Clearly state your research questions and goals before conducting an analysis like this.

Competing risks states a different 'world you believe in'. Do you want curves or effect estimates for a world where, when people are treated for this disease, they cannot possibly get cured?

That's what a competing risk analysis would do and it doesn't sound very helpful.

The other way around would be, in a world where after treatment you cannot possibly die, how often (and quickly) would you get cured?

This is actually why I believe Multi-State Models (MSM) would be more appropriate.  Individuals do get cured, but either relapse or get reinfected and can die.  However, my first attempts (using msm package in R) I was getting an overflow error.

Most survival analyses try to determine the effectiveness of treatment.  What is the durability of the positive response (cure) to the treatment?  The survival function is most simply summarized by a Kaplan-Meier curve where each successive failure decrements the remaining fraction surviving.  Competing risks make the Kaplan-Meier percentage surviving inaccurate because patients may be subtracted from the pool before they have a chance to fail. 

Disease specific mortality is not a competing risk because that event is scored as a failure either prior to or at the time of death.  Other cause mortality and loss to follow-up are events that skew the survival curve and should be included. 

The other factors you mention depend on when you start the survival clock.  For treatments of finite duration, we usually begin on the last day of treatment.  If patients require indefinite treatment to keep the disease in check, then you start the clock at the beginning of treatment and plot the growth of survivors which is tempered by a background failure rate. 

A definitive physical analog that has been well studied is the manufacture of specific radionuclides by bombardment activation of a parent nuclide. 

Do not turn towards more elobarate, sophisticated methods just because you can and you are curious what comes out of them.

What is the reason? What are you trying to give an answer to?

Your research will be much more efficient if you state that for yourself first.

This statement is actually one of my problems with survival analysis: 

"Most survival analyses try to determine the effectiveness of treatment."

If the Kaplan-Meier curve is measuring the differences between survival/mortality, as its outcome (binary outcome).  I do have deaths but I have twice as many cures.  Analyzing mortality is definitely important but it seems to me that ignoring the cure state throws away a lot of data.

That is actually the stimulus for the question...  If you have alternative outcomes during survival analysis (e.g. death due to your disease of interest, death due to other diseases) you use competing risks.  In my case I want to add "cure" as an alternative.

Dear Donald,

I guess You mix "event" and "action": an event is state of patient, an action is treatment.

For example:

You have some patient who have no treatment .

competing events-

1 Death (due to disease),

2 Death (due to other causes),

3 Cure

Others are censored

You have some patient who had complete treatment .

competing events-

1 Death (due to disease),

2 Death (due to other causes),

3 Cure

Others are censored

The questions are - how long is treatment? From what timepoint You should start follow-up? Completed Treatment (but did not cure)- they have chance of cure after some months...?, on those questions depend how to handle data from patient with fail treatment and Completed  and Continuing Treatment

=====

If You follow-up patients with cure ( to death), so it is more complicated model (multistate)

Anyway events chart is appreciated.

Maybe you can organize your data in terms of cure models, for example, mixture cure models. 

I have a suggestion.

The management of healthy people is the standard, the disease is the risk, the disease degree is the risk level.

You might get unexpected results. You could do the same thing.