1opt}$( t ) and x$_{1}$$_{A}$ are calculated for the SIR model. V. RESONANCES IN THE STOCHASTIC SVIR MODEL We now apply some of our results to an important and widely used stochastic epidemic model, the Susceptibles- Vaccinated-Infected-Recovered (SVIR) model. The model is sketched in Fig. 1. In the absence of vaccination, ξ$_{0}$ ( t ) = 0, the SVIR model reduces to the stochastic SIR model with population turnover, which was originally in- troduced to describe the spread of measles, mumps, and rubella, see [1, 3]. In the SIR model, susceptible in- dividuals are brought in, individuals in all population groups leave (for example, die), a susceptible individual can become infected upon contacting an infected indi- vidual, and an infected individual can recover. If we set X$_{1}$ = S, X$_{2}$ = I , and X$_{3}$ = R , the rates of the cor- responding processes are: (i) influx of the susceptibles, W ( X , r ) = µN for r$_{1}$ = 1 , r$_{i}$$_{̸}$$_{=1}$ = 0, (ii) leaving, with the same rate for all populations, W ( X , r ) = µX$_{i}$ for r$_{i}$ = − 1 , r$_{j}$$_{̸}$$_{=}$$_{i}$ = 0, (iii) infection, W ( X , r ) = βX$_{1}$X$_{2}$/N for r$_{1}$ = − 1 , r$_{2}$ = 1 , r$_{i}$$_{̸}$$_{=1}$$_{,}$$_{2}$ = 0, and (iv) recovery of the infected, W ( X , r ) = γX$_{2}$ for r$_{2}$ = − 1 , r$_{3}$ = 1 , r$_{i}$$_{̸}$$_{=2}$$_{,}$$_{3}$ = 0, Fig. (1). The Fourier spectrum of the logarithmic susceptibility ˜ χ ( ω ) is plotted in Fig. 3 (a). It corresponds to the optimal extinction path shown in Fig. 2. As one can see, the spectrum has a peak at the characteristic frequency of oscillations of the system in the absence of vaccination ω$_{0}$ (for the chosen parameter values ω$_{0}$ ≈ 5 . 2 µ ). We now consider the effect of the resonant peak in ˜ χ ( ω ) on vaccination. The dependence of the scaled change of the disease extinction barrier s (1) ext = Q $^{(1)}$/N on vaccina- tion period T is shown in Fig. 3 (b). The solid line in Fig. 3 (b) shows the behavior of s (1) ext where there is no limit on vaccine accumulation or, equivalently, for such periods where the limitation does not come into play and