Say we have the following set up. At time t=0 there are N infected patients. There is a treatment which, if taken until t=T, cures 100%. However, some patients will be cured before t=T while others will die before t=T. I need to analyse times to death. How do I treat those patients that have cured? They are not really censored, in the way I understand it, since we in fact know that they are cured. I am looking for (references to) exact instructions on how to construct distribution function for death times. Many thanks.
Added after an answer from DWin I realise that hazards, which I understand to mean (in discrete time framework) the probability of dying at time $t=n$ conditional on having survived and not cured by time $t=n-1$. But what about unconditional probabilities?
If I didn't have "cures" and a conditional probability of dying at time $t=n$ is $d_n$, then the unconditional probability of dying, say, at time $t=2$ is $P(D=2) = (1-d_1) \cdot d_2$.
But with "cures", do I not need to take into account the fact that at each time $t$ I now have three outcomes: (1) dead, (2) not dead but not cured, (3) cured? And is the probability of dying at $t=2$ now given by $P(D=2)=(1-d_1)\cdot(1-c_1)\cdot d_2$ where $c_n$ is the conditional probability of curing at time $t=n$?
From playing around with numbers, this seems to be the right way to go in the sense that the cumulative probabilities constructed in this way recover the correct number of deaths that accrued by time $t$. But how to recover hazard rates from a cumulative distribution function constructed in this way is not clear to me.