Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 207 / 6 / c / Solution

The two event types are disjoint and exhaustive among first events, so their cumulative incidence functions sum to the probability of any event by $t$. Equivalently their derivatives add to $S(t)(h_A(t)+h_B(t))=-S^{\prime }(t)$. ThereforeFor the usual positive disease-death intensity ${\theta }_A>0$, the limit is one: eventually every individual has an event, even though treatment-side-effect events cease after $\tau$. If the permitted boundary ${\theta }_A=0$ is included, however,Then a fraction $e^{-{\theta }_B\tau }$ remains event-free forever, because all hazards vanish after $\tau$. This includes the no-event case ${\theta }_A={\theta }_B=0$. Thus an assertion that the limit always equals one requires ${\theta }_A>0$.