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Accurate and timely data collection on infectious disease incidence is vital for monitoring as well as for decision making such as taking preventative action. Unfortunately, many such data sets are observational with a data collection mechanism that is flawed, resulting in counts of disease incidence that are missing/censored. Here we illustrate flexibility in using Bayesian hierarchical models for correcting such data. In particular, we use such models to correct under-reported tuberculosis incidence data in Brazil, as well as to correct delays in reporting of dengue data that feed into a warning system, in Rio de Janeiro.