The reinfection matter has largely been debunked. Most experts believe it is due to false negative test results the first time around.
As far as trusting people goes, I am certainly not in favour of anything so foolish. When they get their negative test, they get a certificate. Failure to show the certificate results in jail.
It amuses me that Bill Gates (who I do not consider any kind of saint), having grown bored with being the richest man in the world, decided to try to buy some good will and enhance his reputation by doing charitable work. He would’ve done far better (at least as far as his reputation goes) to have retired to a private island and screwed young prostitutes for the rest of his life. Nobody would have said anything bad about him. Instead he tries to save the world (for his own ego, admittedly) and people hate him.