Summary: In modeling disease transmission there is much emphasis on how many people are infected but less attention is paid to when the infections occur, and the unrealistic assumption of constant infectiousness is often made. We propose a missing data formulation that enables us to apply the ECME algorithm to estimate a discretized intensity function of an inhomogeneous Poisson process. This approach requires interval-censored data only, but known infection times can be incorporated as well. We apply the proposed method to transmission data on severe respiratory syndrome (SARS) collected in Singapore and Hong Kong. The resulting estimates show that the rate of infection as a function of time may have more than one peak. By fitting a two-environment proportional intensity model to the Singapore data, we estimate that the rate of infection in an (unisolated) hospital environment is almost ten times greater than occurs in a nonhospital environment. This lends support to the theory that the SARS epidemic in Singapore was mainly driven by hospital-acquired infections. Estimates of individual infectivity reveal that three persons commonly regarded as “superspreaders” actually do not have unusually high individual infectiousness. The observed superspreading events seem to have been caused by environmental rather than biological factors.