I am not a hair dresser but I used to do red's on my hair when I was a receptionist at a salon. Red's are the hardest color to maintain. I used to have the same problem even though it was touched up more often. I wash my hair every day and that does not work well with red's. My hair color used to rinse out in the shower too. We switched to brown's with red in them. For some reason at least with me the red faded slower when mixed with the brown.
I'm not sure what line my hairdresser uses, but I do know the salon changes the line from time to time. I had a vibrant red once that I really loved, but it literally washed out in my shower! Maybe my hair isn't the right texture to really hold the red for a long time. Course I only get it coloured every other cut. So every 3 months. Maybe that's the problem!
The red washing out may have very little to do with texture of your hair, I was told to wash my hair with conditioner and only shampoo it a few times a week. This did seem to help a little.
I loved having red hair but now that I am no longer working in a salon it is way too expensive to maintain. I only had to pay 10% of product cost and the tip for the person doing my hair. I don't bother with hair color at all right now.