Hang on, so you're going to do a half in 4 weeks? First ten days you are resting from the race you've just done, then you're running, then you're resting for ten days. So thats your twenty days or 4 weeks: a grand total of 2 runs altogether.
And on that you want to run a half marathon? Nothing to stop you, but I have to ask, why exactly? It won't be a very pleasant experience. You'd be better of gradually getting your runs up to three times a week or something and getting a good mileage base, focussuing on short events like 5ks or 10s if you want to race once in a while.