We study the dynamics of epidemics in a networked metapopulation model. In each subpopulation, representing a locality, the disease propagates according to a modified susceptible-exposed-infected-recovered (SEIR) dynamics. In the modified SEIR dynamics, individuals reduce their number of contacts as a function of the weighted sum of cumulative number of cases within the locality and in neighboring localities. We consider a scenario with two localities where disease originates in one locality and is exported to the neighboring locality via travel of exposed (latently infected) individuals. We establish a lower bound on the outbreak size at the origin as a function of the speed of spread. Using the lower bound on the outbreak size at the origin, we establish an upper bound on the outbreak size at the importing locality as a function of the speed of spread and the level of preparedness for the low mobility regime. We evaluate the critical levels of preparedness that stop the disease from spreading at the importing locality. Finally, we show how the benefit of preparedness diminishes under high mobility rates. Our results highlight the importance of preparedness at localities where cases are beginning to rise such that localities can help stop local outbreaks when they respond to the severity of outbreaks in neighboring localities.