One Advantage to High Infection Rates
The obvious disadvantage is the risk of overwhelming the hospitals, but it occurred to me that there may also be one advantage.
The latest evidence suggests that immunity after infection probably lasts at least four months. We don't know how much longer it lasts. Suppose that over four months everyone who can get infected does. At that point everyone is immune, so the virus has nobody to infect and vanishes away.
That is an extreme case, but four months is also an estimate of the current lower bound for how long immunity lasts. More plausibly, assume that the infection rate is high enough to push the population to herd immunity before the immunity of a significant number of those infected first wears off. At that point the number of infections starts to go down, since each infected person passes the disease on to fewer than one other. If it goes down far enough before the number of no longer immune people gets high enough to push the population back below herd immunity, it may reach the point where further transmission can be controlled by test and trace.
Suppose, on the other hand, that we hold the initial infection rate low enough so we eventually that reach a point at which people are losing immunity as fast as people are getting it, while the number immune is still below the herd immunity level. In that case the disease goes on forever, or at least until a vaccine becomes available.
Obviously the argument does not hold if a vaccine is going to show up shortly, which at this point seems likely. It also does not hold if immunity is permanent, as might be the case. But it does imply that under some circumstances a higher infection rate is better, in the long run, than a lower.
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