Logistical constraints lead to an intermediate optimum in outbreak response vaccination

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.Author summary: 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.