Although it's obviously causing you a lot of grief, the fact that the forum isn't overrun by people reporting this, or pitching in to say "known problem - do this" means it's something fairly rare, possibly unique to your setup.
I know that's no great help to fixing it, but "This is a very serious problem if anyone wants to use this software for important monitoring" isn't fair or accurate
Zoneminder is a persistent little bu**er when it comes to IP cameras dropping off the stream, and re-getting them. There may be a short pause before it retries, but it comes back.
I don't know whether your use of FFMPEG might be complicating things. Not pointing any fingers, just saying, it's an extra layer of stuff to go wrong. What version of FFMPEG are you using?
You haven't said what camera brand you are using -- might be relevant, and also, if this suddenly started happening: What did you change
Something totally unrelated may have upset things, upgraded a library?
And have the cameras, and the PC, been given a power-down-reboot since it started (as mine worked fine until they'd been up for a couple of months, and then started, EVERY ONE OF THEM, dropping connections, faster and faster, until almost unusable).