Not necessarily died, I think. I can think of three reasons they'd turn off the siren, all to do with the fact its not an emergency any more.
1) The patient is stabilised and out of any near danger;
2) The patient dies, and is beyond resuscitation;
3) The patient has recovered and it's found that there was nothing really wrong with him.
No. 3 could be the case if he had simply fainted; they'd take him to hospital anyway to get checked out. If he had "died", they'd have tried to resuscitate him, which can take up to 10 minutes or more, and I guess they'd only turn the siren off if they failed.
X-Ray should be able to offer better insights to this one.
