Pathogen-pathogen interactions represent a critical but little-understood feature of infectious disease dynamics. In particular, experimental evidence suggests that influenza viruses and respiratory syncytial viruses (RSV) compete with each other, such that infection with one confers temporary protection against the other. However, such interactions are challenging to study using common epidemiologic methods. Here, we use a mathematical modeling approach, in conjunction with detailed surveillance data from Hong Kong, to infer the strength and duration of the interaction between influenza and RSV. We further utilize our model to evaluate the potential of live attenuated influenza vaccines (LAIV) as a novel strategy to control RSV epidemics. We find evidence of strong, bidirectional interference, such that infection with either virus yields almost complete protection against infection with the other for two to four months. Furthermore, we demonstrate that the likely impact of widespread LAIV use depends on underlying viral circulation patterns. More broadly, we highlight the utility of mathematical models as a tool to characterize interactions and predict the indirect effects of vaccines.