This is a good point. In fact, the most effective form of population control turns out to be girl's education (both sexual education and otherwise), which reduc

This is a good point. In fact, the most effective form of population control turns out to be girl's education (both sexual education and otherwise), which reduces the birth rate. High burdens of disease prevent girl's education from happening, especially in regions where there already exists an education imbalance based on gender. Girl's education has a variety of positive non-population control effects as well (consider that without educating women, 1/2 of a nation's population will not be as productive as they would be otherwise). So the current situation is high birth rates with high death rates, which keeps population in a grim equilibrium while the rest of society's resources are spent trying to keep as many people alive as possible rather than engaged in productive development. Reducing the burden of disease is a necessary first step in freeing up these resources to be used in other ways, like girl's education, which will help to alleviate many of the problems of a "sudden" population boom that you mention.