According to this dictionary about the etymology of English words:
LUNATIC, lunatique, F. lunaticus, L. From Luna, the moon, because that kind of madness called lunacy increases with the moon. Luna qu. lucina, is from luceo, to shine.
Related, this other dictionary states:
lunatic: L. Luna, 'moon'; one too much under the influence of the moon, which, according to astrology, caused insanity.
So it seems inherent in the word that there is an intermittent pattern in the "disease".
Does this "lunar effect" have any scientific basis? Wikipedia states:
Two studies found evidence that those with mental disorders i.e. Schizophrenia generally exhibit 1.8% of increased violent or aggressive episodes during the full Moon, but a more recent study found no such correlation to that of nonschizophrenic human beings. An analysis of mental-health data found a significant effect of Moon phases, but only on schizophrenic patients. Such effects are not necessarily related directly to the appearance of the Moon.
An article in the Scientific American magazine provides an interesting analysis of this. It starts with some background:
ACROSS THE CENTURIES, many a person has uttered the phrase “There must be a full moon out there” in an attempt to explain weird happenings at night. Indeed, the Roman goddess of the moon bore a name that remains familiar to us today: Luna, prefix of the word “lunatic.” Greek philosopher Aristotle and Roman historian Pliny the Elder suggested that the brain was the “moistest” organ in the body and thereby most susceptible to the pernicious influences of the moon, which triggers the tides. Belief in the “lunar lunacy effect,” or “Transylvania effect,” as it is sometimes called, persisted in Europe through the Middle Ages, when humans were widely reputed to transmogrify into werewolves or vampires during a full moon.
Towards the end, it states:
According to Raison, the lunar lunacy effect may possess a small kernel of truth in that it may once have been genuine. Raison conjectures that before the advent of outdoor lighting in modern times, the bright light of the full moon deprived people who were living outside—including many who had severe mental disorders—of sleep. Because sleep deprivation often triggers erratic behavior in people with certain psychological conditions, such as bipolar disorder (formerly called manic depression), the full moon may have been linked to a heightened rate of bizarre behaviors in long-bygone eras. So the lunar lunacy effect is, in Raison and his colleagues’ terms, a “cultural fossil.”
I could not find the original references about this in Aristotle and Pliny the Elder.