A study published in the American Journal of Infection Control found that the bathtub was the most likely site for Staphylococcus aureus contamination to occur in the bathroom (and one of the most likely places in the house). Staphylococcus aureus, or staph, is a bacteria that causes skin infections.
Though bleach is the only way to definitively eliminate staph in your house, it is also toxic. Of course. According to Science Daily, essential oils such as thyme can be effective in killing staph bacteria.* Scrub with a cleaner (like this one), rinse, and dry the tub with a clean towel. If you don't want to purchase expensive cleaners, you can make your own! Follow this tutorial by One Good Thing by Jillee to make reusable disinfectant wipes with thyme essential oil.
*If you're interested, The Atlantic also addresses similar experiments that demonstrate the effectiveness of using essential oils to fight drug-resistant bacteria.