The person who also caught the 'disease' was Jihwa, not the brother. Jihwa is the one who is jealous of Nakyum and is plotting to have him killed. I suspect the father saw the two boys kissing and that's why he said the illness was contagious. We now know Jihwa has loved Seung-ho since they were children.
Also they have never mentioned who Lord Song is. It could be the father or someone else entirely. Either way it had a huge triggering effect on Seung-ho.

wow okay so reading the recent chapter it makes sense on why he hates his father so much, being forced home due to his "illness" must of been him either getting tortured or worse to cause him to become like this, it would make sense for him to ignore his father's wishes & request to see him because he holds that grudge against him. i see comments that the dad could be homophobic since his brother has caught that "illness" too but how come the brother wasn't treated the way Seungho was? last time we saw his brother he looked to be normal, i wonder if the dad has disliked him from the beginning in the chapter you can see the dad clinging onto the back on his neck with a grip. idk maybe im looking too deeply into it but i'm curious to know why his dad may feel a bit different between the two sons & completely blames Seungho for it. Plus this part isn't a theory but who's Lord Song did i miss something?