UPDATE: I wrote this, discussing that I don’t really know the justification for the law of mass action, however, comments from Martin and Helen suggest that a derivation is possible using moment closure/mean field methods. I recently found this article:
Use, misuse and extensions of “ideal gas” models of animal encounter. JM Hutchinson, PM Waser. 2007. Biological Reviews. 82:335-359.
I haven’t have time to read it yet, but from the title it certainly sounds like it answers some of my questions.
Yesterday, I came across this paper from PNAS: Parameter-free model discrimination criterion based on steady-state coplanarity by Heather A. Harrington, Kenneth L. Ho, Thomas Thorne and Michael P.H. Strumpf.
The paper outlines a method for testing the mass-action assumption of a model without non-linear fitting or parameter estimation. Instead, the method constructs a transformation of the model variables so that all the steady-state solutions lie on a common plane irrespective of the parameter values. The method then describes how to test if empirical data satisfies this relationship so as to reject (or fail to reject) the mass-action assumption. Sounds awesome!
One of the reasons I like this contribution is that I’ve always found mass-action to be a bit confusing, and consequently, I think developing simple methods to test the validity of this assumption is a step in the right direction. Thinking about how to properly represent interacting types of individuals in a model is hard because there are lots of different factors at play (see below). For me, mass-action has always seemed a bit like a magic rabbit from out of the hat; just multiply the variables; don’t sweat the details of how the lion stalks its prey; just sit back and enjoy the show.
Before getting too far along, let’s state the law:
Defn. Let be the density of species 1, let be the density of species 2, and let be the number of interactions that occur between individuals of the different species per unit time. Then, the law of mass-action states that .
In understanding models, I find it much more straight forward to explain processes that just involve one type of individual – be it the logistic growth of a species residing on one patch of a metapopulation, or the constant per capita maturation rates of juveniles to adulthood. It’s much harder for me to think about interactions: infectious individuals that contact susceptibles, who then become infected, and predators that catch prey, and then eat them. Because in reality:
Person A walks around, sneezes, then touches the door handle that person B later touches; Person C and D sit next to each other on the train, breathing the same air.
There are lots of different transmission routes, but to make progress on understanding mass-action, you want to think about what happens on average, where the average is taken across all the different transmission routes. In reality, also consider that:
Person A was getting a coffee; Person B was going to a meeting; and Persons C and D were going to work.
You want to think about averaging over all of a person’s daily activities, and as such, all the people in the population might be thought of as being uniformly distributed across the entire domain. Then, the number of susceptibles in the population that find themselves in the same little as an infectious person is probably .
Part of it is, I don’t think I understand how I am supposed to conceptualize the movement of individuals in such a population. Individuals are going to move around, but at every point in time the density of the S’s and the I’s still needs to be uniform. Let’s call this the uniformity requirement. I’ve always heard that a corollary of the assumption of mass-action was an assumption that individuals move randomly. I can believe that this type of movement rule might be sufficient to satisfy the uniformity requirement, however, I can’t really believe that people move randomly, or for that matter, that lions and gazelles do either. I think I’d be more willing to understand the uniformity requirement as being met by any kind of movement where the net result of all the movements of the S’s, and of the I’s, results in no net change in the density of S(t) and I(t) over the domain.
That’s why I find mass-action a bit confusing. With that as a lead in:
How do you interpret the mass-action assumption? Do you have a simple and satisfying way of thinking about it?
This paper is relevant since the author’s derive a mechanistic movement model and determine the corresponding functional response:
How linear features alter predator movement and the functional response by Hannah McKenzie, Evelyn Merrill, Raymond Spiteri and Mark Lewis.