In short: medical staff make you stick around after getting a vaccination to make sure that you do not die from untreated unanticipated allergic responses and to fulfill a duty to report any symptoms that arise after a vaccination (think of it as quality assurance).
Here’s a good story about several people suddenly coming down with reactions: https://www.cbs46.com/news/severe-allergic-reaction-to-covid-19-vaccine-reported/article_2c432b97-78ed-5e14-896f-8d01f2c85605.html
Any source of a vaccine is charged with the responsibility of ensuring that the entire process is safe for everyone, which means that quality control and quality assurance (often abbreviated to QA and QC) measures are undertaken.
What happens is that each report of symptoms by patients immediately afterwards is sent to a central data store, and when a certain number of them are submitted, a short review process begins. If the reactions are severe, such as facial swelling or difficulty breathing, the response may change from “review” to “action” – such as warnings of reaction paired with requests for information from other patients receiving the vaccination.
Sometimes the flow of reports can result in a stop-sale order or restriction on further distribution of the vaccine under investigation. Sometimes, as in the case of medications (there is a similar reporting mechanism available for reporting adverse reactions and side effects of over-the-counter and prescribed medications), a black box warning is placed in effect and can escalate into a withdrawal of the item from the market (a recall).
Most reactions on record happened within a very short period of time, so fifteen to twenty minutes of waiting time should be factored in when planning for a vaccination.
So plan for a short time doom-scrolling through the Twitter, and enjoy your wait.
December 25, 2020