There are many infectious disease that are spread by insects - such as malaria, dengue, and West Nile virus. However, these infectious are relatively uncommon, especially in the United States.
What is more common, however, is that the break in the skin caused by the insect bite can be a site for bacteria to get in and cause a local infection. This is especially the case if you scratched the area, which opens it up more to infection.
If the area of swelling and redness is small, you can disinfect it with antimicrobial soap and apply an antibacterial cream and see if that takes care of it.
However, you should pay close attention and have a low threshold for calling your doctor
to be seen. In particular, if the redness is spreading further around the insect bite, if there is significant pain, or if you have fever
, or if you notice any pus coming from the site, you should go see your doctor. These are all symptoms of a likely evolving skin infection which will probably require treatment with antibiotics. Your doctor will be able to take a look at the area and help you make this decision.
Good luck and I hope you are feeling better soon.