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Isolation separates recognized infected or ill individuals from susceptible contacts to limit transmission. Quarantine instead concerns exposed individuals who may or may not be infected.

Quarantine restricts contacts of exposed people whose infection status is uncertain and monitors them for illness, reducing opportunities for transmission before infection is recognized. Recognized cases are managed by isolation rather than treated as merely uncertain exposures. Duration must be measured from the last possible infection-producing exposure, or otherwise account for uncertainty in exposure time. The mathematical calculations below concern the incubation period among infected people who develop symptoms, not a complete infectiousness model.

With a credible hard endpoint $M$, the truncated distribution hasA duration $q\ge M$ then covers all symptom-onset times under that model, provided the exposure time is correctly specified. If a nonzero late-onset probability $\epsilon$ is accepted, solveThe calculation is appropriate when external biological evidence genuinely supports a hard maximum, or when a deliberately bounded model is justified for the population being considered. Truncation caused by a study's ascertainment window instead requires an observation correction: it does not prove that the disease has a hard biological endpoint. Nor does $\hat{M}=t_{(n)}$ guarantee that future cases will develop symptoms by the current sample maximum. Endpoint and nuisance-parameter uncertainty need to enter a conservative duration calculation.

For an untruncated gamma distribution, lognormal distribution or other model with an unbounded upper tail, no finite duration eliminates the modeled late-onset probability. Choose a tail tolerance and solveThis is appropriate when a hard endpoint is unsupported and rare long incubation times remain possible. One can estimate a high quantile, assess its uncertainty and balance the modeled residual probability against the costs of longer restrictions. Different tail families can give very different high quantiles even when they fit the central observations similarly, which makes the model dependence seen in part b operationally relevant.

If an individual has already remained symptom-free for $a$ time units after a known exposure, a conditional additional duration $b$ instead uses $S(a+b)/S(a)\le \epsilon$, where $S$ is the relevant survivor function and $S(a)>0$. This conditions on being an infected eventual symptomatic case. Incubation data alone do not describe asymptomatic infection, the timing of infectiousness or compliance with restrictions; symptom surveillance, testing where informative and a transmission model may therefore be needed to translate late onset into an actual release-risk calculation. Use a hard bound only when it is supported; otherwise quantify and allow for the upper-tail uncertainty.

Write $S_0=S_{initial}$ and $I_0=I_{initial}$. Until quarantine and while $I,S>0$, division of the equations givesIntegration from the initial state yields the first integralLet $S_{\ast }$ be the susceptible count at the first time $I=I_{\ast }$. It is the root of an equation reached on the decreasing-$S$ outbreak trajectory ofFor a strictly rising threshold crossing this is the larger admissible root of an equation, with $a/r<S_{\ast }<S_0$; if the threshold equals the peak, $S_{\ast }=a/r$. The assumption that quarantine is reached selects the physically relevant branch and excludes a merely hypothetical declining-branch crossing after an earlier one.

Since there is no incubation period here, all removals from $S$ have already become infection cases by quarantine. No further susceptible individuals are infected after isolation. The requested number among the initially susceptible individuals is therefore

The term $rIS$ is a mass-action incidence rate: each infective individual makes potentially transmitting contacts with susceptible individuals, and the total rate is proportional to the product of the two compartment sizes. An infection transfers one animal from $S$ to $I$, accounting for opposite signs in their equations. The per-capita mortality rate of an infective individual is $a$, giving transfer $aI$ from $I$ to $R$.

There are no births, recoveries or additional deaths in this model. Adding the three equations gives $\dot{S}+\dot{I}+\dot{R}=0$, so the total initial biological population is conserved when the dead compartment is included. Before quarantine, $S$ decreases whenever $I,S>0$; after effective isolation no further transmission is permitted.

At the retained order define $K=1+\tau r(I-S)$. Neglecting terms of order $a\tau$ in the incidence correction givesDividing and expanding consistently,Thus the susceptible count $S_{\ast }$ at the first threshold crossing obeys the same retained-order equation as before:It is important to distinguish completed case transfers at quarantine from all infections already initiated. Let the quarantine time be $T$. Isolation prevents exposures after $T$, but animals exposed during $(T-\tau ,T)$ still finish their incubation period during $(T,T+\tau )$. Their number, in the printed bookkeeping, isThe infection cases transferred by time $T$ among the initial $S$ class number $S_0-S_{\ast }$. Adding those pending infection cases gives the eventual number who catch the infectious disease:This adds only infections caused by exposures before isolation; it does not allow secondary transmission after quarantine. If one asks instead only for the transfers completed by the instant of quarantine, the answer is $S_0-S_{\ast }$. The distinction is necessary because the model keeps incubating animals in its $S$ compartment.

Quarantine restricts contacts of exposed individuals whose infection status is uncertain, with monitoring to detect illness. It differs from isolation of recognized infected or ill individuals. An incubation period quantile can enter a duration calculation, but assumptions about exposure time, symptom detection and infectiousness are needed before the calculation describes transmission risk.

Before isolation, the closed SIR model has first integral $I+S-(a/r)\log S$. If quarantine occurs at the first crossing $I=I_{\ast }$, the corresponding susceptible count $S_{\ast }$ is the root on the outbreak branch of $I_{\ast }+S_{\ast }-(a/r)\log S_{\ast }=I_0+S_0-(a/r)\log S_0$. Immediate prevention of further transmission makes the number infected among the initial susceptible class equal to $S_0-S_{\ast }$.