I guess I was just worried that, say, my dad would get better for a day or two but then catch a mutated strain from me, getting sick again. Then we'd all be trapped in a cycle of sickness as we gave each other new germs.
It didn't make a ton of sense in my mind but I know that Covid mutates quickly and it's perfectly possible to catch it again months later.
It sounds like you have some misconceptions about how reinfection works in the first place.
Coronaviruses "mutate quickly" in general, relatively speaking, but it's falling antibody titres (and to an extent changes in the antibody types being produced) that drive reinfection, not mutation - most mutations don't even change the resulting protein product at all, and those that do usually don't affect antibody binding (this is necessary - the viral proteins still have to have roughly the same forms in order to function, so mutations that can
both inhibit binding
and still produce a working protein are rare). Even when they do, the effect is most likely partial, and with the broad spectrum of differnetly-targeted antibodies produced in an actual infection, inhibiting antibody binding in one place still doesn't change the binding anywhere else. You can also tell this from the fact that the Omicron type is
substantially different from the original strain, but still hasn't fully neutralized antibodies that were made to target it, although they seem to be much less effective. Most of the currently circulating variants are just not different enough from earlier Omicron-type variants for there to be much difference in antibody binding. (Of course, the considerable differences we see between Omicron and the original are enough, but that's not something that happens easily or often.)
Another line of argument that could help: viruses go through an
enormous number of generations just inside one single person. The viruses inside you at the end of an infection have experienced a huge amount of evolution toward "infecting this guy in particular" compared to the ones that went in. If it were that easy to mutate away from your antibodies while the immune system is active and producing them at high levels, you'd never be able to clear an infection at all. (Well, the innate immune system would still help, but it'd have a hard time of it.)
Now, there is a caveat, which is that, if the body is
still specifically producing antibodies targeted toward the original strain even while being infected by an Omicron-type strain, which can happen, then those antibodies just don't work very well and you can still be infected; but that has nothing to do with
ongoing mutation so the speed of mutation doesn't apply. You could be hit by the same exact Omicron strain again and again and fall for it every time. That shouldn't apply to you, though, since you've never had it before, unless you already had antibodies to earlier strains.