We propose a Poisson-based model that uses both infallible data and fallible data subject to misclassification in the form of false negatives that yield visibility bias. We than derive maximum likelihood estimators for the Poisson rate parameter of interest and the misclassification parameter under two different sampling scenarios. We also derive expressions for the information matrices and the asymptotic variances of the maximum likelihood estimators for the rate parameter and the maximum likelihood estimators for the false-negative parameter. Finally, we also study our new models via a simulation experiment and then apply our new estimation procedures to a real data set.