# Model definition: estimate_dist()

# 1 Overview

`estimate_dist()` fits a parametric delay distribution from interval-censored linelist data via MCMC. The model accounts for double interval censoring of the primary and secondary events, right truncation due to a finite observation period, and optional left truncation. The likelihood is provided by the primarycensored package[1]; see its documentation for derivations and a fuller account of why these adjustments are needed[2,3]. The conceptual motivation in EpiNow2 is covered in `vignette("delays")`, and a worked example in `vignette("estimate_dist_workflow")`. For analyses needing time-varying delays, partial pooling across strata, or regression on covariates, `epidist` builds on the same likelihood and is the recommended next step.

# 2 Data and notation

Each observation \(i = 1, \ldots, N\) is a record of a primary event (e.g. infection) and a secondary event (e.g. symptom onset). Both events are observed only as intervals. Let \(P_i = [p^{-}_i, p^{+}_i)\) be the primary event window and \(S_i = [s^{-}_i, s^{+}_i)\) be the secondary event window. We refer to \(w^{P}_i = p^{+}_i - p^{-}_i\) as the primary window width and to \(w^{S}_i = s^{+}_i - s^{-}_i\) as the secondary window width. At daily resolution both are typically equal to one.

The observed delay interval is \([d^{-}_i, d^{+}_i) = [s^{-}_i - p^{-}_i,\, s^{+}_i - p^{-}_i)\), the difference between the secondary window and the start of the primary window. Each observation also has a relative observation time \(D_i\) and an optional left truncation point \(L_i\), both measured from \(p^{-}_i\). \(D_i\) is the time at which the data were extracted relative to the primary event start and acts as the right truncation point. Per-observation weights \(n_i\) allow identical \((d^{-}_i, d^{+}_i, w^{P}_i, D_i)\) combinations to be aggregated; this affects only computational efficiency, not the likelihood.

The continuous, unobserved delay \(T_i\) between the true primary event time \(T^{P}_i\) and the true secondary event time \(T^{S}_i\) is assumed to follow a parametric distribution with cumulative distribution function \(F(\cdot \,|\, \theta)\), where \(\theta\) is the parameter vector. We denote the primary event distribution within its window by \(f_P\). The model treats \(\theta\) as unknown and \(f_P\) as known.

# 3 Likelihood

## 3.1 Continuous formulation

If \(T^{P}_i\) has density \(f_P\) on \([p^{-}_i, p^{+}_i)\), the distribution of the delay \(T_i\) from the primary event to a fixed reference point (e.g. \(p^{-}_i\)) is the convolution of \(f_P\) with the underlying delay distribution. The corresponding cumulative distribution function adjusted for primary event censoring is

\[\begin{equation} F^{*}(t \,|\, \theta, w^{P}_i) = \int_{p^{-}_i}^{p^{+}_i} F(t - u \,|\, \theta) f_P(u) \, du. \end{equation}\]

Without primary censoring the distribution of the discrete observed delay would simply be obtained by differencing \(F\) at the secondary window boundaries. With primary censoring, \(F\) is replaced by \(F^{*}\) throughout. For uniform \(f_P\) this can be evaluated analytically for the lognormal, gamma, and Weibull delay distributions; for other combinations the integral is evaluated numerically. The vendored Stan code dispatches to either branch through `check_for_analytical()` in `primarycensored.stan`.

## 3.2 Truncation

The model handles right truncation at \(D_i\) and optional left truncation at \(L_i\) by renormalising \(F^{*}\):

\[\begin{equation} F^{*}_{[L_i, D_i]}(t \,|\, \theta, w^{P}_i) = \frac{F^{*}(t \,|\, \theta, w^{P}_i) - F^{*}(L_i \,|\, \theta, w^{P}_i)} {F^{*}(D_i \,|\, \theta, w^{P}_i) - F^{*}(L_i \,|\, \theta, w^{P}_i)} \end{equation}\]

for \(L_i \leq t \leq D_i\) and zero outside the interval. \(D_i\) may be set to \(\infty\) to indicate no right truncation; in that case the denominator reduces to \(1 - F^{*}(L_i \,|\, \theta, w^{P}_i)\). \(L_i\) defaults to \(0\) and is not currently exposed at the R interface.

## 3.3 Discrete observation likelihood

The probability mass for observation \(i\) is the difference of the truncated, primary-censored CDF at the secondary window boundaries:

\[\begin{equation} \Pr(d^{-}_i \leq T_i < d^{+}_i \,|\, \theta, w^{P}_i, L_i, D_i) = F^{*}_{[L_i, D_i]}(d^{+}_i \,|\, \theta, w^{P}_i) - F^{*}_{[L_i, D_i]}(d^{-}_i \,|\, \theta, w^{P}_i). \end{equation}\]

The aggregated log-likelihood used in `estimate_dist.stan` is

\[\begin{equation} \log \mathcal{L}(\theta) = \sum_{i=1}^{N} n_i \log \Pr(d^{-}_i \leq T_i < d^{+}_i \,|\, \theta, w^{P}_i, L_i, D_i), \end{equation}\]

implemented as the call to `primarycensored_lpmf()` inside the model block.

## 3.4 Untruncated approximation

When \(D_i\) is much larger than the largest observed delay, the right truncation factor is numerically indistinguishable from one and the renormalisation in the equation above can be skipped. `estimate_dist()` applies this approximation by setting \(D_i\) to \(\infty\) whenever \(D_i > c\, \max_j d^{+}_j\), where \(c\) is the `obs_time_threshold` argument and defaults to \(2\) following the same heuristic used by `epidist`. Setting `obs_time_threshold = Inf` disables the approximation.

# 4 Primary event distribution

Two primary event distributions are supported:

- Uniform (default): \(f_P\) is uniform on \([p^{-}_i, p^{+}_i)\). This is appropriate for daily reporting where the within-day timing of the primary event is unknown and assumed equally likely at any point in the window.
- Exponential growth: \(f_P\) is proportional to \(\exp(r u)\) on \([p^{-}_i, p^{+}_i)\), with growth rate \(r\) supplied by the user via `primary_params`. This adjusts for the bias introduced when the primary event rate is changing rapidly within the window, for example in the early phase of an outbreak[2]. The growth rate is treated as fixed data and is not estimated.

The primary event parameters are passed to Stan in the `primary_id` and `primary_params` data fields and are not part of \(\theta\).

# 5 Delay families and parameterisations

The model supports five delay families, selected by the `dist` argument. Parameter names match the natural parameterisation used elsewhere in EpiNow2.

`dist` | Parameters | Density support
--- | --- | ---
`"lognormal"` | `meanlog`, `sdlog` | \(T > 0\)
`"gamma"` | `shape`, `rate` | \(T > 0\)
`"normal"` | `mean`, `sd` | \(T \in \mathbb{R}\)
`"exp"` | `rate` | \(T > 0\)
`"weibull"` | `shape`, `scale` | \(T > 0\)

For `"lognormal"`, `"gamma"`, and `"weibull"` with a uniform primary, the primary-censored CDF \(F^{*}\) has a closed form and is computed analytically. The other combinations use numerical integration over the primary window.

## 5.1 Priors

Priors are passed via the `priors` argument as a named list of `dist_spec` objects, with names matching the parameter names of the chosen family. Each prior is converted into a `Normal()` density on the parameter and applied independently. Lower bounds are imposed for parameters that must be positive (e.g. `sdlog`, `rate`, `shape`, `scale`); the truncation is handled by `params_lp()` in `inst/stan/functions/params.stan`.

The default priors are family-specific:

\[\begin{align} \text{lognormal:}\quad \text{meanlog} &\sim \mathrm{Normal}(1, 1), & \text{sdlog} &\sim \mathrm{Normal}(0.5, 0.5), \\ \text{gamma:}\quad \text{shape} &\sim \mathrm{Normal}(2, 2), & \text{rate} &\sim \mathrm{Normal}(0.5, 0.5), \\ \text{normal:}\quad \text{mean} &\sim \mathrm{Normal}(5, 5), & \text{sd} &\sim \mathrm{Normal}(1, 1), \\ \text{exponential:}\quad \text{rate} &\sim \mathrm{Normal}(0.5, 0.5), \\ \text{Weibull:}\quad \text{shape} &\sim \mathrm{Normal}(2, 2), & \text{scale} &\sim \mathrm{Normal}(5, 5). \end{align}\]

These defaults are deliberately wide and are intended as starting points rather than fits to any particular delay. Users are encouraged to set priors that reflect domain knowledge about the delay being estimated; see `vignette("estimate_dist_workflow")` for guidance on translating prior beliefs into the lognormal parameterisation.

# References

1.
Abbott, S., Brand, S., Azam, J. M., Pearson, C., Funk, S., & Charniga, K. (2026). Primarycensored: Primary event censored distributions. https://doi.org/10.5281/zenodo.13632839
2.
Park, S. W., Akhmetzhanov, A. R., Charniga, K., Cori, A., Davies, N. G., Dushoff, J., Funk, S., Gostic, K., Grenfell, B., Linton, N. M., Lipsitch, M., Lison, A., Overton, C. E., Ward, T., & Abbott, S. (2024). Estimating epidemiological delay distributions for infectious diseases. medRxiv. https://doi.org/10.1101/2024.01.12.24301247
3.
Charniga, K., Park, S. W., Akhmetzhanov, A. R., Cori, A., Dushoff, J., Funk, S., Gostic, K. M., Linton, N. M., Lison, A., Overton, C. E., Pulliam, J. R. C., Ward, T., Cauchemez, S., & Abbott, S. (2024). Best practices for estimating and reporting epidemiological delay distributions of infectious diseases. PLoS Comput. Biol., 20(10), e1012520. https://doi.org/10.1371/journal.pcbi.1012520