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Assume the incubation times are independent observations from the same upper-truncated distribution. Conditional on lying below $M$, the gamma distribution density is $g(t;a,s)/G(M;a,s)$ on its support. With $t_{(n)}={\max }_i{t}_i$, the likelihood function isIn particular, the normalizing factor $G(M;a,s{)}^{-n}$ cannot be omitted. On the admissible region the log-likelihood isThe profile likelihood eliminates the nuisance parameters $a,s$ by fitting them separately at each candidate endpoint:For fixed $a,s$ and $M>t_{(n)}$,Equivalently, for $t_{(n)}<M_1<M_2$, every nuisance-parameter fit has $L(a,s,M_1)>L(a,s,M_2)$. Taking suprema shows that $L_p$ is nonincreasing. If the nuisance maximum at $M_2$ is attained and finite, evaluating its maximizing parameters at $M_1$ shows strict decrease there as well. Values below the largest observation are inadmissible. Thus upper-endpoint estimation in a truncated distribution gives the usual boundary estimateThere is a support-convention qualification: the literal strict support $t<M$ excludes $M=t_{(n)}$, so it gives a supremum approached as $M\downarrow t_{(n)}$, rather than an attained maximum-likelihood estimator. Using the equivalent closed-endpoint density convention for a continuous distribution yields the usual boundary estimate above. The argument presumes a meaningful finite nuisance fit; degenerate samples can instead have an unbounded density likelihood. Since $M$ changes the support, regular interior-parameter asymptotic formulas are not automatically applicable.

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.

The inverse-square law gives $f_s=L_0/(4\pi r_s^2)$, so a star is observed exactly whenWriting $a=R/r_0$, integration of the shape-three gamma density givesTherefore the fully normalized truncated distribution is