Identifiability is the property in mathematical modelling that determines if model parameters can be uniquely estimated from data. For infectious disease models, failure to ensure identifiability can lead to misleading parameter estimates and unreliable policy recommendations. We examine the identifiability of a modified SIR model that accounts for under-reporting and pre-existing immunity in the population. We provide a mathematical proof of the unidentifiability of jointly estimating three parameters: the fraction under-reporting, the proportion of the population with prior immunity, and the community transmission rate, when only reported case data are available. We then show, analytically and with a simulation study, that the identifiability of all three parameters is achieved if the reported incidence is complemented with sample survey data of prior immunity or prevalence during the outbreak. Our results show the limitations of parameter inference in partially observed epidemics and the importance of identifiability analysis when developing and applying models for public health decision making.

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