This dissertation discusses the properties of point process models for epidemic diseases and other clustered phenomena. We present (1) a novel computationally efficient estimator for the parameters of conditional intensity functions used to model point process data, (2) a comparison of compartmental models and Hawkes-type models for predicting the spread of COVID-19, (3) a potential outcomes framework for point process data, and (4) a novel methodology for bounding the complexity of sparse Boolean-valued tensors represented as point processes, discussed here in the context of tomographic images of fractured silicon materials.