If you read the mad dog context carefully, then it is clear that the mad dog represents what often happens when race is involved in Maycomb. The average citizen, who is rational and reasonable most of the time, goes crazy when blacks are involved. Therefore, the mad dog represents the racism in Maycomb. That we are on the right track is confirmed by what Atticus says right before the mad dog incident. He hopes that Jem and Scout don't catch "Maycomb's disease." Listen to what Atticus says:
You know what’s going to happen as well as I do, Jack, and I hope and pray I can get Jem and Scout through it without bitterness, and most of all, without catching Maycomb’s usual disease. Why reasonable people go stark raving mad when anything involving a Negro comes up, is something I don’t pretend to understand... I just hope that Jem and Scout come to me for their answers instead of listening to the town. I hope they trust me enough... Jean Louise?”
The language, Atticus uses cannot be mistaken. Madness is madness whether it is in dogs or humans.
Now when we look at the mad dog scene, it is Atticus who puts down the dog. This point is significant as well. As Miss Maudie says elsewhere, Atticus is one of those men God has given to do what others won't or can't do. Here he is the voice reason.