The bad news is that putting a lid on it doesn't result in any substantial reduction in contamination, according to their recent paper published in the American Journal of Infection Control. The good news: Adding a disinfectant to the toilet bowl before flushing and using disinfectant dispensers in the tank significantly reduce cross-contamination.
The first experiments examining whether toilet plumes contained contaminated particles were done in the 1950s, and the notion that disease could be spread this way was popularized in a 1975 study. In 2022, physicists and engineers at the University of Colorado, Boulder, managed to visualize toilet plumes of tiny airborne particles ejected from toilets during a flush using a combination of green lasers and cameras. It made for some pretty vivid video footage:
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