You may. Although it may or may not make sense to bring a lawsuit depending on the nature of the STD. Here's an article from nolo.com which gives a pretty good discussion of the subject.
https://www.nolo.com/legal-encyclopedia/sexually-transmitted-diseases-stds-lawsuits.html
Best regards,
Jonathan R. Ratchik, Esq.
Kramer, Dunleavy & Ratchik, PLLC
61 Broadway, Suite 2220
New York, NY 10006
(212) 226-6662
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