  In adversarial patrolling games, a mobile Defender strives to discover
intrusions at vulnerable targets initiated by an Attacker. The Attacker's
utility is traditionally defined as the probability of completing an attack,
possibly weighted by target costs. However, in many real-world scenarios, the
actual damage caused by the Attacker depends on the \emph{time} elapsed since
the attack's initiation to its detection. We introduce a formal model for such
scenarios, and we show that the Defender always has an \emph{optimal} strategy
achieving maximal protection. We also prove that \emph{finite-memory}
Defender's strategies are sufficient for achieving protection arbitrarily close
to the optimum. Then, we design an efficient \emph{strategy synthesis}
algorithm based on differentiable programming and gradient descent.
