Survival Curves with a (potential) intermediate illness Suppose initially that I am interested in the death rate of males and females that enter a hospital.
I could fit survival curves and stratify by group (male or female) and obtain two separate survival curves.
I have performed a similar analysis for death rate of patients in a hospital. The patients all start in the same state, but they may be diagnosed with a disease during their stay. This is an example of a simple multi-state model.
There appears to be a clear distinction between the death rates of patients who have the disease compared to those who do not. However, I think my estimates are wrong.
My data looks as follows:
Patient Time Death Disease
      1   12     1       0
      2   23     1       1
      3   4      0       0
      4   9      0       1

Note for this made up example patients do not leave the Disease state (it is an untreatable disease).
I have fitted separate models for each of the states: Disease = 0 and Disease =1. This is what I did for the male/female example above, however, in the male/female example, the groups were fixed.
I.e., it was easy to see which female patients died and which were right censored, and the same for male patients.
In the disease model, patients who leave the hospital with no disease could catch the disease later in their lives. I.e., patients with Death, Disease = 0 are considered right censored for the model with Disease = 0, however, these patients are still susceptible to the disease. I think this simplistic approach is underestimating the number of right censored patients in the Disease = 1 group.
Are my estimates biased since the grouping factor that I am using is time dependent?
 A: As @carbocation suggests in a comment, you need to analyze these data with a formal multi-state model. As the multi-state vignette says on page 8 of the current version:

A common mistake with competing risks is to use the Kaplan-Meier separately on each event type while treating other event types as censored.

That supports your sense that a simpler analysis with separate models can lead to bias.
In terms of modeling, you say:

for this made up example patients do not leave the Disease state.

If you model this way, you have a competing-risk model with a transition from start state to Death competing with a transition to Disease. Section 3.1 of the vignette illustrates that type of analysis.
That model, however, would not address the "distinction between the death rates of patients who have the disease compared to those who do not." For that you also need to model a transition from Disease to Death, which requires specifying start times and stop times for the intervals corresponding to each transition. That requires re-formatting the data to combine the "counting-process" data format, explained in the vignette on time-dependent coefficients, with the multi-category event indicator needed for multi-state models, in which the reference category represents censoring at the end of the interval.
