Mental illness probably isn't the right word for it, but hydrophobia is indisputably a disease of the brain. The virus travels from the bite wound up your CNS; when it reaches your brain you die in agony.

Mental illness isn't the right word for it. And hydrophobia is a symptom of a brain disease more accurately.

Hydrophobia is a common (albeit bit archaic?) name for the disease generally, inspired by the symptom.

That definition is more than a bit archaic.