HS is not a contagious disease in any way. It's not an infectious disease, and it's not something like staph or other types of STDs that can be passed from person to person. There are a lot of married couples where one person has HS and the other doesn't, and that never changes throughout their whole lives.
Whereas HS can look like an infection, because there can be pus and drainage, there's also probably some role for normal skin bacteria. That makes the disease worse for some people. It is not something that can spread to others through sexual means or others, though.