It was supposed to end last night
Last week, I had a chance to check out the Destiny alpha ahead of time, and enjoyed it quite a bit. Live during their E3 press conference, Sony announced that everyone who signed up in time, would be able to play the same alpha from Thursday until Sunday -- which was yesterday.
The thing is, you can still boot up the alpha and play it. Confirming our suspicions, Bungie tweeted out the following message, "There's a rumor that the Destiny Alpha might be extended to allow dangerous experiments. Play beyond the official close at your own risk!"
So if you're home sick today or whatnot, give it a go and see if you can get online!