# 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.

# Why does aggregation of macroparasites matter?

Note:  This is the third post in a series about aggregation of macroparasites.  You can see the first post here and the second post here.

In the last two posts, we established that macroparasites are pretty much always aggregately distributed among hosts, and this aggregation can result from several ecological processes (e.g., variation in infection rates among hosts).  This week, we will answer the next obvious question: who cares?!  Why is macroparasite aggregation important enough to study?  There are many reasons, but we’ll focus on these three things:

1. Individual host fitness
3. Regulation of host populations

Individual Fitness:

As we’ve already discussed, the more parasites that a host is infected with, the more likely that host is to suffer negative fitness consequences.  Highly infected hosts might have lower fecundity, slower growth rates, or higher mortality rates, for instance.  So, when we’re considering individual-level fitness, we need to consider individual-level parasite loads.  That is, assuming that all hosts harbor some mean number of parasites is probably not going to cut it.

They say that repeating something over and over is a good teaching technique. How about using the same graph in three different posts? 😛 Remember, “awfulness” – a proxy for mortality rates, for instance – doesn’t necessarily increase linearly with parasite abundance.

Having a bunch of parasites might also make a host more susceptible to future infection – the “vicious circle” of disease where having parasites leads to lower body condition which leads to higher susceptibility to parasites which leads to MORE parasites, etc.  Once upon a time, I made a cartoon about that. Parasite transmission – superspreaders!

Say you’re about to go eat some delicious sushi, and the chef lets you decide between a fish with just a few parasites and a fish with tons of parasites.  Which one do you pick?  (Yes, I know, I’m evil.)  Obviously, you’re more likely to get infected by the parasites – assuming that they’re trophically transmitted and you can serve as the next host – if you eat a huge dose of parasites. This brings us to the topic of superspreaders, which we have discussed once or twice on this blog already.  If we’re talking about just one host species, superspreader hosts are responsible for a disproportionate amount of the parasite/pathogen transmission.  For instance, superspreaders may be individuals that 1) are heavily infected, 2) are shedding more ‘infectious particles’ (i.e., parasites) than other hosts, or 3) both.  (Other options exist – like individuals with normal infection levels who have many social contacts and thus transmit more infectious particles than others).  These individuals are the ones that exist in the tail of the negative binomial distribution: Superspreaders are important because if we can identify which individuals are the superspreaders, we can target them for disease management.  For instance, if we can easily recognize the wormiest hosts – by doing fecal egg counts, etc. – then we can target those individuals with anthelmintic drugs.

Regulation of Host Populations I could talk about regulation of host populations by parasites all day.  Like here.  But let’s just focus on how aggregation of parasites affects host populations.  And for that, let’s visit the very important Anderson and May (1978) paper.  I was originally going to make this post very mathy, but instead, I’m just going to summarize one main point of Anderson and May (1978), and you guys can check out the PDF here for the beautiful details.

If you start with a host population that has exponential growth in the absence of parasites, no parasite aggregation, no affects of parasites on host mortality or fecundity, etc., you can get two basic kinds of population behavior – damped oscillations to a constant population sizes or population cycles.  The potential problem with this basic, simplified model is that it is neutrally stable – meaning that if you perturb the system from an equilibrium, it shifts to a different equilibrium.  And that means that the model is structurally unstable: a small change in parameters (e.g., infection rate) can cause the model to shift from one qualitative behavior to another, and that may not be biologically realistic.

Even though the basic model has some less than ideal characteristics, we can use it as a baseline model to see what happens when we add complications to the model.  In a series of two papers, Anderson and May (1978) added a bunch of complications: overdispersion of parasites, underdispersion of parasites, parasite-induced host mortality, density-dependent parasite population growth, and other things.  Some of these things made the model dynamics more stable, and some of them make the model dynamics more unstable.  And the relevant point for this post is that aggregation of parasites is a stabilizing force in host population dynamics.

So, aggregation of parasites among hosts is important because individual-level parasite loads determine individual host fitness and transmission potential, and the individual-level impacts scale up to affect transmission in host populations and also the stability of host population dynamics.  Next week, we’ll talk about how to model macroparasite aggregation.  Stay tuned!!

References: Anderson, R. M., and R. M. May. 1978. Regulation and stability of host-parasite population interactions. I. Regulatory processes. Journal of Animal Ecology 47:219-247.

# Why are macroparasites aggregately distributed among hosts?

Note: This is the second post in a series about macroparasite aggregation.  You may need to read the first post to understand this one.

Last week, I told you guys that macroparasites are (almost) always aggregately distributed among hosts.  This week, I want to talk about why macroparasites are aggregately distributed.  That is, we want to know which ecological processes cause parasite aggregation.

First, let’s think about what determines the number of macroparasites on any given host.  To increase the number of macroparasites on a host, new parasites either need to immigrate to the host or new parasites need to be born on the host.  To decrease the number of macroparasites on a host, some of the current parasites either need to emigrate or die.  Therefore, to understand why some hosts have few or many parasites, we need to know more about parasite births/deaths and immigration/emigration.

When we talk about parasite “immigration,” we’re talking about the rate at which hosts get infected.  Let’s imagine for a minute that every host has the same probability of becoming infected.  For instance, think back to our discussion of density-dependent transmission (you might need to do a little review here).  In a regular old SIR model, the rate that a given susceptible individual becomes infected is B*I, where B is the transmission rate and I is the number or density of infected individuals.  If every host has the same probability of becoming infected, then B is the same for all hosts.  For instance, let’s say that it doesn’t matter if you’re Bob, Sally, or Thor, you have a 10% chance of getting infected by this parasite on any given day of your life (B = 0.1).  In a scenario like that, we expect parasites to have Poisson distributions among hosts, with mean/variance = 1.  Explaining why exactly that is the case is a bit hard – it’s just how Poisson processes work – but the key point is that by using some basic statistics, we know that when infection is a random process with a fixed probability of infection, parasites will be Poisson-distributed.

Think back again to that previous post and remember that transmission rate B was a “conglomerate” term, where B = c*v, c was the average per capita contact rate between individuals, and v was average probability of successful transmission of the parasite, given that a contact occurred.  And let’s say that each time step, a host has a 50% chance of contacting another host (c=0.5), and there is a 20% chance that a contact will lead to successful transmission of the parasite (v=0.2).  To generate a Poisson distribution of parasites using those values, you need to know that the mean in a Poisson distribution (lambda) is also called the rate parameter, and it is equal to some rate, r, multiplied by the length of time that the process has been occurring, t.  In our case, r = B = c*v = 0.5*0.2 = 0.1.  And then we can just pick a t – say, 20 time steps.  So, let’s generate a Poisson distribution for this infection process that has been running for 20 time steps:

parasites<-rpois(n=1000, lambda=0.5*0.2*20)

var(parasites)/mean(parasites) ##I get 1.02

hist(parasites, right=FALSE, ylab=”Number of Hosts w/ X Parasites”)

Before, all hosts (Sally, Bob, and Thor) had the same 0.5*0.2 = 0.1 = 10% chance of getting infected each day.  But what if the hosts don’t have the same probability of getting infected?  That is, what if B varies among hosts?  In that case, a regular Poisson distribution is no longer appropriate.  What we need instead is a compound distribution, where infection is still a random Poisson process, but now the rate parameter (r*t = lambda) isn’t a constant.  Instead, the rate parameter itself is a random variable.  This may seem like it’s getting unbearably complicated, but we can illustrate this idea with a very simple example.  Let’s say that the rate parameter, r*t, is no longer a constant 0.1, but now it is a random variable that follows a gamma distribution.

rate<-rgamma(10000, shape=1, scale=10)

parasites<-rpois(n=10000, lambda=rate)

##parasites<-rpois(n=10000, lambda=rgamma(10000, shape=1, scale=10)) ##same

var(parasites)/mean(parasites) ##I get 11.07

hist(parasites, right=FALSE, ylab=”Number of Hosts w/ X Parasites”)

What just happened?  We let the rate of infection vary among hosts in a gamma-distributed way, and then the distribution of parasites among hosts went from being something with a variance/mean ratio of 1 to a variance/mean ratio > 1.  The distribution became overdispersed/aggregated!!! And as it turns out, a Poisson-Gamma compound distribution is the same thing as a negative binomial distribution.  You can learn more about parasites and Poisson-Gamma distributions in May (1978).

When infection varies among hosts, does the distribution of infection rates need to be gamma-distributed?  Nope!  For instance, a Poisson-Poisson compound distribution (also called a Neyman Type A distribution) will also be overdispersed.  You guys can try that one in R.  Also, you can learn more about parasites and the Neyman Type A distribution in Anderson and Gordon (1982).

That was a lot of stats.  Now let’s get back to the biology.  What we’ve just shown is that when parasite immigration rates (i.e., infection rates) are constant among hosts, parasites will have a Poisson distribution among hosts.  They will not be aggregately distributed.  But when infection rates vary among hosts – say, infection rates have gamma distributions – parasites will be aggregately distributed among hosts.  Basically, any kind of variation in infection rates acts as an aggregating process.  There are also processes that act to un-aggregate parasites, like density-dependent parasite mortality, but you’ll have to go check out Anderson and Gordon (1982) if you want to know more about that.

Finally, we want to know why infection rates might vary among hosts.  Clearly, such variation is important, but what causes it?  If parasites/infective stages are not uniformly distributed in the environment – if they have a gamma distribution among patches, for instance – and then host contact with parasites within a patch is random (Poisson), you get a Poisson-Gamma distribution, which is the same as a negative binomial distribution (May 1978).  In other words, heterogeneity in parasite distributions in the environment can case variation in infection rates.

There has also been a lot of interest in variation in infection rates due to variation in host susceptibility.  For instance, when the host’s immune response is related to normally-distributed host body condition, you can theoretically get aggregation of parasites among hosts (Morrill and Forbes 2012).  Infection rates might also vary according to host sex, host age, host genotype, host behavior, and a bunch of other things that you can read more about in this book chapter.

Everything we just talked about was related to parasite immigration.  I should point out that whenever you have parasite births on hosts, you also end up with aggregated distributions (Anderson and Gordon 1982).  Many macroparasites don’t multiply on their hosts, though – they reproduce and then broadcast their infective stages into the environment – so births on hosts can’t explain why most macroparasites have aggregated distributions.

Ok, to recap: any time infection rates vary among hosts or parasites reproduce and multiply on hosts, you can get aggregation of parasites among hosts.  And variation in infection rates can be caused by a variety of abiotic and biotic processes, like variation in host immunity.

Now you know why macroparasites have aggregated distributions!!  Next week, we’ll talk about why aggregated distributions are important.  Stay tuned!

References:

Anderson, R. M., and D. M. Gordon. 1982. Processes influencing the distribution of parasite numbers within host populations with special emphasis on parasite-induced host mortalities. Parasitology 85:373-398.

May, R.M. 1978. Host-parasitoid systems in patchy environments: A phenomenological model. Journal of Animal Ecology 47: 833-844.

Morrill A, Forbes MR. 2012. Random parasite encounters coupled with condition-linked immunity of hosts generate parasite aggregation. International Journal for Parasitology 42(7):701-6..

# 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 21st 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