Competing risks model with transient surgical mortality ($S(t)=e^{-a\min (t,\tau )-bt}$)

If one cause-specific hazard acts at constant rate $a$ only until time $\tau$, and another acts at constant rate $b>0$ indefinitely, overall survival is $S(t)=e^{-a\min (t,\tau )-bt}$. The first cause has ultimate cumulative incidence function $a(1-e^{-(a+b)\tau })/(a+b)$; the second accounts for all remaining eventual failures. This illustrates how a transient competing hazard modifies a persistent cause's cumulative risk.