[ddebernardy]

> I kind of assume the virus grows exponentially inside the body and would dwarf the initial constant.

That assumes your immune system wouldn't kick in during the asymptomatic phase or a time close to exiting the latter. But your immune system would actually kick in as soon as it detects the infection, which would plausibly be much earlier. That would effectively buying you time to figure out which antibodies to produce before things become out of control.