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:
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=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!
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..