Hi everyone! I have been doing research on the TKF91 model and its successors for over a year now. Recently, I have been tasked with writing a research proposal, on mathematical / statistical / computational aspects of indel models, for a general math-professor audience.

It seems that, for Markovian indel models at least, continuous-time models (TKF91, TKF92, long indel, Poisson indel, etc.) are much more popular than discrete-time models. (I only know of one discrete-time indel model, due to David Koslicki in his PhD thesis. And this work has only 7 citations.) I have been wondering why this is the case. Why have continuous-time models been so much more popular than discrete-time models? And relatedly, why did TKF choose continuous time over discrete time for their 1991 paper?

If I know that discrete-time models are actually valid, then I can avoid continuous-time models in my proposal (to make things less confusing to the reader). In addition, depending on the virtues of continuous time versus discrete time, it may make a good research proposal to study discrete-time indel models until we understand them as well as continuous-time indel models.

One possible argument for continuous time may be that continuous-time Markov chains are simpler than discrete-time countable-state Markov chains. For instance, in continuous time there are no periodicity issues (in continuous time, ergodic is equivalent to being both irreducible and positive-recurrent, whereas in discrete time for a countable state space, ergodic is equivalent to irreducible, aperiodic and positive-recurrent). However, most people (including me) would say discrete time is simpler than continuous time – the continuous-time theory has a few more technical complications.

Another possible argument may be that continuous-time models are more realistic. For instance, in continuous-time models we can have arbitrary branch lengths, whereas in discrete-time models branch lengths are multiples of a fixed constant. However, I see almost no problems with choosing the constant small enough so we can have a dense-enough range of possible branch lengths as we want.

So both arguments are unconvincing to me.