I am not sure if I fully understand your question. I take it you were positive for an STI, had treatment and then had sex with an untreated partner. Let me know if I got that wrong.
If the person you had sex with was not treated, depending on the STI it could be in their throat and could pass with oral contact. Given this it is possible that it could be passed back to you.
I would suggest getting in touch with the clinic who did the testing for you or the public health people that contacted you. They would know what STI you had and would be able to give you the best info on what to do.
After getting treatment for STI like gonorrhea and chlamydia you are usually good to go after 1 week. You just need to be careful if having sex with an untreated partner as most STI do not show symptoms and can pass with oral sex, you wouldn’t know if they had it or if it was passed back to you. With these STI just wait 1 week after your partner has been treated before having sex.
If someone had syphilis it would be the same expect people need to wait for 2 weeks before having sex.
I find in these situations if there is any doubt we usually suggest both people get treated again or we do a retest in a few weeks (time period depends on the STI and testing method)
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