We use the world population and average life expectancy to calculate births in a year. To connect these point estimates at the varying times in history we use a bounded exponential growth model for births.

I'm still not completely getting it. Let's say in some year there are a billion people alive and the average life expectancy is 50 years. How are these two pieces of information sufficient to determine how many children were born in that year?

You also need the previous year's population. With population and life expectancy, you can approximate the per capita death rate, and then it's a straightforward linear equation.

What about infant mortality? If a million babies are born in the year 234BC, and die before 233BC rolls around, they won't be counted in the population for either year.

That's what I'm wondering about. The infant mortality rates in pre-industrial times were astoundingly high, but I don't see anything that tries to explicitly account for them. I guess they get rolled in with life expectancy (bringing the mean way down), but it would be nice to see an explicit formula for births_t.