With rapid transmission, the coronavirus disease 2019 (COVID-19) causes more than 2 million deaths worldwide. Human-to-human transmission is most common for COVID-19, primarily via respiratory droplets or aerosols from an infected person. This makes contact tracing, which records the travel paths of confirmed patients in detail, a highly effective disease control measure. Spatial point process models for contact tracing data can provide useful insights on the mechanism of patient visit occurrence. Here, we propose a new point process model that can provide the probabilistic mechanism of patient visit occurrence, which can provide useful epidemiological information such as a warning system for local hot spots. Analysis of the contact tracing data is challenging as patient visits show strong clustering patterns while the clusters of the events may have complex attraction-repulsion behavior. To account for such behaviors, we develop a novel interaction Neyman-Scott process that regards observed patient visit events as offsprings generated from a parent spreading event.