content_image/430182.jpg content_image/430183.jpg content_image/430184.jpg
time when χ$_{T}$ ( t ) is minimal and Δ t is determined by the average vaccination rate Ξ. In other words, the vaccina- tion rate ξ ( t ) has the form of periodic rectangular pulses of width Δ t , centered at t$_{min}$ + nT , n = 0 , ± 1 , ± 2 , ... . The pulse width is 

Since the vaccination rate is limited by the rate of elementary transitions without vaccine, we have ξ$_{0}$$_{m}$ ∣ w $^{(1)}$( x$_{A}$ , r ) ∣ ≲ t − 1 r . Then for weak vaccination, Ξ T ≪ 1, from Eq. (20) Δ t ≪ t$_{r}$ . Therefore, χ$_{T}$ ( t ) = χ$_{T}$ ( t$_{min}$ ) during the pulse of ξ ( t ), to the leading order in Ξ T [we note that χ$_{T}$ ( t ) may vary on a time scale shorter than t$_{r}$ , see below; however, this time scale is always long compared to Δ t for sufficiently weak modulation]. The resulting change of the action is again given by Eq.(17). 

## C. Vaccination protocol for periodically modulated systems 

Optimal vaccination in periodically modulated systems requires a separate consideration. Here, there is only one optimal extinction path per period T in the absence of vaccination. When the average vaccination rate Ξ is low, vaccination with the same period T will only weakly per- turb this path. To first order in Ξ, the linear in ξ term in the action still has the form of Eq. (15), but without min- imization over t$_{0}$ . Since ξ ( t ) ≥ 0, the minimum of action is still achieved for ξ ( t ) = Ξ T ∑ n δ ( t − t$_{min}$ + nT ), but now t$_{min}$ is uniquely determined by the strong modula- tion. In other words, the modulation uniquely determines the phase of the optimal vaccination pulses. The result- ing expression for s (1) ext for the optimal vaccination pro- tocol has the form of Eq. (17). If the vaccination pulses are applied at a wrong time, i.e. if the phase difference between the vaccination and the modulation differs from the optimal one, the vaccination will be not as efficient and may even be harmful: it may prolong the lifetime of the endemic state by increasing the disease extinction barrier Q . 

## IV. DISEASE EXTINCTION BARRIER AS A FUNCTION OF VACCINATION PERIOD 

The vaccination-induced reduction of the disease ex- tinction barrier Q (1) = Ns (1) $_{ext}$, as given by Eqs. (17), de- pends on the interrelation between the vaccination period T and the characteristic time scales of the logarithmic susceptibility χ ( t ). Function χ ( t ) may or may not oscil- late in time, but generally χ ( t ) is relatively large within a time interval of the order of the relaxation time of the system t$_{r}$ [20, 34]. To reveal some qualitative features of the effect of vaccination and in particular, its dependence on the vaccination period, we will consider s (1) ext for two types of constraint on this period. 

<img src='content_image/430182.jpg'>

The vaccination period T is naturally limited by the effective lifetime τ v of the vaccine. This lifetime is usually determined by the maximum storage time of the vaccine and/or by the mutation rates of the infectious agent. If τ v is long, τ v ≫ t$_{r}$ , vaccination can be made most ef- ficient by increasing the vaccination period up to ∼ τ v . Indeed, as it follows from Eq. (17), s (1) ext ∝ T in this case. This result is easy to understand. Even though a de- crease of the vaccine pulse frequency Ω = 2 π/T causes a decrease of the prefactor in the disease extinction rate W$_{e}$ , the exponential factor exp( − Ns (1) $_{ext}$) in W$_{e}$ increases sharply. Indeed, it can be seen from Eqs. (5), (12) and (14) that, as the system moves along the optimal path to extinction, χ ( t ) is significant when the system is far from the stationary states x$_{A}$ and x$_{S}$ . The characteristic time scale of this motion is ∼ t$_{r}$ . For T ≫ t$_{r}$ we have min$_{0}$$_{≤}$$_{t}$$_{≤}$$_{T}$ χ$_{T}$ ( t ) ≈ min$_{t}$ χ ( t ) and 

In the opposite limit of τ v ≪ t$_{r}$ , and thus T ≪ t$_{r}$ , we have from Eq. (18) 

In this case the vaccination-induced reduction of the ex- tinction barrier is independent of the vaccination period and is determined by the zero-frequency component of the logarithmic susceptibility. 

An interesting situation may occur in the intermediate range τ v ∼ t$_{r}$ if, in the mean-field description, the system approaches the endemic state in an oscillatory manner. In this case function χ ( t ) is also expected to oscillate. The oscillations are well-pronounced if their typical fre- quency is ω$_{0}$ ≫ t − 1 r . It is clear from Eq. (18) that a strong effect on disease extinction can be achieved by tuning the vaccination frequency Ω = 2 π/T or its overtones in res- onance with ω$_{0}$ . An example of such a resonance will be discussed in Sec. V. 

A different situation occurs if the total amount of vac- cine that can be accumulated is limited. This limitation implies that Ξ T ≤ M (note that M is the limit on the ensemble-averaged amount of the accumulated vaccine). Such a constraint is typical for live vaccines, as it may be dangerous to store too much vaccine in this case. The actual average vaccination rate in this case is now T - dependent. We use the notation Ξ$_{a}$ for this rate, with Ξ$_{a}$ = min(Ξ , M/T ). This is Ξ$_{a}$ that should be used now in Eqs. (21) and (22) for s (1) ext in the limits T ≫ t$_{r}$ and T ≪ t$_{r}$ , respectively. 

## 1. Limited lifetime of the vaccine 

<img src='content_image/430183.jpg'>

<img src='content_image/430184.jpg'>

## 2. Limited vaccine accumulation 

