Ok, hmm. At the risk of sounding flippant, I think it's way too early for you to assume striking out across the board.
To illustrate: it seems like you've applied to schools that are pretty selective- i.e. have acceptance rates that are on average around ten percent. If you had, say, a ten percent chance of getting into any given school, the chances that you would be rejected from the first five schools you hear back from are better than not: .9^5=~59%
However, the changes of getting rejected everywhere with those same odds are considerably lower: .9^15=~20.5%. From what it sounds like, you have a pretty competitive profile, so I don't think it's unreasonable to assume you have at least a 10% chance of getting into any given school, making for an ex ante probability of getting in to at least one place at 79.4% at the beginning of the cycle and 65.1% now that you've (presumably) been rejected by five schools.
Obviously these probabilities are dependent on the quality of the applicant (again, it seems like you are more qualified than average) and these calculations are extremely rough, but I hope they give you a sense of why you have reason to keep hope. Even very qualified applicants can expect to get rejections galore in an environment this brutally competitive.