= Solution
A <microparasite> typically multiplies within an infected <host> on a much shorter time scale than between-host transmission. A useful first model classifies <hosts> by infection status. A <macroparasite>, such as an adult parasitic worm, instead requires a count of <parasites> per <host>: two infected <hosts> can carry very different <within-host parasite burdens>. These are modelling distinctions rather than a rule determined only by an organism's physical size.
For a <microparasite> with no lasting <immunity>, a homogeneous <SIS model> is
$$
\dot I=\beta\frac{SI}{N}-\gamma I,
\qquad S+I=N.
$$
The <basic reproduction number> is $\beta/\gamma$, and the <prevalence> $I/N$ describes the infected-host fraction. A <SIR model> adds an immune class, and a <SEIR model> adds a latent class. Contact structure, age, susceptibility, <recovery rate> and infectiousness can require several such classes. Then a <next-generation matrix> or operator, rather than a single averaged contact rate, sets the invasion threshold. <Stochastic epidemic models> are relevant to early extinction and finite populations; network or spatial models represent nonuniform contact opportunities.
For a simple <macroparasite burden model>, let $p_j$ be the fraction of <hosts> carrying $j$ adult worms. Assume acquisitions at rate $\lambda=\beta L$, independent adult-worm losses at rate $\mu j$, and a number $L$ of infective stages in a well-mixed environmental reservoir. The <immigration-death worm-burden model> has
$$
\dot p_j=\lambda p_{j-1}+\mu(j+1)p_{j+1}-(\lambda+\mu j)p_j,
\qquad p_{-1}=0.
$$
This <birth-death process> needs the burden distribution or its <probability generating function>, not just an infected/not-infected partition. Its <mean> $m=\sum_jjp_j$ satisfies $\dot m=\beta L-\mu m$. With <host> number $N$ fixed and each adult producing infective stages at rate $\sigma$, one possible environmental closure is
$$
\dot L=\sigma Nm-(d_L+\beta N)L.
$$
For this asexual, unsaturated transmission approximation, one worm's <basic reproduction number> is $(\sigma/\mu)\,\beta N/(d_L+\beta N)$. Growth above the invasion threshold requires additional density regulation if a finite endemic burden is sought. Sex-dependent mating would change this closure.
With constant external exposure $\lambda$, the <equilibrium> burden follows a <Poisson distribution> with <mean> $m=\lambda/\mu$, giving infected-host <prevalence> $1-e^{-m}$. If exposure rates vary across <hosts> as a <gamma distribution>, the <Poisson-gamma mixture> gives a <negative binomial distribution> with <mean> $m$ and aggregation parameter $k$:
$$
\operatorname{Var}(j)=m+\frac{m^2}{k},
\qquad \Pr(j>0)=1-\left(1+\frac{m}{k}\right)^{-k}.
$$
The same <mean> burden can therefore coexist with very different <prevalence> and highly infected subgroups. <Aggregation of macroparasite burdens> affects treatment coverage, burden-dependent <host> mortality, <immunity> and <parasite> fecundity. For sexually reproducing worms, <hosts> need both sexes for fertile output; a <mean> alone may not determine it. With independent equally likely sexes and Poisson burden, the <probability> of both sexes is $(1-e^{-m/2})^2$, illustrating <mating-limited macroparasite transmission> and its low-burden bottleneck.
Both <microparasite> and <macroparasite> models need heterogeneous exposure and <host> responses. The special extra difficulty for <macroparasites> is translating a distributed worm burden, and possibly worm sex or developmental stage, into transmission and <host> harm. Conversely, complicated within-host <microparasite> processes can also require more than a binary <host> state when the simple fast-within-host approximation fails.
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