as we get fitter it takes us longer to increase our heart rate. as it is lower to start with, also the fitter we become the better the stroke volume of the heart so therefore more oxygen per beat. To improve stamina you could try fartlek running, its a interval training technique that you incorporate into your runs, so you should be doing a faster but bit as long, one steady for a longer duration, one interval training and if at all possible one run with a mixture of flat and hill. Don't forget to include a bit of core training as well. There are a lot of good half marathon training runs on the internet I did a half marathon about 20 years ago in the Falkland islands got a time of 1.48