Underlying dynamic event processes unfolding in continuous time give rise to spatiotemporal patterns that are sometimes observable at only a few discrete times. Such event processes may be modulated simultaneously over several spatial (e.g., latitude and longitude) and temporal (e.g., age, calendar time, and cohort) dimensions. The ecological challenge is to understand the dynamic latent processes that were integrated over several dimensions (space and time) to produce the observed pattern: a so‐called inverse problem. An example of such a problem is characterizing epidemiological rate processes from spatially referenced age‐specific prevalence data for a wildlife disease such as chronic wasting disease (CWD). With age‐specific prevalence data, the exact infection times are not observed, which complicates the direct estimation of rates. However, the relationship between the observed data and the unobserved rate variables can be described with likelihood equations. Typically, for problems with multiple timescales, the likelihoods are integral equations without closed forms. The complexity of the likelihoods often makes traditional maximum‐likelihood approaches untenable. Here, using seven years of hunter‐harvest prevalence data from the CWD epidemic in white‐tailed deer (Odocoileus virginianus) in Wisconsin, USA, we develop and explore a Bayesian approach that allows for a detailed examination of factors modulating the infection rates over space, age, and time, and their interactions. Our approach relies on the Bayesian ability to borrow strength from neighbors in both space and time. Synthesizing a number of areas of event time analysis (current‐status data, age/period/cohort models, Bayesian spatial shared frailty models), our general framework has very broad ecological applicability beyond disease prevalence data to a number of important ecological event time analyses, including general survival studies with multiple time dimensions for which existing methodology is limited. We observed strong associations of infection rates with age, gender, and location. The infection rate appears to be increasing with time. We could not detect growth hotspots, or location by time interactions, which suggests that spatial variation in infection rates is determined primarily by when the disease arrives locally, rather than how fast it grows. We emphasize assumptions and the potential consequences of their violations.