Immigration-death worm-burden model (${\dot{p}}_j=\lambda p_{j-1}+\mu (j+1)p_{j+1}-(\lambda +\mu j)p_j$)

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.