The innate immune system can protect cells from viral infection, suggesting that pathogenic viruses must outpace the innate defense they trigger in order to spread in the host. To study this race, we have developed mathematical models that describe how a susceptible host cell population splits upon virus infection and the ensuing interferon response into infected and protected subpopulations. Confronting these models with time-lapse imaging data of dengue virus infection of lung epithelial cells, we show that the rate at which infected cells transition to productive virus replication controls whether the virus will spread. We corroborate this prediction by quantifying replication dynamics in individual cells. An attenuated dengue virus mutant that is readily recognized by innate immune sensors differs from wildtype virus by having a long transition to productive replication whereas the replication rate itself is hardly affected. These quantitative findings suggest that antiviral drugs that inhibit the formation of replication organelles by dengue virus - and other plus-strand RNA viruses – will synergize with the innate immune response of the host. [Joint work with Soheil Rastgou, Alessia Ruggieri and Ralf Bartenschlager]