As for the vaccine quote, I would ask Gates about that. But I have a little theory:
Before vaccines, families used to have more kids than today---because more bodies were needed to work the farm or whatever. And infant mortality, before the mid-20th century, was extremely high throughout the world--at least 100 of 1,000 babies died before their first birthday in those days. Moreover, before vaccines, many children and young adults died, or were rendered insolvent, because of such preventable diseases as polio and diphtheria, thereby necessitating the creation of "new bodies" to support the family.
Therefore, with vaccines, less children would die of preventable diseases--hence, there is less of a need for "bodies" to support the family. This would (and did) have an exponential effect on birth rates.
I understand that there is a civil liberties issue relating to vaccines and how the government requires their usage.