Understanding the infection and recovery rate from parasitic infections is valuable for public health planning. Two challenges in modeling these rates are (1) infection status is only observed at discrete times even though infection and recovery take place in continuous time and (2) detectability of infection is imperfect. We address these issues through a Bayesian hierarchical model based on a random effects Weibull distribution. The model incorporates heterogeneity of the infection and recovery rate among individuals and allows for imperfect detectability. We estimate the model by a Markov chain Monte Carlo algorithm with data augmentation. We present simulation studies and an application to an infection study about the parasite Giardia lamblia among children in Kenya.