  Real-time vaccination following an outbreak can effectively mitigate the
damage caused by an infectious disease. However, in many cases, available
resources are insufficient to vaccinate the entire at-risk population,
logistics result in delayed vaccine deployment, and the interaction between
members of different cities facilitates a wide spatial spread of infection.
Limited vaccine, time delays, and interaction (or coupling) of cities lead to
tradeoffs that impact the overall magnitude of the epidemic. These tradeoffs
mandate investigation of optimal strategies that minimize the severity of the
epidemic by prioritizing allocation of vaccine to specific subpopulations. We
use an SIR model to describe the disease dynamics of an epidemic which breaks
out in one city and spreads to another. We solve a master equation to determine
the resulting probability distribution of the final epidemic size. We then
identify tradeoffs between vaccine, time delay, and coupling, and we determine
the optimal vaccination protocols resulting from these tradeoffs.
