There are very very strong historical precedents indicating that a decrease in childhood mortality leads to a decline in birth rates. You should google this.
"The child survival hypothesis states that if child mortality is reduced, then eventually fertility reduction follows, with the net effect of lower growth of population. In populations living under low socioeconomic conditions, other factors have also been observed. The question arises whether fertility reduction could be influenced by family planning. Bangladesh data have demonstrated that if not a single child died in a family then the average total fertility rate (TFR) was 2.6 children; when 1 child died the number was 4.7 children; 2 child deaths meant 6.2 children; and more than 3 child deaths boosted the TFR to 8.3 children."[0]