The rat is only a product of your mind, an image created by your subconcious, representing all that is dirty, immoral, etc that you have done and are. This is your subconcious trying to get away from that and lead you to better yourself. It is showing you your errors in a way you can realise and fight them.
You just have to look beyond the horizon. You are Jekyll and the rat is your Mr. Hyde.
Defeating the rat means bettering yourself, becoming a kind, mannered person. Kill your dark side, kill the rat!