TL;DR: "Breakthrough infections" is a misleading term, let's not use it. Not sure what the ideal term is, maybe post-vax infections.
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Variants do muddy the water a bit, but let's set them aside for a moment. No vaccine is 100% effective. The wide range of available Covid vaccines are quite effective, with the predominant ones used in the US, Pfizer and Moderna, being an astonishing 95% efficacious. Which, reminder, means not that 5% of vaccinated people are destined to eventually get Covid; but rather, the incidence of getting Covid will be 95% lower in vaccinated people.
So, variants aside, the efficacy rate indicates that there will always be some fully vaccinated people who get Covid. Why? The vast majority of the time, it is because, for one reason or another, the vaccine did not generate an adequate immune response. Sometimes because the person is already immunocompromised (more likely if elderly, or if on immunosuppressants, or perhaps exposure to a massive viral load), but also sometimes happens for other reasons to younger, healthy people.
My point: in those cases, nothing has been "broken through" by the nasty Covid virus. Rather, there was no strong defense in place, Covid waltzed right into their body. No breaking was required. This no more involves breaking-through than a non-vaccinated, immunologically "naive" person getting Covid.
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Another thing that should be emphasized is that even when breakthrough infections occur, the severity is markedly less. Many are entirely asymptomatic, and found only by random testing. Others are very mild. The efficacy of vaccines against severe Covid and death is stunningly high, in the 99+% range. That gets back to the flawed levee analogy, and also maybe the nature of Covid variants, as compared to other things, especially the familiar disease-resistant antibiotics. It it much less all-or-nothing. A person with a weaker response, or a variant that is more vaccine-evasive, usually involves far less risk of a dire outcome than for an unvaccinated person.
So variants are worrisome, but not cause for panic. Of course, the more people that get vaccinated, the less change for variants to emerge.
In the meantime, let's stop referring to post-vax infections are "breakthrough".