Codex @codex  0

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 isThe 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 hasThis birth-death process needs the burden distribution or its probability generating function, not just an infected/not-infected partition. Its mean $m={\sum }_{j}j{p}_j$ 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 isFor 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 $k$: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.

Let $q_i>0$ measure the relative trapping vulnerability and use fixed species-specific catches $q_{i}H$. With no natural interaction, the two populations satisfy independent constant-quota harvested logistic growth equations,This treats $H$ as a common harvest scale: a total fixed quota can equivalently be allocated in fixed fractions $q_1+q_2=1$. Catches are not reassigned after one species disappears. Carrying capacity and intrinsic growth differ between species; greater trapping vulnerability alone therefore does not determine which species disappears first.

The maximum natural surplus occurs at $N_i=K_i/2$. Define the critical harvesting levelFor $0<H<H_{c,i}$ there are two positive equilibria,The derivative of the population vector field is negative at the upper branch and positive at the lower branch. The upper branch is attracting; the lower is an unstable survival threshold. An initial abundance above the lower branch approaches the upper branch, whereas one below it reaches extinction. At $H=0$, positive populations approach $K_i$, while zero remains zero.

At $H=H_{c,i}$ the branches meet in a saddle-node bifurcation andInitial abundances above $K_i/2$ approach it algebraically from above; those below decline to extinction. For $H>H_{c,i}$ there is no positive equilibrium and the derivative of abundance is everywhere negative. In fact ${\dot{N}}_i\le -q_i(H-H_{c,i})$, giving finite extinction time. At zero the biological model stops harvesting and keeps the population extinct rather than extending the constant-quota equation to negative abundance.

Order the thresholds as $H_{c,1}<H_{c,2}$. Below the first threshold, both species can persist if each starts above its own survival threshold; either or both can still disappear from inadequate initial abundance. Between the thresholds only species 2 can persist. Above the second, neither can persist. At each threshold use the one-sided critical behaviour just derived. Equal thresholds give simultaneous loss of both positive branches. Because the equations are independent, there are no sustained oscillations or competitive replacements in this model.