= Immigration-death worm-burden model
{title2=$\dot p_j=\lambda p_{j-1}+\mu(j+1)p_{j+1}-(\lambda+\mu j)p_j$}
<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.
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