Tinea Corporis and Capitis ~ Carriers?

My 5 yr old daughter has been diagnosed with tinea corporis and one spot of tinea capitis ~ she had it roughly for 3 weeks.  She has been put on Griseofulvin, starting yesterday, since topical anti-fungal creams were not working.

I have been washing sheets, not having her use the same towel more than once, have sprayed lysol on everything ~ couches, surfaces, doorknobs, beds, pillows, brushes, anything she touches.  I have been in constant contact, even using the same hairbrush and have not contracted it.

My question is this:  my doctor said that it is a low level contagen and since no one else has shown signs of it, we cannot pass it on, and we are not carriers ~ ie;  we cannot give it to anyone else.  Is this true?  Would it not be on our clothing?  Can we still not spread it, even though we are not getting infected?