  This work proposes a method to compute the maximum value obtained by a state
function along trajectories of a Delay Differential Equation (DDE). An example
of this task is finding the maximum number of infected people in an epidemic
model with a nonzero incubation period. The variables of this peak estimation
problem include the stopping time and the original history (restricted to a
class of admissible histories). The original nonconvex DDE peak estimation
problem is approximated by an infinite-dimensional Linear Program (LP) in
occupation measures, inspired by existing measure-based methods in peak
estimation and optimal control. This LP is approximated from above by a
sequence of Semidefinite Programs (SDPs) through the moment-Sum of Squares
(SOS) hierarchy. Effectiveness of this scheme in providing peak estimates for
DDEs is demonstrated with provided examples
