Are stochastic simulations in continuous time really that hard?

It may be pretty obvious by now that I am a theoretician, I use the language of math to formulate my questions and mathematical analysis and computer simulations to explore the nature of the models I devise. Currently I am going through a period where I largely focus on explore the dynamics of finite populations in continuous time. Typically I first formulate a deterministic continuous time model of the biological system I am interested in. This involves a series of coupled ordinary differential equation assuming infinite population sizes. I then use this deterministic model to derive the corresponding stochastic continuous time model for a finite population. The main difference between the deterministic model and the stochastic model is that demographic stochasticity now plays a role (we assume that the environment stays constant). Basically who is born, who dies, and who procreates is stochastic.

These type of stochastic models have several obvious advantages. For example, they are an exact stochastic representation of the deterministic model. This means one can use these models to validate deterministic predictions (as it turns out sometimes population dynamics in finite populations can be radically different from the deterministic predictions, see e.g. Pineda-Krch et al . 2007). Another advantage is that most biological processes occur in continuous time and discrete population models are often used as approximations to continuous processes (e.g. epidemiological chain models, see e.g. Becker 1980 , Biometrics 36: 249-254). An additional bonus is that deriving this type of stochastic model from its deterministic counterpart is pretty straight forward (of course it partly depends on the complexity of your deterministic model). I think that any graduate student in biology should be expected to acquire these skills during the course of his or her graduate studies. I would certainly expect this from my students and I would also be the first one push for a graduate class on this topic. Oddly and unfortunately there are currently very few text books addressing this topic (no, I am no about to write one). Many of the classical text book in mathematical biology are simply silent on how one would go about deriving the stochastic counterpart of a deterministic continuous time population model.

Until recently I have not been clear about why this is the case. Recently I think I might have found a clue. Ben Bolker, who is an prof in theoretical ecology, in the Zoology Department at the University of Florida, is working on a book called Ecological models and data in R. I have been slowly pursuing his book for a while now, the draft version is actually available in its entirety online where he says that he would be extremely happy to get feedback, now that’s what I call a Bazaar mode of operation. One of the first chapters I read was the last chapter, Dynamic models. The clue I am getting at comes at the end of this chapter in the following statement:

One can build dynamical models that are stochastic, discrete-valued (and hence more sensible for populations) and run in continuous time, picking random numbers for the waiting times until the next event (birth, death, immigration, infection, etc.). The basic algorithm for simulating these models is called the Gillespie algorithm (Gillespie, 1977), but it, and the advanced methods required to estimate parameters based on such models, are beyond the scope of this chapter (Gibson and Renshaw, 1998, 2001). [Ch 11, pp 6-7 in the 27 December 2006 draft version]

Considering that this book is geared to the upper-level graduate students with some pretty serious math in it, the only conclusion I can reach is that the topic of stochastic models in continuous time is deemed as too advanced for this book (and, to be entirely fair, also the parameter estimations required for this type of model). This is unfortunate because the Gillespie algorithm is very simple and would fit in quite nicely into the context of the book. In the large scope of things the omission of stochastic continuous time models in most text books on mathematical biology may be the simply be due the idea that this topic is to advanced for graduate students to tackle. I beg to differ.