Macroparasite burden model ($p_j=\mathrm{Pr}(\text{host burden}=j)$)

A macroparasite burden model tracks the distribution of within-host parasite burdens and couples it to transmission or environmental infectious stages. Mean burden alone is insufficient when worm mating, nonlinear fecundity, immunity or host mortality depend on that distribution.

For sexually reproducing worms, an infected host need not contain both sexes. Independent random sexes split a Poisson total burden of mean $m$ into independent male and female Poisson distributions of mean $m/2$, giving the displayed probability of a fertile mixed-sex burden. For an arbitrary burden probability generating function $G$, this probability is $1-2G(1/2)+G(0)$. Its low-burden behaviour can introduce an Allee effect in transmission, so an asexual mean-burden invasion calculation is not automatically valid.

If host-specific equilibrium mean burden has a gamma distribution, conditional Poisson distributions produce a Poisson-gamma mixture, hence a negative binomial distribution. With mean $m$ and shape $k$, its zero-burden probability is $(1+m/k{)}^{-k}$. Decreasing $k$ increases aggregation: a smaller infected-host prevalence can accompany the same mean burden and more heavily infected individuals.

Independent worm acquisition at rate $\lambda$ and loss at rate $\mu j$ produce a birth-death process for within-host parasite burden. Summing its forward equations against $j$ gives $d\mathbb{E}[j]/dt=\lambda -\mu \mathbb{E}[j]$. At constant exposure, detailed balance $\lambda p_j=\mu (j+1)p_{j+1}$ yields a Poisson distribution with mean $\lambda /\mu$. Heterogeneous acquisition rates change the pooled burden distribution even when each individual host follows this model.