Intensive care unit (ICU) patients are highly susceptible to hospital-acquired infections due to their poor health and many invasive therapeutic treatments. The effect on mortality of acquiring such infections is, however, poorly understood. Our goal is to quantify this using data from the National Surveillance Study of Nosocomial Infections in ICUs (Belgium). This is challenging because of the presence of time-dependent confounders, such as mechanical ventilation, which lie on the causal path from infection to mortality. Standard statistical analyses may be severely misleading in such settings and have shown contradictory results. Inverse probability weighting for marginal structural models may instead be used but is not directly applicable because these models parameterize the effect of acquiring infection on a given day in ICU, versus "never" acquiring infection in ICU, and this is ill-defined when ICU discharge precedes that day. Additional complications arise from the informative censoring of the survival time by hospital discharge and the instability of the inverse weighting estimation procedure. We accommodate this by introducing a new class of marginal structural models for so-called partial exposure regimes. These describe the effect on the hazard of death of acquiring infection on a given day s, versus not acquiring infection "up to that day," had patients stayed in the ICU for at least s days.