It's been a while, but if I recall his argument is something along the lines of "if parents know their kids won't die to disease they'll have less, since they know the ones they do have will be alive to take care of them when they're old and frail."

As if people are basing their choice for how many kids to have solely on getting taken care of when they're old. It's all a bunch of junk, and we know the real reason he pushes vaccines, but I don't see that he's directly admitted that the reason is depopulation.