The Stochastic Firefighter Problem

Abstract: The dynamics of infectious diseases spread is crucial in determining their risk and offering ways to contain them. We study sequential vaccination of individuals in networks. In the original (deterministic) version of the Firefighter problem, a fire breaks out at some node of a given graph. At each time step, b nodes can be protected by a firefighter and then the fire spreads to all unprotected neighbors of the nodes on fire. The process ends when the fire can no longer spread. We extend the Firefighter problem to a probabilistic setting, where the infection is stochastic. We devise a simple policy that only vaccinates neighbors of infected nodes and is optimal on regular trees and on general graphs for a sufficiently large budget. We derive methods for calculating upper and lower bounds of the expected number of infected individuals, as well as provide estimates on the budget needed for containment in expectation. We calculate these explicitly on trees, d-dimensional grids, and Erd\H{o}s R\'{e}nyi graphs. Finally, we construct a state-dependent budget allocation strategy and demonstrate its superiority over constant budget allocation on real networks following a first order acquaintance vaccination policy.