Although not precisely related to Ancient Rome, Michel Foucault wrote about the socio-historical genealogy of mental illness (or "madness" and "unreason") in his book History of Madness. Perhaps a Foucauldian answer to your question is that mental illness, as we moderns understand and construct it, simply did not exist. Here I will provide my own answer to a previous but related question, and I hope this will provide some interesting thoughts / material should you wish to follow up on it.

In his book, History of Madness, Michel Foucault attempts to provide a socio-historical account of how the notion of 'unreason' has shifted and become reconfigured over time in the Western European context. Foucault makes a complex and varied argument. One possible reading (and answer to your question) is that 'unreason' was, until around the 18th century, a particular form of knowledge or 'reason' that was legitimated and constituted within cultural and social contexts (e.g. mystics, seers, shamans, etc). Foucault argues that around the 18th century the notion of 'madness' began to emerge in conjunction with psychiatry and 'authoritative' or expert knowledges that subjected unreason to a sort of clinical gaze that constituted it as invalid and a sign of mental deficit in the subject, and so forth. The lines between unreason, reason, and madness became entangled with emerging forms of psychiatric and medical knowledge that focussed not on the body but on mental states, that is to say, the mind as an object of inquiry. Systems of categories and classifications emerged to provide a sort of knowledge-power infrastructure to support the development of what madness is and how we should deal with it. What we observe is that unreason becomes a subjugated form of knowledge - it becomes the purview of artists and specially sanctioned spiritual figures perhaps. Madness becomes a social problem, and one that can be dealt with through both old and emerging practices and techniques, such as cellular discipline (locking people up), interventions (psychiatric help), and medical procedures (such as lobotomies), to name a few.

Slightly related question: would "madness" in young, other wise healthy people have been viewed differently from "madness" that came as a result of old age? Were the attitudes colored by the fact that madness related to old age would be a problem largely confined to the upper classes?