What Causes Persistent Rash Around My Mouth?

hi there. i have a rash on my face around my chin / mouth area, went to the doctors a month ago she said it looked like a fungal rash from the way it had spread, prescribed clotrimazole, the rash still hasnt gone, im female 35. now my sone looks like he is showing signs of it,

is it not what she said?

how come my son as it?

is it infectious?

the only thing my son and i have shared is a hand towel