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vivster said:
Couldn't it also be that in some people the infection was so weak that there are no antibodies to detect anymore?

It's probably because months have passed already, and the rapid tests are poor at detecting low antibody tithers (see what happened in New York, where they found out their antibody tests were useless unless applied between 2 - 4 weeks after infection, on average). The numbers were 61% in Bergamo by the end of April, and 25.6% in Lombardy by early May.

https://bergamo.corriere.it/notizie/cronaca/20_aprile_30/coronavirus-bergamo-primi-risultati-test-61percento-positivi-85e8e36c-8a6b-11ea-94d3-9879860c12b6.shtml

https://www.milanotoday.it/attualita/coronavirus/tst-sierologici-positivi.html

Does this mean these people are vulnerable again already? Well, we know celullar, rather than humoral, is the main source of immunity against coronaviruses, and these tests might not even be going for the longest lasting neutralizing antibodies (which would be igA in the respiratory cavities).