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How serious is a leaking roof in a typical American home, in a typical location with all 4 seasons? The roof in question has failed to the point that parts of the inner ceiling are visibly sagging, and water actually drips onto the floor during rainstorms. And there are audible sounds of bats in the attic. How long until this house becomes unlivable? How long until black mold becomes a health problem? How soon until this problem really snowballs? The family in this house doesn't open any windows either, not even during a hot summer.
My instinct tells me this is really serious, because unlike a log cabin, a modern home is not designed to dry out quickly after each rain shower or storm. Much less airflow, you have drywall, insulation, lumber everywhere to hold the moisture and grow mold in between storms.

IMO, it's past that point already based on what you describe.  I'm sure the attic is filled with bats, mold, all kinds of vermin, etc. ... and that it's just a matter of time before the ceiling gives way.  No, modern homes will not just dry out on their own, and even if they dry out, the mold will still be there, as will the bats, vermin, etc.  It could probably go at any time.  Most roofing companies will come out and inspect the roof with the thought of providing a estimate.  If the wood on the root is rotten, they may not want to go up there because it would be dangerous, but I think a roofing company could give them (for free) a good idea of the current extent of the damage, since all those considerations would factor in to their estimate.

People put off replacing their roofs because it's too expensive and they simply can't afford it.  For my house, the quote was around $15,000 for a new roof.  Some shingles claim to be good for 30 years, others for 20 or 15, and some newer ones even claim 50.  I'm due for one within the next 3-4 years and am not looking forward to it.  I'm thinking of doing the metal roofing ... which should theoretically last forever ... but then it's going to cost twice as much ... but then if go with 30-year shingles, and I'm still around and still own the house 30 years from now, I'd be looking at doing it again.

You don't want to get a new root too soon because it's expensive, but if you put if off even a little too long, the expense of replacing the roof goes way up, where you likely have to replace the wood / OSB on the roof, clean out the attic from mold (very expensive), and replace the ceiling drywall.

A more interesting question is how this topic even remotely approaches being posted in the anonymous forum?



One also wonders why this "mythical family" living anonymously in a leaky roof house won't open their windows?

Sounds like a sensitive/personal question to me.