Background

When a novel pathogen arises in a population, time may be needed before surveillance systems are set up and can identify infected individuals. As such, the start of an epidemic may differ from the date of first reported case, and epidemiologists may need to take into account the impact of unobserved generations of infections on their analyses.

In the case of EpiEstim, the first estimates of the reproduction number (R_t) are sensitive to the recent incidence history. Indeed, the denominator of the estimator depends on the term: Λ_t = ∑_τ < tw_t − τI_τ, where (I_t)_t denotes incidence and (w_j)_j the serial interval distribution. If the first cases are not observed, this sum will be truncated, hence the denominator will be underestimated and infectiousness overestimated.

In Brizzi, O’Driscoll and Dorigatti (10.1093/cid/ciac138), we proposed a simple exponential growth model to impute missed generations of infections, leading to a reduction in bias in early R_t estimates. This can now be implemented through EpiEstim’s estimate_R function, by setting the backimputation_window to an integer greater than 2, specifying the number of observations used to fit the exponential growth model.

In short, the method combines the strength of the exponential growth to estimate the basic reproduction number R₀ together with the capabilities of EpiEstim to detect changes in transmission dynamics.

Why do we need backimputation?

A simple toy scenario

To quickly introduce early overestimation issue, consider the case of constant incidence, for example I_t = 10. We would expect R_t to be constantly close to 1. However, estimates of R_t are decreasing, suggesting a reduction in pathogen infectiousness.

incid_constant <- rep(10, 50)
config <- make_config(list(mean_si = 7, std_si = 4))
estimate_R(incid = incid_constant, config = config, method = "parametric_si") |> 
    plot("R")
#> Default config will estimate R on weekly sliding windows.
#>     To change this change the t_start and t_end arguments.

The problem here is that EpiEstim implicitly assumes the 10 cases in day 2 to have been generated by cases in day 1! If this seems like an unlikely scenario, perform backimputation by setting imputation_window to an integer larger than 2. When working with daily incidence, we suggest to set this window to at least 5 observations.

estimate_R(
    incid = incid_constant,
    backimputation_window = 7,
    config = config,
    method = "parametric_si") |> 
    plot("R")
#> Default config will estimate R on weekly sliding windows.
#>     To change this change the t_start and t_end arguments.

In this case, the estimates are constant, as expected.

Real world scenario: UK COVID deaths

Let us now consider some real world data: UK COVID-19 deaths from 2020. As reporting of deaths is more accurate and reliable than case reporting, it is unlikely that a large proportion of people died of COVID before the first reported death. However, we can use this dataset to show how left-censoring incidence may affect traditional EpiEstim estimates, and how the adjustment behaves more robustly. We focus on the first 50 days of reports.

data(covid_deaths_2020_uk)
max_t <- 50
incid_covid <- covid_deaths_2020_uk$incidence$Incidence[1:max_t]
config_covid <- make_config(list(si_distr = covid_deaths_2020_uk$si_distr))

No left-censoring: early R_t estimates remain similar

We first consider a scenario where the back-imputation shouldn’t be necessary: i.e. when no generations of infections (or deaths) are missed. We would hope that performing back-imputation would not drastically lower the estimates for the reproduction number. Indeed, this is the case.

no_truncation_no_imputation <- estimate_R(
                                          incid = incid_covid,
                                          method = "non_parametric_si",
                                          config =  config_covid
)
#> Default config will estimate R on weekly sliding windows.
#>     To change this change the t_start and t_end arguments.
no_truncation_with_imputation <- estimate_R(
                                            backimputation_window = 8,
                                            incid = incid_covid,
                                            method = "non_parametric_si",
                                            config =  config_covid
)
#> Default config will estimate R on weekly sliding windows.
#>     To change this change the t_start and t_end arguments.
gridExtra::grid.arrange(
  plot(no_truncation_no_imputation, "R") + ggtitle("No imputation"),
  plot(no_truncation_with_imputation, "R") + ggtitle("With imputation")
)

The reduction in R_t estimates is strongest in the early stages, and vanishes with time.

time <- no_truncation_no_imputation$R[, c("t_start", "t_end")]
estimates <- data.frame(
  x = time$t_end,
  original = no_truncation_no_imputation$R[, "Mean(R)"],
  adjusted = no_truncation_with_imputation$R[, "Mean(R)"]
)

ggplot(estimates, aes(x=x, y=(adjusted-original)/original * 100 )) +
  geom_line(colour="#5983AB") +
  theme_epiestim() +
  labs(
     title = "Percent reduction in mean R_t estimate due to imputation",
     x = "Midpoint of sliding window",
     y = "Percentage reduction"
  )
#> Warning: The `size` argument of `element_line()` is deprecated as of ggplot2 3.4.0.
#> ℹ Please use the `linewidth` argument instead.
#> This warning is displayed once every 8 hours.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
#> generated.

In this case, the shape of the R_t estimates are similar for the two methods. Still, the imputation reduces the first R_t estimate from 3.8 to 3.1.

Left-censoring: back-imputation reduces bias.

Now let us assume the first two weeks of deaths were not observed. How would the two methods compare?


trunc_t <- 14

# fit EpiEstim
truncation_no_imputation <- estimate_R(
                                     incid = incid_covid[ -(1:trunc_t)], 
                                     method = "non_parametric_si",
                                     config = config_covid
)
#> Default config will estimate R on weekly sliding windows.
#>     To change this change the t_start and t_end arguments.

truncation_with_imputation <- estimate_R(
                                   backimputation_window = 10,
                                   incid = incid_covid[ -(1:trunc_t)],
                                   method = "non_parametric_si",
                                   config = config_covid
)
#> Default config will estimate R on weekly sliding windows.
#>     To change this change the t_start and t_end arguments.

# specify classes
truncation_with_imputation$R$imputation <- TRUE
truncation_no_imputation$R$imputation <- FALSE

res1 <- truncation_with_imputation$R
res2 <- truncation_no_imputation$R

test <- data.frame(x = (res1$t_start + res1$t_end)/2, 
                     y = res1$`Mean(R)`,
                     ymin = res1$`Quantile.0.025(R)`,
                     ymax = res1$`Quantile.0.975(R)`, 
                     imputation = res1$imputation)
  
test2 <- data.frame(x = (res2$t_start + res2$t_end)/2, 
                     y = res2$`Mean(R)`,
                     ymin = res2$`Quantile.0.025(R)`,
                     ymax = res2$`Quantile.0.975(R)`, 
                     imputation = res2$imputation)
  
ggplot(test, aes( 
                          x = x,
                          y = y,
                          ymin = ymin,
                          ymax = ymax,
                          color = imputation,
                          fill = imputation,
                          linetype = imputation
                          )) + 
        geom_ribbon(alpha = .3, color = NA) +
        geom_ribbon(data = test2, alpha = .3, color = NA) +
        geom_line() +
        scale_color_manual(values=c("#5983AB", "#E5A3A3")) +
        scale_fill_manual(values=c("#5983AB", "#E5A3A3")) +
        labs(x = "Midpoint of sliding window", y = "EpiEstim Posterior Mean") +
        theme_epiestim() +
        geom_line(data = test2) +
        labs(x = "Midpoint of sliding window", y = "EpiEstim Posterior Mean") +
        theme_bw() +
        theme(legend.position = "bottom")

With truncated deaths data, the imputation procedure brought the R_t mean estimate from 3.9 to 2.8 . This is closer to the estimate obtained from the complete data at the same time point (3).

Caveats

The back imputation process is not always necessary, and is not suggested in the following cases.

Surveillance is timely set up and case reporting is precise: the first reported cases are likely the first generations of infections.
The majority of new cases are imported, hence were not generated by unseen local infections. In this case, refer to the “Specifying imported cases” section of the original demo.
Cases are sparsely reported, with frequent 0 cases. The exponential growth regime may not have started, or other reporting biases may lead to sensitive estimates of the growth rate.

References

Andrea Brizzi, Megan O’Driscoll, Ilaria Dorigatti, Refining Reproduction Number Estimates to Account for Unobserved Generations of Infection in Emerging Epidemics, Clinical Infectious Diseases, Volume 75, Issue 1, 1 July 2022, Pages e114–e121, https://doi.org/10.1093/cid/ciac138

Dealing with missed generations of infections with EpiEstim