Get a fish to teach you chaos theory!

Why does the number e appear in models of exponential growth?

Yes the approved vaccines are safe. If you balance the risks, then taking the vaccine is much safer than taking your chances with COVID-19.

Because the vaccine may not stop transmission of the virus.

The efficacy of vaccines is estimated from trials, using statistical techniques.

Like any medical drug, vaccines are tested in large-scale randomised controlled trials.

Modelling shows that old and vulnerable people should be prioritised.

We can't say for sure, but everything depends on how many people agree to be vaccinated.