  In the maximum $n$-times coverage problem, we are provided a set of elements,
a weight for each element, and a set of overlays where each overlay specifies
an element specific coverage of zero or more times. The goal is to select up to
$k$ overlays such that the sum of the weights of elements that are covered at
least $n$ times is maximized. We also define the min-cost $n$-times coverage
problem where the objective is to select the minimum set of overlays such that
the sum of the weights of elements that are covered at least $n$ times is at
least $\tau$. We show that the $n$-times coverage objective is not submodular,
and we present an efficient solution by sequential greedy optimization. We
frame the design of a peptide vaccine for COVID-19 as maximum $n$-times
coverage using machine learning defined candidate peptide sets, and show that
our solution is superior to 29 other published COVID-19 peptide vaccine designs
in predicted population coverage and the expected number of peptides displayed
by each individual's HLA molecules.
