When multiple infections are possible during an individual's lifetime, as with influenza, a host's history of infection and immunity will determine the result of future exposures. In turn, the suite of varying individual infection histories will shape the population level dynamics of the disease. Exploring the consequences of precisely how immunity is acquired using mathematical models has proven challenging though: if n strains have circulated previously, there are 2(n) combinations of past infection to consider. However, by using an age-structured mathematical model of a disease with multiple strains, we can examine the population immune profile without explicitly keeping track of all possible infection histories. This framework allows previously unknown consequences of assumptions about immune acquisition to be observed. In particular, we see that 'original antigenic sin' can reduce immunity in some age groups: these immune blind spots could be responsible for the unexpectedly high severity of certain past influenza epidemics.