Specific mortaility. "Should they be excluded?"
In no case! This is a competing risk model. I, unfortunately, have only JMP 10 versions, where is no such model. Therefore, I'll explain the scheme.
Censor code set 0 (you can leave 1, but It will be more understandable in this case).
In one column (CENSOR) we have the following codes:
1 -- dead due to the disease,
2 -- dead not due to the disease,
0 -- alive (censor).
In one (other) column (TIME) we have:
the time to death due to the disease (corresponds to 1 in column CENSOR) or
the death not due to the disease (corresponds to 2) or
the time of the last informing that the patient is alive (corresponds to 0).