If the drug drastically lowers infant mortality then population growth will grind to a halt. 

we see this in countries where infant mortality drops low enough, people have fewer children. Evolutionarily speaking this happens because when the chances of children dying is high people unconsciously have more children and invest less time and energy in each child because that is more successful in getting your genes into the next generation, once infant mortality is low the opposite is true your best bet evolutionarily is having only a few children and investing as much time and energy as possible into them. 

population booms really only happen when a place is transitioning from one to the other. We see this over and over again in populations, infant morality drops and population growth drops a generation later. 

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/zUYk6.png