1  Macroparasite models

1.1  Complex life cycles: vector-borne disease

If we write out the equations for this we get

dIhost dt = a i Nvec Nhost Ivec(1-Ihost) -gIhost (1) dIvec dt = a c Ihost(1-Ivec) - mIvec (2)

with parameters:

Ivec	fraction of vectors infected
Ihost	fraction of hosts infected
a	biting rate of a female mosquito
i	infection probability of a mosquito biting an infected host
c	infection probability of a host being bitten by an infected mosq.
g	host recovery rate
m	mosq. death rate
m = Nvec/Nhost	female mosq. per host

From Anderson & May:

The ratio N_vec/N_host arises as a direct consequence of the fact that female mosquitoes only take a fixed number of blood meals per unit of time. Thus the net rate of transmission is held to an upper limit, irrespective of the absolute densities of mosquitoes and people, by the biting rate times the number of female mosquitoes per person. This term embodies the actual difference between vector and direct transmission (see Anderson and May 1979b, Anderson 1981a).

This makes sense in the following way: if we multiply the first equation above by the total number of hosts (N_host) and the second by the number of vectors (N_vec) to get numbers rather than proportions, we get the total number of bites as a N_vec in both cases (rather than a N_vecN_host). Increasing the number of hosts actually decreases the proportion of the population infected.

With these parameters we can figure out R₀ by following a parasite around the cycle between hosts and vectors: the number of hosts that an infected mosquito infects in its lifetime is (biting rate) × (infec. probability) × lifespan = ai/m, while the number of mosqs that get infected by an infected human during the course of their disease is a m c/g, so the overall R₀ is m a² i c / (gm).

Macdonald (1957) found that ac/m (mosquito biting rate times infectivity times lifespan) was a good index of malaria stability; where it is low (R₀ > 1 but still small), malaria can have periodic outbreaks, while where it is high malaria tends to be endemic.

1.2  Macroparasite models

As I discussed in class, the only weird parts of these equations are the terms that describe transmission from the intermediate to the definitive host via predation, and parasite-induced mortality of the definitive host; the rest you should be able to figure out by thinking about them.

The weird terms,

p H N é ê ë æ ç è P¢ N ö ÷ ø + æ ç è P¢ N ö ÷ ø 2   ù ú û

(6)

and

aH é ê ë æ ç è P H ö ÷ ø + æ ç è P H ö ÷ ø 2   æ ç è 1 + 1 k ö ÷ ø ù ú û ,

(7)

occur because both the predation process and parasite-induced mortality of definitive hosts depend on parasite density.

Taking predation first: the model assumes that the predation rate of definitive hosts on intermediate hosts (per definitive host, per intermediate host, per unit time) is itself a function of parasite load in the intermediate hosts (because of parasite-induced behavior change of some sort that increases vulnerability to predation). So

predation rate = p æ ç è P¢ N ö ÷ ø

(8)

and

total predation rate = é ê ë p æ ç è P¢ N ö ÷ ø ù ú û ×H ×N.

(9)

The average number of parasites that actually get transferred from an intermediate to a definitive host when a predation event occurs is P¢/N, so

total transfer rate = é ê ë p æ ç è P¢ N ö ÷ ø ù ú û ×H ×N × æ ç è P¢ N ö ÷ ø = p H N æ ç è P¢ N ö ÷ ø 2   .

(10)

The tricky piece here is that we shouldn't just translate (P¢/N)² in this expression into (P¢/N)² in the equations, because what we're actually doing when we derive differential equations like those above is taking averages or expectations, and (if we denote averages by putting a bar over a variable) [`((P¢/N)<sup>2</sup>)] ¹ ([`(P¢)]/[`N])<sup>2</sup>. Instead, the expected value of X<sup>2</sup> is (in general) equal to [`X]²+ s²_X, where s² is the variance of X. If there is no variance (all intermediate hosts have exactly the same parasite burden), then the equation [`((P¢/N)<sup>2</sup>)] = ([`(P¢)]/[`N])² is correct; otherwise, just taking the average underestimates the true rate at which parasites are transferred. This is because heavily parasitized intermediate hosts are both more likely to be eaten by definitive hosts and will transfer more parasites when they are eaten.

In order to fix the problem, we have to know something about the variance (=aggregation) of the population. If we assume that the distribution of juvenile parasites within intermediate hosts is Poisson, then the variance equals the mean and

mean2+variance = mean2+mean = é ê ë æ ç è P¢ N ö ÷ ø 2   + æ ç è P¢ N ö ÷ ø ù ú û ,

(11)

which pretty well explains the first ``weird term'' in the equations.

If a parasite population is more aggregated than a Poisson distribution, then we have a little bit more work to do. Let's say that it's negative binomially distributed, and that we know the k parameter. We also know that for the negative binomial distribution, variance equals (mean(1+mean/k)), so

mean2+variance = mean2+mean æ ç è 1+ mean k ö ÷ ø = mean2 æ ç è 1+ 1 k ö ÷ ø +mean = é ê ë æ ç è P H ö ÷ ø 2   æ ç è 1 + 1 k ö ÷ ø + æ ç è P H ö ÷ ø ù ú û

which gives us the second weird term.

Even this (relatively) complicated solution is a compromise, since we're just assuming that the parasite population is aggregated with negative binomial parameter k, rather than letting the models predict the distribution from first principles. However, this is even harder (it represents current research in mathematical ecology/parasitology), but the shortcut works pretty well for some purposes, and it does let us ask questions about the effects of aggregation (changing k) on population dynamics.