First let's deal with the nitpicky semantics. R0 is is the basic reproduction number while R generally denotes the effective reproduction number. I am talking about R. There's indisputable evidence that the vaccination is impacting R. While it's true that the accuracy of estimates for R depends on many factors and certainly there have always been some cases flying under the radar (though that has been studied extensively via antibody studies and other means) proxies like the number of people dying, the number of people hospitalized, while controlling for things like vaccinations are accurate enough to detect this. There are tons of studies, random…
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e.g.: https://www.mdpi.com/2227-9032/9/10/1245/pdf Trying to argue confounding factors vs. the observations is IMO very weak. Ofcourse we can't tell anything with certainty in this domain. It seems like the people who want proof here set the bar impossibly high, no proof will ever satisfy them. EDIT: adding some more references... https://www.nature.com/articles/s41591-021-01316-7 https://www.nbcnews.com/health/health-news/vaccinated-people... https://www.theatlantic.com/ideas/archive/2021/09/the-vaccin... https://www.gavi.org/vaccineswork/mounting-evidence-suggests... https://www.publichealthontario.ca/-/media/documents/ncov/ph...