We have a similar concept on numerical semigroups, borrowed from the idea of run.
For a numerical semigroup, a desert is an run of gaps (non-negative integers not belonging to the semigroup), and it has some connections with the second Feng-Rao number of the semigroup, related to algebro-geometric codes.
In this setting, for a numerical semigroup $S$, then you can count deserts as the cardinality of the Apéry set of -1, that is, the cardinality of $\{s\in S : s+1\not\in S\}$.
In any case, I my feeling is that you should allow infinite intervals in your definition. Then, the Apéry set of 1 would also recover the interval $[-\infty,-1]$.
