Captivity seems unlikely to me - the Austrian girl whose father kept her in their basement had her children with her, and so I seem to think did some of the other women kept prisoners.
Surely, it is more likely that the mother of these children is too poor to bring them up? With all the unemployed, not to mention homeless all countries have these days and the total inadequecy of social security, I find it sadly likely that the parents of these children cannot afford to clothe and feed them.
And yes, contraceptives exist, but the pill is only available, and rightly so, on prescription, so expense kicks in here too, and no form of contraception is 100 % sure.
We all know that any woman consulting a gynaecologist is asked whether she has ever given birth and if so how many children she has - what do you expect the mother in this case to tell her doctor if she consults him or her about sterilisation? She can hardly say, "Oh yes, doctor, I have had three healthy babies in the course of seven years and abandoned them all!"