Toothbrush packages warn us: "Got a cold? Change your toothbrush." Why? Because "germs can hide in toothbrush bristles and lead to reinfection." Can a person really recatch a cold from a toothbrush? No, unless it's someone else's toothbrush (or someone else's cold).
Once you've been infected with a particular virus, your antibodies fight reinfection. Besides, the viral load on the toothbrush's bristles is negligible compared with what's already in your system. However, it is possible for a bacterial reinfection such as strep throat. But even that threat might be mitigated by toothpaste, which sometimes contains antibacterial compounds.
The American Dental Association suggests rinsing off your toothbrush after brushing, storing it in a position that allows it to air dry, and keeping it away from other toothbrushes. And replace your toothbrush every three or four months, especially if bristles are worn!