Seriously? Mao made China healthy? Their drop in family size was caused by economic prosperity? I guess he never heard of China's "one child" policy.
He's just using raw data in an attempt to prove a correlation. There may be in fact such a correlation, but the methodology used to establish such a link is flawed.
What consequence did the export of birth control and family planning play upon these nations? Did their birth rate decline because their prosperity increased, or was it meddlesome first worlders financing abortion and birth control in these places?
Will this stuff reverse if these programs are ended?
I don't know, but I suspect it's not so simple as prosperity convincing people to have less children.
I sat in a work sponsored lecture one time where the professor told us that the richer a country got the lower the birthrate and the more regressive the country was to women the more profound the effect on birthrate. I respected the company and the lecturer and do not doubt his words. Just saying.