Hierarchically cocompact classifying spaces for mapping class groups of surfaces

We define the notion of a hierarchically cocompact classifying space for a family of subgroups of a group. Our main application is to show that the mapping class group $\mbox{Mod}(S)$ of any connected oriented compact surface $S$, possibly with punctures and boundary components and with negative Euler characteristic has a hierarchically cocompact model for the family of virtually cyclic subgroups of dimension at most $\mbox{vcd} \mbox{Mod}(S)+1$. When the surface is closed, we prove that this bound is optimal. In particular, this answers a question of L\"{u}ck for mapping class groups of surfaces.