I have a problem specified in this way, I'll make a fictional example, because the actual data requires quite a bit of domain knowledge to be understood.
There is a series of newborn babies (let's say 10000), affected by a rare condition that makes their chance of survival really small.
We measure the concentration of a certain protein (denoted by variable $x_1$) each day at 9 am for 20 years.
What we observe is that most of the babies die within 1 year, and almost all are gone within 3 years. However there is a very small fraction of these babies that survive all the way (and expected to live a normal life).
I'd like to model if there is a relation between the level of this protein and the chance of survival. But I have both time-dependent covariates and a very small, but crucial, "immortal" fraction. So far I found that the extended cox model can be used to model time-varying covariates, but not a cured fraction.
I found that there are parameteric mixed models that allow for cured fraction but not time-varying covariates.
How would you approach this problem?