After I get over a cold or the flu, do I need to buy a new toothbrush so I don't catch the same illness again?

No, you do not need to get a new toothbrush. The majority of the time you can't get the same virus twice since you develop antibodies that protect you the next time, but bacterial infections are different; you can get them more than once. The following are a list of tips and suggestions to make your toothbrush remain cleaner and last longer. 

• Do not cover or store toothbrushes in closed containers. Keeping toothbrushes covered prevents the bristles from drying out completely, which promotes the growth of bacteria, since it flourishes on a wet surface.

• Never share your toothbrush with anyone. Sharing your toothbrush with someone else increases your risk of infection, due to the possible exchange of saliva, blood and microorganisms.

• Carefully rinse your toothbrush with warm water both before and after brushing your teeth. This helps remove any minute dust or debris particles that may have fallen onto the bristles while air-drying, as well as leftover toothpaste and germs from your mouth.

• Replace your toothbrush every three to four months. Since the bristles become more worn with each use, replacing your toothbrush often will increase their cleaning effectiveness.