@arambaut: a nice paper, and an interesting model. One can imagine many extensions here (e.g. by making an explicit model of the switches between hosts based on a metapopulation model).
@rob_lanfear. yes, there are possible connections here. On my side, I was thinking about modeling the log-rate as an Ornstein Uhlenbeck process with host-dependent mean. But other models would certainly be possible.
However, in the present case, I wonder if autocorrelated clocks would be much better: I am so surprised by the biased divergence time estimates shown in figure 1.c.
I could possibly imagine an interaction with the prior on divergence times here: is it something like a standard coalescent or a birth-death? If yes, then is that expected to correctly describe the structured populations represented by these data?
My idea is this: if the prior on divergence times, for some reason, really wants to push estimates for ancient nodes downwards, then, just because the relaxed clock happens to be more flexible, it will not oppose enough resistance to this bias. Hence what we see in figure 1c.