Macroparasites are aggregately distributed among hosts

We’ve discussed the difference between micro and macroparasites on this blog several times.  To recap: microparasites tend to cause density-independent pathology, while macroparasites tend to cause density-dependent pathology.  In other words, the more macroparasites a host has, the more likely the host is to die or suffer reduced fitness.  A few weeks ago, I illustrated that idea with this graph:

The relationship might not be linear.

I’ve also briefly mentioned that macroparasites tend to be aggregately distributed among hosts.  In a series of four posts, I’m going to go into much greater detail about parasite aggregation.  Specifically, I’ll cover:

1. What is macroparasite aggregation and how do we measure it?  (Today)
2. What causes macroparasite aggregation?
3. Why is macroparasite aggregation important?
4. How do we model macroparasite aggregation?

In 2007, Robert Poulin – the most prolific parasite ecologist of the 21^st century! – published a review paper in Parasitology in which he posed an important question: are there general laws in parasite ecology?  He argued that for something to be an ecological “law,” there must be both a predictable, recurring ecological pattern, and we must understand the mechanism behind that pattern.  Aggregation of macroparasites was one such “law” of parasite ecology that he reviewed.  Next time, we’ll talk about mechanisms, but for today, let’s just ask: what is macroparasite aggregation, and is parasite aggregation a recurring pattern?

Before we can talk about parasite aggregation, I need to introduce some terminology.  Specifically, we need terms that can classify different types of parasite distributions.  If you went out into the field and collected a bunch of hosts and then counted how many parasites each host had, you could easily make a histogram where on the X axis you have “number of parasites” and on the Y axis you have “number of hosts with X parasites.”  That’s the distribution of parasites among hosts.  For instance, in the hypothetical distribution shown below**,  where a hypothetical nine hosts were sampled, one host had zero parasites, three hosts had two parasites, two hosts had five parasites, one host had seven parasites, and two hosts had nine parasites.

One way to think about different probability distributions is to consider their variance to mean ratios.  If you’ve taken a basic stats course, you’re probably already familiar with the mean and the variance of the normal or Gaussian distribution.  Today, we aren’t going to look at the normal distribution, but other distributions also have means and variances.  For instance, in the Poisson distribution, the mean and the variance are equal.  If you want to play along at home, here is some R code that you can use to generate a histogram for a bunch of hypothetical parasites that have a Poisson distribution, where the mean of the distribution is 2 parasites per host, and the variance is also 2.  The rpois() function is going to draw 100 numbers from a random Poisson distribution with mean=2, where ‘lambda’ is the mean.  (Because this is a random number generator, you won’t get the exact same numbers that I get, but the distribution should look similar.)

parasites<-rpois(n=100, lambda=2)

hist(parasites, xlab=”Number of Parasites”, ylab=”Number of Hosts w/ X Parasites”, main=NA)

Remember that we want to consider the variance to mean ratio.  That is, what is the variance divided by the mean?  In the case of the Poisson distribution, because the variance IS the mean, the variance/mean = 1.  Let’s prove that to ourselves with some quick code:

mean(parasites) ##I get 1.94

var(parasites) ##I get 1.936

var(parasites)/mean(parasites) ##I get 0.998

That’s pretty darn close to var/mean = 1!

If you remember back to a few weeks ago, I talked about the negative binomial distribution.  Let’s take a random sample of hypothetical parasites from a negative binomial distribution, plot the histogram, and then look at the variance/mean ratio.  The R function that you want to use is rnbinom(), where this time the mean is called ‘mu’.  For right now, set size=1, and we’ll talk about what the size is below.

parasites<-rnbinom(n=100, mu=2, size=1)

hist(parasites, xlab=”Number of Parasites”, ylab=”Number of Hosts w/ X Parasites”, main=NA)

mean(parasites) ##I get 1.68

var(parasites) ##I get 4.68

var(parasites)/mean(parasites) ##I get 2.788

In the negative binomial distribution, the variance was greater than the mean, so the variance/mean was greater than 1 (it was 2.788).  Whenever the variance/mean is greater than 1, we say that the distribution is overdispersed.  That’s an aggregated distribution.

Finally, we could have a distribution where variance/mean < 1.  That happens with a regular binomial distribution.  In that case, we would have underdispersion.

Ok, so, one more thing about the distributions.  If you go back to the negative binomial distribution, you’ll remember that we had to pick a “size” in the rnbinom() function.  It turns out the negative binomial distribution is defined by two parameters – the mean and the dispersion parameter, k.  “Size” means k in the rnbinom () function.  When k is small, the distribution is overdispersed.  As k increases, the negative binomial distribution converges on the Poisson distribution, and the parasites are no longer overdispersed.  Let’s do that in R.  Remember that variance/mean was greater than 1 (overdispersed distribution) when the mean was 2 and the size, k, was 1.  Now let’s keep the mean at 2 and increase k.

parasites<-rnbinom(n=100, mu=2, size=5) ##k=5

hist(parasites, xlab=”Number of Parasites”, ylab=”Number of Hosts w/ X Parasites”, main=NA)

mean(parasites) ##I get 2.04
var(parasites) ##I get 3.5337
var(parasites)/mean(parasites) ##I get 1.732

Ok, when k was 5, we still had overdispersion.  Let’s increase it more!

parasites<-rnbinom(n=100, mu=2, size=10) ##k=10

hist(parasites, xlab=”Number of Parasites”, ylab=”Number of Hosts w/ X Parasites”, main=NA)

mean(parasites) ##I get 1.9

var(parasites) ##I get 2.05

var(parasites)/mean(parasites) ##I get 1.079

When k=10 or greater, we’ve pretty much converged on the Poisson distribution: the variance/mean is very close to 1.

So, now we know about some important statistical distributions.  The question is: do macroparasites have variance/mean ratios that are greater than 1?  That is, are macroparasite distributions overdispersed/aggregated?

Why yes, yes they are!  In fact, out of 269 parasite distributions from the literature, Shaw and Dobson (1995) only found ONE that didn’t have a variance/mean ratio that was greater than 1.  And the majority of the distributions had k less than 1 – very aggregated distributions!  (Also, see this more recent paper.)  So, yes, it is safe to say that aggregated distributions of macroparasites is a recurring pattern.  If you go out and sample hosts and count their macroparasites, you can predict with near certainty that the parasites will be aggregately distributed among hosts.  Cool!

Next time, we’ll talk about WHY macroparasites are aggregately distributed among hosts.  Stay tuned!!

**Be gentle, World.  I made these graphs at midnight last night.  Judge me not for my failure to prettify my histograms in R before releasing them onto the wild, wild Internet.

References:

Poulin, R. 2007. Are there general laws in parasite ecology? Parasitology 134: 763-776.

Shaw, D. J. and Dobson, A. P. (1995). Patterns of macroparasite abundance and aggregation in wildlife populations: a quantitative review. Parasitology 111: S111–S133