Example data

First, we use the Gillepsie algorithm to generate infectious disease outbreak data (Figure @ref(fig:outbreak)) from a stochastic compartmental model.

outbreak <- simulate_gillespie(seed = 101)

(ref:outbreak) Early on in the epidemic, there is a high rate of growth in new cases. As more people are infected, the rate of growth slows. (Only every 50th case is shown to avoid over-plotting.)

outbreak |>
  filter(case %% 50 == 0) |>
  ggplot(aes(x = ptime, y = case)) +
  geom_point(col = "#56B4E9") +
  labs(x = "Primary event time (day)", y = "Case number") +
  theme_minimal()

(ref:outbreak)

outbreak is a data.frame with the two columns: case and ptime. Here ptime is a numeric column giving the time of infection. In reality, it is more common to recive primary event times as a date rather than a numeric.

head(outbreak)
#>   case      ptime
#> 1    1 0.04884052
#> 2    2 0.06583120
#> 3    3 0.21857827
#> 4    4 0.24963421
#> 5    5 0.30133392
#> 6    6 0.31425010

To generate secondary events, we will use a lognormal distribution (Figure @ref(fig:lognormal)) for the delay between primary and secondary events:

secondary_dist <- data.frame(mu = 1.8, sigma = 0.5)
class(secondary_dist) <- c("lognormal_samples", class(secondary_dist))
secondary_dist <- add_mean_sd(secondary_dist)

obs <- outbreak |>
  simulate_secondary(
    meanlog = secondary_dist[["mu"]],
    sdlog = secondary_dist[["sigma"]]
  )

(ref:lognormal) The lognormal distribution is skewed to the right. Long delay times still have some probability.

ggplot(data.frame(x = c(0, 30)), aes(x = x)) +
  geom_function(
    fun = dlnorm,
    args = list(
      meanlog = secondary_dist[["mu"]],
      sdlog = secondary_dist[["sigma"]]
    )
  ) +
  theme_minimal() +
  labs(
    x = "Delay between primary and secondary event (days)",
    y = "Probability density"
  )

(ref:lognormal)

(ref:delay) Secondary events (in green) occur with a delay drawn from the lognormal distribution (Figure @ref(fig:lognormal)). As with Figure @ref(fig:outbreak), to make this figure easier to read, only every 50th case is shown.

obs |>
  filter(case %% 50 == 0) |>
  ggplot(aes(y = case)) +
  geom_segment(
    aes(x = ptime, xend = stime, y = case, yend = case),
    col = "grey"
  ) +
  geom_point(aes(x = ptime), col = "#56B4E9") +
  geom_point(aes(x = stime), col = "#009E73") +
  labs(x = "Event time (day)", y = "Case number") +
  theme_minimal()

(ref:delay)

obs is now a data.frame with further columns for delay and stime. The secondary event time is simply the primary event time plus the delay:

all(obs$ptime + obs$delay == obs$stime)
#> [1] TRUE

If we were to receive the complete data obs as above then it would be simple to accurately estimate the delay distribution. However, in reality, during an outbreak we almost never receive the data as above.

First, the times of primary and secondary events will usually be censored. This means that rather than exact event times, we observe event times within an interval. Here we suppose that the interval is daily, meaning that only the date of the primary or secondary event, not the exact event time, is reported (Figure @ref(fig:cens)):

obs_cens <- obs |>
  mutate(
    ptime_lwr = floor(.data$ptime),
    ptime_upr = .data$ptime_lwr + 1,
    stime_lwr = floor(.data$stime),
    stime_upr = .data$stime_lwr + 1,
    delay_daily = stime_lwr - ptime_lwr
  )

(ref:cens) Interval censoring of the primary and secondary event times obscures the delay times. A common example of this is when events are reported as daily aggregates. While daily censoring is most common, epidist supports the primary and secondary events having other delay intervals.

obs_cens |>
  filter(case %% 50 == 0, case <= 250) |>
  ggplot(aes(y = case)) +
  geom_segment(
    aes(x = ptime, xend = stime, y = case, yend = case),
    col = "grey"
  ) +
  # The primary event censoring intervals
  geom_errorbarh(
    aes(xmin = ptime_lwr, xmax = ptime_upr, y = case, yend = case),
    col = "#56B4E9", height = 5
  ) +
  # The secondary event censoring intervals
  geom_errorbarh(
    aes(xmin = stime_lwr, xmax = stime_upr, y = case, yend = case),
    col = "#009E73", height = 5
  ) +
  geom_point(aes(x = ptime), fill = "white", col = "#56B4E9", shape = 21) +
  geom_point(aes(x = stime), fill = "white", col = "#009E73", shape = 21) +
  labs(x = "Event time (day)", y = "Case number") +
  theme_minimal()

(ref:cens)

During an outbreak we will usually be estimating delays in real time. The result is that only those cases with a secondary event occurring before some time will be observed. This is called (right) truncation, and biases the observation process towards shorter delays:

obs_time <- 25
obs_cens_trunc <- obs_cens |>
  mutate(obs_time = obs_time) |>
  filter(.data$stime_upr <= .data$obs_time)

Finally, in reality, it’s not possible to observe every case. We suppose that a sample of individuals of size sample_size are observed:

sample_size <- 200

This sample size corresponds to 8.7% of the data.

obs_cens_trunc_samp <- obs_cens_trunc |>
  slice_sample(n = sample_size, replace = FALSE)

Another issue, which epidist currently does not account for, is that sometimes only the secondary event might be observed, and not the primary event. For example, symptom onset may be reported, but start of infection unknown. Discarding events of this type leads to what are called ascertainment biases. Whereas each case is equally likely to appear in the sample above, under ascertainment bias some cases are more likely to appear in the data than others.

With our censored, truncated, and sampled data, we are now ready to try to recover the underlying delay distribution using epidist.

Fit the model and compare estimates

If we had access to the complete and unaltered obs, it would be simple to estimate the delay distribution. However, with only access to obs_cens_trunc_samp, the delay distribution we observe is biased (Figure @ref(fig:obs-est)) and we must use a statistical model.

(ref:obs-est) The histogram of delays from the complete, retrospective data obs_cens match quite closely with the underlying distribution, whereas those from obs_cens_trunc_samp show more significant systematic bias. In this instance the extent of the bias caused by censoring is less than that caused by right truncation. Nonetheless, we always recommend [Charniga et al. (2024); Table 2] adjusting for censoring when it is present.

bind_rows(
  obs_cens |>
    mutate(type = "Complete, retrospective data") |>
    select(delay = delay_daily, type),
  obs_cens_trunc_samp |>
    mutate(type = "Censored, truncated,\nsampled data") |>
    select(delay = delay_daily, type)
) |>
  group_by(type, delay, .drop = FALSE) |>
  summarise(n = n()) |>
  mutate(p = n / sum(n)) |>
  ggplot() +
  geom_col(
    aes(x = delay, y = p, fill = type, group = type),
    position = position_dodge2(preserve = "single")
  ) +
  scale_fill_manual(values = c("#CC79A7", "#0072B2")) +
  geom_function(
    data = data.frame(x = c(0, 30)), aes(x = x),
    fun = dlnorm,
    args = list(
      meanlog = secondary_dist[["mu"]],
      sdlog = secondary_dist[["sigma"]]
    )
  ) +
  labs(
    x = "Delay between primary and secondary event (days)",
    y = "Probability density",
    fill = ""
  ) +
  theme_minimal() +
  theme(legend.position = "bottom")

(ref:obs-est)

The main function you will use for modelling is called epidist()¹. We will fit the model "epidist_latent_model" which uses latent variables for the time of primary and secondary event of each individual². To do so, we first prepare the data using as_epidist_latent_model():

linelist_data <- as_epidist_linelist_data(
  obs_cens_trunc_samp$ptime_lwr,
  obs_cens_trunc_samp$ptime_upr,
  obs_cens_trunc_samp$stime_lwr,
  obs_cens_trunc_samp$stime_upr,
  obs_time = obs_cens_trunc_samp$obs_time
)

data <- as_epidist_latent_model(linelist_data)
class(data)
#> [1] "epidist_latent_model"  "epidist_linelist_data" "tbl_df"               
#> [4] "tbl"                   "data.frame"

The data object now has the class epidist_latent_model. Using this data, we now call epidist() to fit the model. The parameters of the model are inferred using Bayesian inference. In particular, we use the the No-U-Turn Sampler (NUTS) Markov chain Monte Carlo (MCMC) algorithm via the brms R package (Bürkner 2017).

fit <- epidist(
  data = data, chains = 2, cores = 2, refresh = as.integer(interactive())
)

Note that here we use the default rstan backend but we generally recommend using the cmdstanr backend for faster sampling and additional features. This can be set using backend = "cmdstanr" after following the installing CmdStan instructions in the README.

The fit object is a brmsfit object containing MCMC samples from each of the parameters in the model, shown in the table below. Many of these parameters (e.g. swindow and pwindow) are the so called latent variables, and have lengths corresponding to the sample_size.

pars <- fit |>
  variables(reserved = FALSE) |>
  gsub(pattern = "\\[\\d+\\]", replacement = "")

data.frame(
  Parameter = unique(pars), Length = table(pars)
) |>
  gt()

Users familiar with Stan and brms, can work with fit directly. Any tool that supports brms fitted model objects will be compatible with fit.

For example, we can use the built in summary() function

summary(fit)
#>  Family: latent_lognormal 
#>   Links: mu = identity; sigma = log 
#> Formula: delay | vreal(relative_obs_time, pwindow, swindow) ~ 1 
#>          sigma ~ 1
#>    Data: data (Number of observations: 200) 
#>   Draws: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
#>          total post-warmup draws = 2000
#> 
#> Regression Coefficients:
#>                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept           1.75      0.05     1.67     1.85 1.00     2281     1375
#> sigma_Intercept    -0.78      0.07    -0.91    -0.64 1.00     2364     1575
#> 
#> Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
#> and Tail_ESS are effective sample size measures, and Rhat is the potential
#> scale reduction factor on split chains (at convergence, Rhat = 1).

to see that the model has converged and recovered the delay distribution distribution paramters reasonably well.

The epidist package also provides functions to make common post-processing tasks easy. For example, individual predictions of the lognormal delay parameters can be extracted using:

pred <- predict_delay_parameters(fit)

Figure @ref(fig:fitted-lognormal) shows the lognormal delay distribution obtained for 100 draws from the posterior distribution of the mu and sigma parameters. Whereas in Figure @ref(fig:obs-est) the histogram of censored, truncated, sampled data was substantially different to the underlying delay distribution, using epidist() we have obtained a much closer match to the truth.

(ref:fitted-lognormal) The fitted delay distribution (pink lines, 100 draws from the posterior) compared to the true underlying delay distribution (black line). Each pink line represents one possible delay distribution based on the fitted model parameters.

# Sample 100 random draws from the posterior
set.seed(123)
sample_draws <- sample(nrow(pred), 100)

ggplot() +
  # Plot the true distribution
  geom_function(
    data = data.frame(x = c(0, 30)),
    aes(x = x),
    fun = dlnorm,
    linewidth = 1.5,
    args = list(
      meanlog = secondary_dist[["mu"]],
      sdlog = secondary_dist[["sigma"]]
    )
  ) +
  # Plot 100 sampled posterior distributions
  lapply(sample_draws, \(i) {
    geom_function(
      data = data.frame(x = c(0, 30)),
      aes(x = x),
      fun = dlnorm,
      args = list(
        meanlog = pred$mu[i],
        sdlog = pred$sigma[i]
      ),
      alpha = 0.1,
      color = "#CC79A7"
    )
  }) +
  labs(
    x = "Delay between primary and secondary event (days)",
    y = "Probability density"
  ) +
  theme_minimal()

(ref:fitted-lognormal)