The authors have declared that no competing interests exist.

Dynamic models in disease ecology have historically evaluated vaccination strategies under the assumption that they are implemented homogeneously in space and time. However, this approach fails to formally account for operational and logistical constraints inherent in the distribution of vaccination to the population at risk. Thus, feedback between the dynamic processes of vaccine distribution and transmission might be overlooked. Here, we present a spatially explicit, stochastic Susceptible-Infected-Recovered-Vaccinated model that highlights the density-dependence and spatial constraints of various diffusive strategies of vaccination during an outbreak. The model integrates an agent-based process of disease spread with a partial differential process of vaccination deployment. We characterize the vaccination response in terms of a diffusion rate that describes the distribution of vaccination to the population at risk from a central location. This generates an explicit trade-off between slow diffusion, which concentrates effort near the central location, and fast diffusion, which spreads a fixed vaccination effort thinly over a large area. We use stochastic simulation to identify the optimum vaccination diffusion rate as a function of population density, interaction scale, transmissibility, and vaccine intensity. Our results show that, conditional on a timely response, the optimal strategy for minimizing outbreak size is to distribute vaccination resource at an intermediate rate: fast enough to outpace the epidemic, but slow enough to achieve local herd immunity. If the response is delayed, however, the optimal strategy for minimizing outbreak size changes to a rapidly diffusive distribution of vaccination effort. The latter may also result in significantly larger outbreaks, thus suggesting a benefit of allocating resources to timely outbreak detection and response.

It has long been recognized that an epidemic of infectious disease can be prevented if a sufficient proportion of the susceptible population is vaccinated in advance. This logic also holds for vaccine-based outbreak response to stop an outbreak of a novel, or re-emerging pathogen, but with an important caveat. If vaccination is used in response to an outbreak, then it will necessarily take time to achieve the required level of vaccination coverage, during which time the outbreak may continue to spread. Thus, one must consider the logistical and operational constraints of vaccine distribution to assess the ability of outbreak response vaccination to slow or stop an advancing epidemic. We develop a simple mathematical framework for representing vaccine distribution in response to an epidemic and solve for the optimal distribution strategy under realistic constraints of total vaccination effort. Focused deployment near the outbreak epicenter concentrates resources in the area most in need, but may allow the outbreak to spread outside of the response zone. Broad deployment over the whole population may spread vaccination resources too thin, creating shortages and delays at the local scale that fail to prevent the advancing epidemic. Thus we found that, in general, the best strategy is an intermediate optimum that deploys vaccine neither too slow to prevent escape from the outbreak epicenter, nor too fast to spread resources too thin. The specific optimum rate for any given outbreak depends on the infectiousness of the pathogen, the population density, the range of contacts amongst individuals, the timeliness of the response, and the vaccine intensity. This insight only emerges from linking an epidemic model with a realistic model of outbreak response and highlights the need for further work to merge operations research with epidemic models to develop operationally relevant response strategies.

In applied epidemiology, models are increasingly used to inform management decisions on effective responses to a variety of outbreaking diseases, e.g. cholera [

Density-dependent pathogen transmission is a standard assumption in most disease models, represented by a positive correlation between local population density and the probability of individual infection. Few studies, however, consider the effect of density-dependence on outbreak control. In compartmental models of rubella (e.g. [

The utility of a disease response model arguably hinges on its potential to inform management decisions for uncertain future outbreak scenarios under known logistical constraints. Realistic limitations on controls have been frequently neglected in disease transmission models in the interest of analytical simplicity, leading to assumptions of a spatially constant vaccination rate (e.g. [

As with disease transmission, vaccination may be assumed to originate from a focal location from which the response is coordinated: e.g. a central distribution point for house-to-house campaign as is used in polio vaccination or village-to-village strategies used for measles immunization days [

Our model is developed within an original framework that integrates an agent-based simulation of disease spread with a partial differential equation that describes the spatiotemporal distribution of vaccination effort. Prior to this study, spatial models of disease dynamics have often been developed using one of these two frameworks, which represent the Lagrangian and the Eulerian approaches, respectively. We use agent-based simulations to describe the process of transmission as a collection of random infective contacts at the individual level. Simultaneously, we characterized the vaccination response using a compartmental model based on a partial differential equation to describe the process of response as spatiotemporal variations in vaccination coverage at the population level. The combination of these approaches allows us to predict the epidemiological consequences of implementing a robust management decision in response to any realization of a novel outbreak. Our objective is to evaluate the performance of explicit vaccination strategies where the total vaccination effort is constrained on the landscape and its effectiveness varies with local population densities. In particular, we aim to identify context-specific, “optimal” vaccination strategies that minimize expected outbreak size given constrained vaccination effort.

We consider a continuum of vaccination strategies for a fixed amount of vaccination effort. On one end of this range, we have “fast and free” vaccination strategies with a high rate of radial diffusion, which cover a large area quickly, but leave few vaccinators per unit area thereby reducing local vaccination efficiencies. On the other end, we have “slow and steady” vaccination strategies with low rates of radial diffusion, which result in high local coverage but introduce an opportunity cost by delaying the implementation of vaccination efforts in areas far from the initial focus. In other words, by constraining total response effort, the former strategy ensures low intensity of vaccination broadly, while the latter results in high intensity of vaccination locally.

Here, we illustrate that the optimal strategy for vaccine distribution depends on both the population density and the promptness of the outbreak response. We find that a “one size fits all” policy does not exist; there are situations where the optimal strategy for minimizing outbreak size is to concentrate vaccination effort locally to control disease spread, and others where a rapid spatial expansion of vaccination effort is crucial for reducing overall burden. By extension, our context-specific results highlight a tension between direct and indirect protection: if a “slow and steady” strategy can contain an outbreak, then indirect protection can be realized broadly by direct protection concentrated in a small area; by contrast, a “fast and free” strategy distributes direct protection over a broad area, and the resulting indirect protection is local (i.e. neighbors) in scale.

We define our model on a two-dimensional discretized landscape that contains _{0} at time ^{2}^{−1} is the rate at which interaction between individuals decays as a function of their distance apart. During each time step, an infected individual may recover with probability

In the numerical simulations, vaccination is carried out per time step _{0}, which may also represent a central distribution point or medical facility from which the response originates, and variance-covariance matrix

We assume that the total available vaccination effort at the global scale remains constant over time but finite across space, such that ∫_{Ω}

Starting from

We explore the effect of population density and the rate of radial expansion of a vaccination program on the number of individuals either directly protected by vaccination or indirectly protected by herd immunity. On a 101 x 101 square grid, we ran 6000 simulation replicates of an outbreak per population density, ranging from 2000 to 10201 (the latter gives an average density of one individual per grid cell) individuals on the landscape (^{−1} = 2 epidemic generations) after the start of transmission. Assuming that distributing a vaccination response more quickly incurs a larger logistical cost, we define the optimum rate as the slowest rate

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In the absence of vaccination response, our model reflects density-dependence in the transmission process as expected (

a) Fraction of population infected, i.e. outbreak size (red) and duration (blue) as a function of population density in the absence of response effort. 60 density-specific simulation replicates are run until transmission opportunity ceases, each replicate is initialized with a central infected individual and a randomized population of susceptibles. Time step ^{th} and 80^{th} percentile values. b) Time required for different vaccination strategies to reach 90% herd immunity within a circle of radius

In the absence of disease transmission, the time required to achieve strong herd immunity ahead of the outbreak varies nonlinearly with vaccination diffusion rate in a pattern that reflects both spatial constraints and negative density-dependence in the response process (

Comparison of

We first consider a timely vaccination response, for which vaccination begins at the same time as the outbreak. At low population densities (

Given population density, 60 simulation replicates were run for each vaccination diffusion rate ^{th} order polynomial regression curves to smooth out simulation noise. Small circles of corresponding colors show the sizes of all major outbreaks in the events of occurrence.

Within the density range where infection readily establishes and spreads (^{th} order polynomial regression of the proportion infected as a function of vaccination diffusion rate in order to smooth over simulation noise.

Population densities are simulated from 2000 to 10201 at intervals of 200. Time step

At high population density (

We verified that intermediate vaccination diffusion rates are optimal to control outbreaks in moderately-to-highly dense populations for a range of interaction scale, transmission rate, and vaccine intensity parameters (see

We also optimized vaccination strategies when the response is delayed, lagging behind the outbreak by 2 epidemic generations. When population density

Here, we present a linked epidemic and vaccination distribution model that evaluates the performance of spatially explicit vaccination distribution strategies. The model integrates an agent-based SIRV simulation of a stochastic outbreak with a continuous-time diffusion model that describes the process of vaccination diffusion. Vaccination coverage is spatially constrained and individual vaccination rate depends both on the local density of susceptibles (vaccine demand, or need) and the area over which vaccination resources are distributed. We found that, for a timely vaccination response, the optimal rate of vaccination diffusion depends on population density. Under our model setting, at low densities (

Based on sensitivity analysis, the optimality of intermediate rates of vaccination in high-density contexts appears to be a general pattern that emerges from the dynamic interactions between transmission and response processes. For all parameter combinations considered, we found a range of susceptible densities over which the outbreak size is minimized by vaccination diffusion at an intermediate rate. Thus, we find that the emergence of an intermediate optimum diffusion rate as population density increases was consistent; however, the population density at which this intermediate optimum emerges and the relative benefit of the optimum depends on the epidemiological context.

By modeling vaccination as a dynamic process, our results address the inherent constraints and tradeoffs in deploying control over space and time. Conventionally, management though vaccination is described in terms of the herd immunity threshold: the proportion of the population that needs to be immune in order to prevent epidemic spread. For instance, this threshold is 1 − 1/_{0} in the classic mean-field models [

By formalizing the spatial limitation and temporal (i.e. transient) dynamics of vaccine delivery in our model, we are able to find an optimal strategy in terms of the action to implement (e.g. diffuse vaccination effort at a specific rate) rather than an outcome to achieve (e.g. total coverage). Furthermore, our representation of an explicit vaccination process reveals the intrinsic tradeoff in the consequence of any action (i.e. area of coverage versus treatment intensity), such that vaccinating a population as extensively as possible can be suboptimal in dense demographic settings. If we consider that population density may change over time, for example when suburban, sparsely-inhabited areas grow, these results would further suggest that the optimal response strategy is likely to differ between outbreaks that happen at different stages of urban development, changing over time from “slow-and-steady” to intermediate rate vaccination.

An emergent pattern of our model is the effect of initial response time on the optimal vaccination diffusion strategy. As expected, timely response surpasses delayed response in terms of the fraction of the population protected under all density conditions; the distribution of outbreak sizes for a timely response is consistently lower than that for a delayed response. Interestingly, however, the recommendation from this model is that the best strategy for a timely response is different from that of a delayed response. In particular, a timely response performs best when vaccination is distributed more slowly, with more effort initially concentrated in the local vicinity of the outbreak epicenter, thus areas (or individuals) far from the epicenter indirectly benefit from the stronger effort to locally contain the outbreak. This containment was not possible with a delayed response in our simulations; therefore, the best option is to distribute protection as quickly as possible to the whole population. To the extent that the slower vaccination diffusion rate recommended for a timely response is less expensive to implement, our results would suggest that the outcome could be improved by allocating that savings to enhanced surveillance and rapid outbreak detection, yielding a smaller outbreak overall.

There have been analogous studies of control of pests and invasive species that have highlighted the benefit of focusing effort on the invasion front [

To achieve mechanistic clarity, we wrote our model in a general form and made a number of simplifying assumptions regarding the response process. First, we fixed the center of vaccination effort to the initial disease epicenter throughout the simulation. Second, the total amount of vaccination resource is subject to a conservation constraint, such that the value of its spatial integral equals one at all time steps; thus, we have presumed that any type of depleting resource (e.g. vaccine, syringe) is promptly replenished during each transmission cycle and never increases, e.g. due to an influx of additional resources. Third, since we defined vaccination strategies only by constant diffusion rates, the distribution of response effort is insensitive to local context-dependent factors. This suggests an implicit vaccination priority in the campaign, one that always favors individuals close to the epicenter rather than targeting those in more remote, higher-risk areas. Fourth, we implemented response in homogenously distributed populations with low degrees of clustering.

While these assumptions are simplistic relative to logistical complexity of vaccination response, they do present a low-dimensional representation of key operational constraints that are seen in real vaccination campaigns. House-to-house oral polio vaccination campaigns are among the best documented [

In all scenarios presented here, we assume that the same vaccine delivery strategy is applied in all areas, which the model expresses using a constant diffusion coefficient for the deployment of vaccination effort. This describes a campaign that covers a fixed distance in random direction per unit time, which may be difficult to achieve in more remote regions. However, in practice, it is common to use mixed strategies, with fixed post delivery in dense urban centers and mobile strategies in more rural areas [

Future extensions of our model where the existing assumptions are relaxed can be used to explore vaccination delivery systems of increased realism and complexity. These developments are technically approachable using our current framework, which has an analogous form to classic models of animal home range and territory formation. Equations that combine diffusion and advection processes have been used to describe either attractive or avoidant behavioral responses of territorial mammals to local stimuli, such as resource availability [

Our study represents a simple framework for studying disease response under logistical constraints. We showed that vaccination strategies informed by an operational understanding of vaccine allocation are critical to optimizing public health objectives. Future work that builds on our implementation of a spatially constrained vaccination process is particularly relevant for reliably predicting the consequence of intervention campaigns, as it would identify necessary tradeoffs and prioritizations in vaccine distribution.

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Given population density, 60 simulation replicates were run for each vaccination diffusion rate ^{th} order polynomial regression curves to smooth out simulation noise.

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Population densities are simulated from 2000 to 10201 at intervals of 200. Time step

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(A, B) show, respectively, the fraction of the population infected at the end of an outbreak under a timely response times, i.e. implemented at ^{th} order polynomial regression curves. Small circles show the sizes of all major outbreaks in the events of occurrence. In (B, D, F), the optimal diffusion rate indicates the minimum vaccination diffusion rate

(TIF)

(A, B) show, respectively, the fraction of the population infected at the end of an outbreak under a timely response times, i.e. implemented at ^{th} order polynomial regression curves. Small circles show the sizes of all major outbreaks in the events of occurrence. In (B, D, F), the optimal diffusion rate indicates the minimum vaccination diffusion rate

(TIF)

(A, B) show, respectively, the fraction of the population infected at the end of an outbreak under a timely response times, i.e. implemented at ^{th} order polynomial regression curves. Small circles show the sizes of all major outbreaks in the events of occurrence. In (B, D, F), the optimal diffusion rate indicates the minimum vaccination diffusion rate

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Results are collected and tallied from three sets of sensitivity analysis (

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Population is initialized with

(MP4)

We thank Amalie McKee and Brian Lambert for help and comments.