How do we model aggregately distributed macroparasites?

Note:  This is the last post in a series of posts about aggregation of macroparasites.  You can see the first three posts here, here, and here.

So far, we’ve established that macroparasites are typically aggregately distributed among hosts, and this aggregation is important.  For instance, overdispersed parasite distributions can stabilize host population dynamics.  Therefore, when we model macroparasite transmission, we should probably include parasite aggregation in the models.  But how can we do that?  And more importantly, how can we do it in a biologically relevant way?

As we’ve discussed before, we usually model microparasite transmission with compartmental models, where we divide the host population into Susceptible hosts (S), Infected hosts (I), and resistant hosts (R).  To model macroparasite transmission, we can further divide up the infected hosts (I).  For instance, we can have hosts with one parasite (p1), hosts with two parasites (p2), host with three parasites (p3), … host with 500 parasites (p500),…and hosts with pn parasites.  As you might imagine, you end up with a LOT of equations to keep track of when you do this.  Also, if the parameters (e.g., mortality rate) differ among hosts with different numbers of parasites, you might need to estimate and keep track of a lot of parameters.

What you might want instead are just two equations: one for the total number of hosts, N, and one for the total number of adult parasites, P.  And that’s actually pretty easy to do!  But, it isn’t a closed system of equations.  That is, N and P depend on the proportion or number of hosts with each number of parasites, so you still need to deal with all that pn stuff.  Anderson and May (1978) very cleverly got around that problem, though.  If you know the mean number of parasites per host (P/N) and k (the aggregation parameter of the negative binomial distribution), you can easily figure out the theoretical proportion of hosts with n parasites.  If you plug in that pn(t) equation, you end up with a closed system.  Yay!

That Anderson and May (1978) model did great things.  It is very, very useful.  But it isn’t good for everything.  Specifically, when people use the model to predict what will happen under various disease management strategies, they usually assume that management reduces the mean number of parasites per host (P/N), but does not change k, the aggregation parameter.  But as we talked about before, biological processes that might be changed by management strategies affect the degree of parasite aggregation.  So, assuming a constant k may not be very realistic.

Part of this issue gets into realm of phenomenological vs. mechanistic models.  We know that the negative binomial distribution is a very good way to describe the pattern of macroparasite distributions.  It’s a good phenomenological model.  But the concern is that it might not be good to model macroparasite distributions in a phenomenological way if the goal of the model is to see what happens when we alter the underlying ecological processes.  In that case, we probably want to allow aggregation of macroparasites to emerge from mechanistic processes in the model.

If you’re trying to model parasite transmission and you don’t want to use a constant k, there are ways to let k vary with time or other variables.  There are also ways to let aggregation emerge from mechanistic processes, especially if you’re using agent based models.  But I think we’re going to see a lot more about this in the future, because people are still concerned that macroparasite models aren’t accurately depicting real systems.  For instance, see the recent Yakob et al. (2014) paper in the International Journal of Parasitology.

Ok!  That’s it for macroparasite aggregation for now!  Next week, I’ll switch topics and talk about Godzilla’s parasites.  Stay tuned!

References:

Anderson, R.M., May, R.M., 1978. Regulation and stability of host–parasite population interactions I. Regulatory processes.. J. Animal Ecol. 47, 219–247.

Yakoba, L., R.J. Soares Magalhães, D.J. Graya, G. Milinovicha, N. Wardropb, R. Dunningc, J. Barendregt, F. Bieri, G.M. Williams, A.C.A. Clements. 2014. Modelling parasite aggregation: disentangling statistical and ecological approaches. International Journal for Parasitology 44: 339-342.