Public Holiday has a 14-day recovery time, so can't occur in that time. You can't tell in-game how far into that it is, keep track on paper or on a third-party archive like elitebgs.app
After that, if the conditions are met (which may itself take a day or may take several months, depending on player activity), either:
- Public Holiday will recur for another two days
- Another event will occur (Pirate Attack is quite likely, I would expect ... others less so but possible). If this happens, Public Holiday cannot recur until after this event has ended and its 14-day recovery has expired.
What conditions cause Public Holiday? Well, no-one outside Frontier knows - likely it has multiple things which make it more imminent and multiple things which make it less imminent, and you want much more of the first than the second. Good research topic if someone's looking to make a name for themselves.
(Public Holiday is an Event state, not an Economy or Security one, so it doesn't show up on the state sliders.)