• Serial interval (SI): time between symptom onset of infector and infectee
• Generation interval (GI): time between infections
• The SI can be negative: the infectee’s symptoms can appear before the infector’s, for example with presymptomatic transmission
• The GI is what some transmission models need; the SI is usually what we get to observe

• Infection and transmission times are latent — we almost never see them directly
• Symptom onsets are recorded to a day at best, and “onset” is itself imprecise: definitions vary, and recall is biased
• So we never fully observe it — we come to how we currently handle that next
• Estimating a serial interval is really estimating a delay from incomplete data

• Feeds \(R_t\) through the renewal equation, and links the growth rate \(r\) to the reproduction number \(R\)
• Beyond renewal: used to reconstruct infection times from onsets by deconvolution
• Also feeds forecasting and contact tracing
• But the SI is a proxy for the generation interval, the quantity some transmission models need
• And do we want the measure itself? The transmission-pair data are what is valuable; the serial interval is just one summary of them

• Adjust for within-interval censoring of both events (always)
• Right truncation is the correction for dynamical (epidemic-phase) bias
• Fit several distributions and compare with model-fit statistics
• Report uncertainty, and share de-identified line lists and code

• For now treat the serial interval as a single event-to-event delay, ignoring that it is itself composed — we return to that later
• Daily or weekly data put both the primary (infector) and secondary (infectee) events in windows, not at points
• In real time, long delays have not happened yet
• Build the observation model up in three steps:
  1. primary interval censoring
  2. secondary interval censoring
  3. right truncation

• The infector’s event falls in a window \([0, w_P]\)
• It need not be uniform, but a uniform primary is a fair assumption, especially when the window is short
• Mix the delay over where the event sat in the window — a convolution on the CDF

\[ F_{\mathrm{PEC}}(t) = \int_{0}^{w_P} g(\tau)\, F(t - \tau)\, d\tau \]

• The infectee’s onset is also recorded to the day
• The probability in a daily bin is a difference of the primary-censored CDF
• This doubles as a probability mass function for a continuous delay in daily discrete-time models (EpiNow2, EpiEstim, epinowcast)

\[ \Pr\!\big(T \in [n, n + w_S]\big) = F_{\mathrm{PEC}}(n + w_S) - F_{\mathrm{PEC}}(n) \]

• In real time we only see a pair if the secondary event is before the cutoff \(C\)
• Condition on being observed: divide by the CDF at the truncation horizon
• Short delays are over-represented early in an epidemic

\[ L^{\mathrm{RT}} = \frac{F_{\mathrm{PEC}}(n + w_S) - F_{\mathrm{PEC}}(n)} {F_{\mathrm{PEC}}(C - P)} \]

• Really a wrapper for any delay distribution: it adds the primary and secondary censoring and truncation around whatever family you choose
• Analytic primary-censored CDFs for gamma, lognormal, and Weibull with a uniform primary; numerical integration otherwise
• Signed support (normal, logistic, Laplace, Cauchy, Gumbel) for delays that can be negative
• And a non-parametric, hazard-based estimator (#312) — it needs an eye on it, and is not clearly a good idea: for onward reuse we have little guidance yet on how to estimate or report these

• The same delay \(f\) enters three of our tools in different forms:
  • epidist — individual-level delays from line lists
  • EpiNow2 — delays in estimate_dist, estimate_truncation, estimate_infections
  • epinowcast — population-level nowcasting: \(f\)’s hazard modified over time, or straight discrete-time hazards; with uncertainty on the counts
• How to choose between them is an open question — is there guidance in the time-to-event literature?

epidist — individual likelihood \[ \textstyle\prod_i f(t_i) \]

EpiNow2 — convolve incidence \[ C_t = \sum_{\tau} f(\tau)\, I_{t-\tau} \]

epinowcast — \(f\)’s hazard, modified (or free) \[ \operatorname{logit} h_d = \operatorname{logit} h_d^{f} + \text{effects} \] \[ \mathbb{E}[n_{t,d}] = \lambda_t\, h_d\textstyle\prod_{d'<d}(1-h_{d'}) \]

• A common-source outbreak shares one exposure window
• The distribution of exposure times within the source may itself differ — it need not be uniform
• Estimate the incubation period from onset dates; the same censoring and dynamical-bias issues apply
• A Legionnaires’ model in use by UKHSA does this, deconvolving onsets into a release time and an incubation period (Egan & Hall 2011)
• It extends naturally to multiple candidate sources

• The censored CDF is the delay density \(f\) convolved with the primary-event window \(H_w\):

\[ F_{\mathrm{cens}}(q) = \int_0^q H_w(q-u)\, f(u)\, du \]

• Expanding the window as exponentials reduces this to Laplace-type transforms \(T_f(\xi;\tau)=\int_0^\tau e^{\xi u} f(u)\,du\) of \(f\):

\[ F_{\mathrm{cens}}(q) = \sum_k a_k(q)\, T_f(\xi_k; q) \]

• The window (uniform, exponentially tilted, logistic, Gumbel) sets the \(a_k, \xi_k\); the delay family sets whether \(T_f\) is analytic

• We treated the SI as one event-to-event delay; it can be thought of as a composition of delays
• Onsets are what we see; infection and transmission are latent

\[ GT = LP + U \] \[ SI = GT + (IP_2 - IP_1) \]

• The composition is where the biology can enter

• A shared pathogen load \(V(t)\) in the infector drives three event hazards
  • infectiousness onset \(h_I\)
  • symptom onset \(h_S\)
  • transmission \(p_{\text{trans}}\)
• So \(LP\), \(U\), and \(IP_1\) are coupled by the viral load, not separate marginals
• The delays stay mutually consistent, and the parameters keep a biological meaning
• Builds on the within-host to Burr-distribution idea of Jamieson & Hall (2024)

• A rise-and-fall pathogen load with four interpretable parameters
  • growth rate \(r\)
  • peak time \(m\)
  • decay rate \(d\)
  • transition sharpness \(\kappa\)

\[ V(t) = A\, e^{rt} \left\{1 + \tfrac{r}{d}\,e^{\kappa(t-m)}\right\}^{-(r+d)/\kappa} \]

• In reality it is often more complex, with substantial individual-level variation (Russell et al. 2024)

• Each event is a stochastic activation with a hazard set by the load

\[ F_Z(t) = 1 - \exp\!\left\{-\!\int_0^t h_Z(s)\,ds\right\} \]

• \(V(t)\) is shaped by \(r, m, d, \kappa\); a per-event scale \(\beta_Z\) sets how strongly each event responds to load

Each delay reads off the same \(V(t)\):

infectiousness onset \(\to LP\) \[ h_I(t) = \beta_I\, V(t) \]

symptom onset \(\to IP\) \[ h_S(t) = \beta_S\, V(t) \]

transmission \(\to U\) (so \(GT = LP + U\)) \[ \lambda_T(u \mid LP{=}\ell) = c\,p_{\text{trans}} = c\,\beta_T\, V(\ell + u) \]

• Dutch household transmission pairs, Delta replaced by Omicron
• Shorter mean generation time for Omicron than Delta
• Lower WAIC and LOOIC than a lognormal generation-time model, for both strains

• Both models reproduce the modal interval and the tails
• For Omicron the within-host informed model captures the peak and both tails better than the lognormal
• The fit stays comparable while keeping a biological link to the latent generation time

• 34 high-confidence mpox transmission pairs
• External viral-load data build informative priors on the trajectory parameters \((\log r, \log d, m, \log\kappa)\)
• The decline \(\log d\) and peak time \(m\) shift under the viral-load prior: external information constrains weakly identified parameters
• A natural next step: jointly fit the transmission pairs and the viral-load data, rather than passing one in as a prior

• The event-time hierarchy holds for any hazard form; the within-host \(V(t)\) is one choice
• A generic alternative: contact-interval survival models — time from infectiousness onset to infectious contact is a transmission hazard, with no within-host trajectory needed (Kenah 2011)
• This contact interval is exactly our \(U\) (infectiousness onset to transmission), so \(GT = LP + U\) — now with \(U\) from a generic hazard, not \(V(t)\)

• We have focused on censoring and truncation, but transmission data carry more
• Household and cluster data: susceptible depletion shortens the observed generation and serial intervals as susceptibles run out
• Who infected whom is uncertain; coprimary and tertiary cases contaminate pairs
• The reference cohort matters: which event reporting is conditioned on
• Observation is often missing not at random
• Other pathogens compose differently: vector-borne diseases add an extrinsic incubation period in the vector, and further delays

• First sustained person-to-person hantavirus spread: one zoonotic introduction, four human generations, super-spreading (Martínez et al. 2020)
• Index cases give the incubation period; sourced cases give source-onset to onset
• The serial interval is a convolution of transmission timing \(\delta\) and the secondary incubation period — \(\delta\) can be negative
• \(R(t)\) and offspring dispersion come from the same fit

• The incubation period and transmission timing \(\delta\), with the derived generation and serial intervals
• \(\delta\) sits close to source onset, so transmission is near presymptomatic
• The within-host approach could replace all these empirical delays — the incubation period and \(\delta\) — here

• A line list, early in an outbreak — not a serial interval, but the same censoring problem
• The original onset-to-death estimate was a point estimate with no uncertainty (Rosello et al. 2015); the reanalysis adds it
• Each case has four delays, with a per-case identity \(D_{od} = D_{oa} + D_{ad}\)
• Death versus discharge is a competing risk; the split is the case-fatality ratio

• The four delays fitted under double interval censoring, compared across lognormal, gamma, and Weibull
• Gamma fits best (lowest WAIC); dashed lines mark the prior point estimates
• Fitting several families and comparing is the Charniga checklist in action

• As we have seen, many epidemiological delays can be thought of as composed of delays between events
• The serial interval is a generation time plus an incubation difference, observed through primary and secondary censoring and truncation
• The same few biases recur, in different combinations
• We want to write a delay down once, from reusable, tested parts, and reuse it across models

• A Julia package, in effect a version of primarycensored
• Censoring and truncation are just wrappers around a delay
• The result is a distribution you can pdf, rand, and fit with Turing.jl

• In development: compose the delays between events
• Additive chains, competing outcomes by racing hazards, or a data field that selects the sub-model
• Then wrap the whole delay in the observation model
• Covariates can modify it — on the hazard scale, or within individual delay leaves

• Composed delays back different model types
• Individual likelihoods; counts via convolution or reporting hazards; or lowered onto a reaction network for a mechanistic model

• We estimate the current size of the 2026 DRC Bundibugyo (Ebola) outbreak in real time — not a delay, but the same toolkit
• A joint Bayesian renewal model over all these streams, with ascertainment; committed deaths give a lower bound

An ecosystem for estimating and using delays:

• epidist — individual-level delay estimation
• primarycensored — censored, truncated delays
• CensoredDistributions.jl — composing delays
• EpiNow2 and epinowcast — \(R_t\) and nowcasting
• EpiBranch.jl — branching-process models