Seasonal influenza imposes a significant worldwide health burden each year. Mathematical models help us to understand how changes in vaccination affect this burden. Here, we develop a new dynamic transmission model which directly tracks the four dominant seasonal influenza strains/lineages, and use it to retrospectively examine the impact of the switch from a targeted to a universal influenza immunization program (UIIP) in the Canadian province of Ontario in 2000. According to our model results, averaged over the first four seasons post-UIIP, the rates of influenza-associated health outcomes in Ontario were reduced to about half of their pre-UIIP values. This is conservative compared to the results of a study estimating the UIIP impact from administrative data, though that study finds age-specific trends similar to those presented here. The strain interaction in our model, together with its flexible parameter calibration scheme, make it readily extensible to studying scenarios beyond the one explored here.