Influenza and Respiratory Syncytial Virus (RSV) interact within their host posing the concern for impacts on heterologous viruses following vaccination. We aimed to estimate the population level impact of their interaction. We developed a dynamic age-stratified two-pathogen mathematical model that includes pathogen interaction through competition for infection and enhanced severity of dual infections. We used parallel tempering to fit its parameters to 11 years of enhanced hospital-based surveillance for acute respiratory illnesses (ARI) in children under 5 years old in Nha Trang, Vietnam. The data supported either a 41% (95%CrI: 36-54) reduction in susceptibility following infection and for 10.0 days (95%CrI 7.1-12.8) thereafter, or no change in susceptibility following infection. We estimate that co-infection increased the probability for an infection in <2y old children to be reported 7.2 fold (95%CrI 5.0-11.4); or 16.6 fold (95%CrI 14.5-18.4) in the moderate or low interaction scenarios. Absence of either pathogen was not to the detriment of the other. We find stronger evidence for severity enhancing than for acquisition limiting interaction. In this setting vaccination against either pathogen is unlikely to have a major detrimental effect on the burden of disease caused by the other.