Reactive vaccination in the presence of disease hotspots

Reactive vaccination has recently been adopted as an outbreak response tool for cholera and other infectious diseases. Owing to the global shortage of oral cholera vaccine, health officials must quickly decide who and where to distribute limited vaccine. Targeted vaccination in transmission hotspots (i.e. areas with high transmission efficiency) may be a potential approach to efficiently allocate vaccine, however its effectiveness will likely be context-dependent. We compared strategies for allocating vaccine across multiple areas with heterogeneous transmission efficiency. We constructed metapopulation models of a cholera-like disease and compared simulated epidemics where: vaccine is targeted at areas of high or low transmission efficiency, where vaccine is distributed across the population, and where no vaccine is used. We find that connectivity between populations, transmission efficiency, vaccination timing and the amount of vaccine available all shape the performance of different allocation strategies. In highly connected settings (e.g. cities) when vaccinating early in the epidemic, targeting limited vaccine at transmission hotspots is often optimal. Once vaccination is delayed, targeting the hotspot is rarely optimal, and strategies that either spread vaccine between areas or those targeted at non-hotspots will avert more cases. Although hotspots may be an intuitive outbreak control target, we show that, in many situations, the hotspot-epidemic proceeds so fast that hotspot-targeted reactive vaccination will prevent relatively few cases, and vaccination shared across areas where transmission can be sustained is often best.

Uncontrolled epidemic dynamics are a function of connectivity and patch-specific transmission efficiency (R i , the 'local' basic reproductive number ). Simulated uncontrolled epidemics lasted from 12 to 86 weeks, with highly connected epidemics lasting longer as the number of nonhotspot patches increased (Table S1). The final size of these epidemics ranged from 163,704 to 1,677,912, with more heterogeneity in larger systems (i.e. those with more patches, Table S2).
As a proportion of the total population size, uncontrolled epidemics infected 7-78% of the total population. Local epidemics (i.e. within a single patch) lasted as short as 20 days. Local final epidemic sizes ranged from 3 to 446,322 persons infected. The time to hotspot peak incidence, the first local epidemic peak that will occur in all scenarios, ranged from 25 to 240 days.   Figure S1: Final size and duration of uncontrolled epidemics in 2-patch system  Figure S2: Hotspot (top) and non-hotspot (bottom) epidemic curves with sections coloured by percent of epidemic elapsed (two-patch model). Panels A, B and C show the curves for highly-connected, weakly connected, and unconnected settings shown in Figure 3 of the main text. 4

Estimation of Metapopulation Reproductive Number
In the main manuscript we use the reproductive numbers corresponding to unconnected populations (i.e. the reproductive number for a population had c jj = 1) to help with comparisons.
We can find the basic reproductive number for the whole metapopulation using the next generation matrix, [1] which is made up of two parts, the transmission matrix (T), and transition matrix (Σ). For our simple system with two populations the matrices are as follows: The next generation matrix, K, is defined as K = −TΣ −1 , and R is defined as the dominant eigenvalue of K. Thus, in this simple system, we use the characteristic polynomial (and the quadratic formula) to get R, as R = 1 2 tr(K) + tr 2 (K) − 4 det(K)

Final Size for Proactive Vaccination
We can use a simple probabilistic representation of the final size of an epidemic to explore the impact of proactive vaccination strategies with two populations. Here we derive the final size assuming the epidemic follows a Poisson process in a manner similar to previous studies [2,3].
The final size in location i, Z i ∞ , can be represented by where S i 0 represents the initial susceptibles in population i, R i,j represents the number of secondary infections in location i caused by an individual in location j , and N i is the population in location i, and n is the number of locations (in this situation n = 2).
For simplicity, we explore models where the epidemic connectivity is defined by a single parameter α, that represents the proportion of subsequent infections an infectious individual makes outside his or her home area. 6 The final epidemic size after proactive vaccination is represented by an 'all or nothing' vaccine [4] that reduces the initial susceptible in the population as follows: where v i represent the number vaccinated in area i and δ is the vaccine efficacy (V E S ). It   Figure S4: Comparison of simple reactive vaccination strategies in a 2-patch system with varying connectivity, R hotspot and R non−hotspot . Colours represent the best strategy (red for hotspot-targeted, blue for non-hotspot-targeted, and green for pro-rata), and color intensity represent the percent difference between that and the worst strategy (Θ = F Sworst−F S best   Figure S6: Comparison of simple reactive vaccination strategies in a 2-patch system with varying connectivity, R hotspot and R non−hotspot after 40 years of recurring epidemics every 4 years. These simulations assumed an exponential susceptible replacement distribution with a rate of 1/5 + 1/50, thus every 4-years 58.5% of individuals who were immune from the previous epidemic become susceptible again. Colours represent the best strategy (red for hotspottargeted, blue for non-hotspot-targeted, and green for pro-rata), and color intensity represent the percent difference between that and the worst strategy (Θ = F Sworst−F S best   Figure S7: Comparison of vaccination targeted at the hotspot and vaccination targeted at the non-hotspot in a 2-patch system with varying connectivity, R hotspot and R non−hotspot . Colours represent the best strategy (red for hotspot-targeted and blue for non-hotspot-targeted) and color intensity represent the percent difference between that and the worst strategy (Θ = F Sworst−F S best F Sworst ).

Multiple Non-hotspots
Now we consider metapopulations with 3-5 patches. In each set of simulations we allow there to be a single hotspot and multiple non-hotspot with identical R's. All sub-populations in these simulations are 500,000 individuals. We consider 4 different targeting approaches here; (1) targeting a single hotspot, (2) targeting a single non-hotspot, (3) sharing vaccine between all non-hotspots, and (4) pro-rata vaccination.   Figure S8: Overview of relative performance of vaccination strategies in 3-patch system. Panels illustrate best vaccination strategies (as measured by Θ) as a function of (1) availability of vaccine (y-axis, as a % of the total population), (2) the timing of the vaccination campaign (xaxis, as the percent of total cases infected in an uncontrolled epidemic), and (3) the transmission efficiency (as measured by the (local) reproductive number, R) in each patch. The color of each grid cell represents the preferred strategy at that vaccine availability level and vaccination campaign timing (green → pro-rata, blue→ single non-hotspot targeting, purple→ shared nonhotspot targeting, red→hotspot targeting), with the color intensity representing, Θ, or how much better that strategy is than the worst strategy (darker colors representing situations where the best decision is far better than the worst).

Optimal Allocations in Two Patch Models
While we only considered simple vaccination allocation strategies in the main text, here we show the optimal allocation in 2 patch systems. In all cases, the aim is to minimise the final epidemic size subject to limited vaccine supply. More formally, we wish to: where l represents area l, V i (u) represents the number vaccinated at time u in area l, and V max represents the maximum number of full vaccine courses available. In the case of a two patch system this reduces to a one-dimensional constrained optimisation problem, and we used the function minimize scalar in the SciPy library for python [6].   Figure S12: Optimal proportion of vaccine allocated to hotspot. Panels illustrate optimal proportion of vaccine allocated to the hotspot as a function of (1) availability of vaccine (yaxis, as a % of the total population), (2) the speed of the vaccination campaign (x-axis, as the percent of total cases infected in an uncontrolled epidemic), and (3) the transmission efficiency (as measured by the (local) reproductive number, R i ) in each patch.  Figure S14: Overview of relative performance of vaccination strategies in 2-patch system with an imperfect vaccine (VE=80%).