Chris, might I suggest that you walk before you hit the 3-4 minute mark. Warm up first with a brisk walk, stretch, then begin.
If you know you are really going to suffer at the 3 - 4 minute mark then why not run for 2 minutes, walk, recover and then run for another 2 minutes.
This is just a suggestion of course and you can ignore it but I wouldn't get to the stage when you feel you are going to cough up a lung, not unless you are sprinting for the finish line.
If you can dedicate 20 minutes then try this:
5 minute warm up and stretch
Run 2 minutes, walk 1 minute x 3 times
Cool down for 5 minutes and stretch
You will find that each time you go out it will get easier and you can increase the time you are jogging/running.
Another tip is to avoid doing the same route every time, especially if it is on a camber, alternate the route if at all possible.
With time it will become easier, you will reap the rewards and you will want to do more.
Do not be tempted to rush this nor go too far or too fast too soon.
Good luck and see you on a half marathon very soon