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An infection caused by bacteria.

An infection involves invasion and multiplication of an infectious agent in a host. Bacteria are one possible cause; an agent capable of causing disease is a pathogen. The National Cancer Institute defines infection through invasion and growth in the body. ^[ref].

In a SIR model, acquiring infection moves a member of the susceptible compartment into the infectious compartment. That immediate transition is a model assumption: the SEIR model instead includes an exposed stage before infectiousness. Specify these states when translating a biological process into compartmental models.

Compartmental models divide a biological population into states and specify transfers between them. A compartment might contain susceptible individuals, infective individuals or recovered individuals. The model must state what a compartment represents and whether its variable is a count or a fraction.

For a closed, homogeneously mixed SIR model, let $N=S+I+R$ and use force of infection $\beta I/N$ and recovery rate $\gamma$. ThenEvery transfer leaves one compartment and enters another, so summing the equations gives $\dot{N}=0$. This conservation of population follows from the specified closed model; demographic gains, deaths or external arrivals can change the total. An SEIR model adds an exposed state to distinguish infection from the ability to transmit.

A deterministic ordinary differential equation describes continuous compartment sizes. A stochastic epidemic model instead treats infection and recovery as random events between integer states. The latter can exhibit epidemic extinction even when the corresponding deterministic trajectory initially grows.

The infectious compartment consists of people currently able to transmit infection. In the closed SIR model, $I(t)$ is their count and $i(t)=I(t)/N$ their fraction of the fixed total $N$. The SIR model has no separate delay between acquiring infection and entering the infectious compartment; the SEIR model adds such a stage.

The latent period is the interval from infection until the infected host becomes infectious. The incubation period instead ends at symptom onset. Neither interval is defined by the time of a diagnostic test.^[ref]

For an illustrative infection beginning on day zero, infectiousness beginning on day two and symptoms beginning on day five give a two-day latent period and five-day incubation period. The host can transmit during the intervening three days despite not yet having symptoms. Changing these relative timings changes which observation-based interventions can interrupt transmission.

In an SEIR model, the exposed compartment contains infected hosts who are not yet infectious. A transition term $\sigma E$ gives independent exponentially distributed latent durations of mean $1/\sigma$. A fixed latent duration $L$ is a different model: all hosts infected together enter the infectious compartment after the same delay. Equal means therefore do not make the two latency assumptions dynamically equivalent.

A microparasite is typically modelled as multiplying rapidly within a host, so infection status rather than the exact organism count is the first between-host state variable. SIS model, SIR model and SEIR model descriptions are examples. Additional within-host structure can still be required.

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.

The SEIR model inserts a latent exposed state:The expected time from infection through the end of infectiousness is $d=1/\sigma +1/\gamma$; the duration during which transmission occurs is $1/\gamma$.

The linear SEIR model with a constant susceptible-to-exposed intensity isAdding the four equations confirms conservation of population: $S+E+I+R=N$.

The SEIR model divides a population into susceptible, exposed, infectious, and recovered compartments.

In a compartmental epidemic model, a susceptible individual is one who can acquire the modeled infection. The susceptible compartment counts these individuals. An exposed but not yet infectious individual belongs to a separate latent class in an SEIR model; models that retain such individuals in a not-yet-infective compartment must state that bookkeeping convention explicitly.