24 Epidemic modeling

24.1 Overview

There exists a growing body of tools for epidemic modelling that lets us conduct fairly complex analyses with minimal effort. This section will provide an overview on how to use these tools to:

• estimate the effective reproduction number R_t and related statistics such as the doubling time
• produce short-term projections of future incidence

It is not intended as an overview of the methodologies and statistical methods underlying these tools, so please refer to the Resources tab for links to some papers covering this. Make sure you have an understanding of the methods before using these tools; this will ensure you can accurately interpret their results.

Below is an example of one of the outputs we’ll be producing in this section.

24.2 Preparation

We will use two different methods and packages for R_t estimation, namely EpiNow and EpiEstim, as well as the projections package for forecasting case incidence.

This code chunk shows the loading of packages required for the analyses. In this handbook we emphasize p_load() from pacman, which installs the package if necessary and loads it for use. You can also load installed packages with library() from base R. See the page on [R basics] for more information on R packages.

pacman::p_load(
   rio,          # File import
   here,         # File locator
   tidyverse,    # Data management + ggplot2 graphics
   epicontacts,  # Analysing transmission networks
   EpiNow2,      # Rt estimation
   EpiEstim,     # Rt estimation
   projections,  # Incidence projections
   incidence2,   # Handling incidence data
   epitrix,      # Useful epi functions
   distcrete     # Discrete delay distributions
)

We will use the cleaned case linelist for all analyses in this section. If you want to follow along, click to download the “clean” linelist (as .rds file). See the [Download handbook and data] page to download all example data used in this handbook.

# import the cleaned linelist
linelist <- import("linelist_cleaned.rds")

24.3 Estimating R_t

EpiNow2 vs. EpiEstim

The reproduction number R is a measure of the transmissibility of a disease and is defined as the expected number of secondary cases per infected case. In a fully susceptible population, this value represents the basic reproduction number R₀. However, as the number of susceptible individuals in a population changes over the course of an outbreak or pandemic, and as various response measures are implemented, the most commonly used measure of transmissibility is the effective reproduction number R_t; this is defined as the expected number of secondary cases per infected case at a given time t.

The EpiNow2 package provides the most sophisticated framework for estimating R_t. It has two key advantages over the other commonly used package, EpiEstim:

• It accounts for delays in reporting and can therefore estimate R_t even when recent data is incomplete.
• It estimates R_t on dates of infection rather than the dates of onset of reporting, which means that the effect of an intervention will be immediately reflected in a change in R_t, rather than with a delay.

However, it also has two key disadvantages:

• It requires knowledge of the generation time distribution (i.e. distribution of delays between infection of a primary and secondary cases), incubation period distribution (i.e. distribution of delays between infection and symptom onset) and any further delay distribution relevant to your data (e.g. if you have dates of reporting, you require the distribution of delays from symptom onset to reporting). While this will allow more accurate estimation of R_t, EpiEstim only requires the serial interval distribution (i.e. the distribution of delays between symptom onset of a primary and a secondary case), which may be the only distribution available to you.
• EpiNow2 is significantly slower than EpiEstim, anecdotally by a factor of about 100-1000! For example, estimating R_t for the sample outbreak considered in this section takes about four hours (this was run for a large number of iterations to ensure high accuracy and could probably be reduced if necessary, however the points stands that the algorithm is slow in general). This may be unfeasible if you are regularly updating your R_t estimates.

Which package you choose to use will therefore depend on the data, time and computational resources available to you.

EpiNow2

Estimating delay distributions

The delay distributions required to run EpiNow2 depend on the data you have. Essentially, you need to be able to describe the delay from the date of infection to the date of the event you want to use to estimate R_t. If you are using dates of onset, this would simply be the incubation period distribution. If you are using dates of reporting, you require the delay from infection to reporting. As this distribution is unlikely to be known directly, EpiNow2 lets you chain multiple delay distributions together; in this case, the delay from infection to symptom onset (e.g. the incubation period, which is likely known) and from symptom onset to reporting (which you can often estimate from the data).

As we have the dates of onset for all our cases in the example linelist, we will only require the incubation period distribution to link our data (e.g. dates of symptom onset) to the date of infection. We can either estimate this distribution from the data or use values from the literature.

A literature estimate of the incubation period of Ebola (taken from this paper) with a mean of 9.1, standard deviation of 7.3 and maximum value of 30 would be specified as follows:

incubation_period_lit <- list(
  mean = log(9.1),
  mean_sd = log(0.1),
  sd = log(7.3),
  sd_sd = log(0.1),
  max = 30
)

Note that EpiNow2 requires these delay distributions to be provided on a log scale, hence the log call around each value (except the max parameter which, confusingly, has to be provided on a natural scale). The mean_sd and sd_sd define the standard deviation of the mean and standard deviation estimates. As these are not known in this case, we choose the fairly arbitrary value of 0.1.

In this analysis, we instead estimate the incubation period distribution from the linelist itself using the function bootstrapped_dist_fit, which will fit a lognormal distribution to the observed delays between infection and onset in the linelist.

## estimate incubation period
incubation_period <- bootstrapped_dist_fit(
  linelist$date_onset - linelist$date_infection,
  dist = "lognormal",
  max_value = 100,
  bootstraps = 1
)

The other distribution we require is the generation time. As we have data on infection times and transmission links, we can estimate this distribution from the linelist by calculating the delay between infection times of infector-infectee pairs. To do this, we use the handy get_pairwise function from the package epicontacts, which allows us to calculate pairwise differences of linelist properties between transmission pairs. We first create an epicontacts object (see Transmission chains page for further details):

## generate contacts
contacts <- linelist %>%
  transmute(
    from = infector,
    to = case_id
  ) %>%
  drop_na()

## generate epicontacts object
epic <- make_epicontacts(
  linelist = linelist,
  contacts = contacts, 
  directed = TRUE
)

We then fit the difference in infection times between transmission pairs, calculated using get_pairwise, to a gamma distribution:

## estimate gamma generation time
generation_time <- bootstrapped_dist_fit(
  get_pairwise(epic, "date_infection"),
  dist = "gamma",
  max_value = 20,
  bootstraps = 1
)

Running EpiNow2

Now we just need to calculate daily incidence from the linelist, which we can do easily with the dplyr functions group_by() and n(). Note that EpiNow2 requires the column names to be date and confirm.

## get incidence from onset dates
cases <- linelist %>%
  group_by(date = date_onset) %>%
  summarise(confirm = n())

We can then estimate R_t using the epinow function. Some notes on the inputs:

• We can provide any number of ‘chained’ delay distributions to the delays argument; we would simply insert them alongside the incubation_period object within the delay_opts function.
• return_output ensures the output is returned within R and not just saved to a file.
• verbose specifies that we want a readout of the progress.
• horizon indicates how many days we want to project future incidence for.
• We pass additional options to the stan argument to specify how long we want to run the inference for. Increasing samples and chains will give you a more accurate estimate that better characterises uncertainty, however will take longer to run.

## run epinow
epinow_res <- epinow(
  reported_cases = cases,
  generation_time = generation_time,
  delays = delay_opts(incubation_period),
  return_output = TRUE,
  verbose = TRUE,
  horizon = 21,
  stan = stan_opts(samples = 750, chains = 4)
)

Analysing outputs

Once the code has finished running, we can plot a summary very easily as follows. Scroll the image to see the full extent.

## plot summary figure
plot(epinow_res)

We can also look at various summary statistics:

## summary table
epinow_res$summary

##                                  measure                  estimate  numeric_estimate
## 1: New confirmed cases by infection date                4 (2 -- 6) <data.table[1x9]>
## 2:        Expected change in daily cases                    Unsure              0.56
## 3:            Effective reproduction no.        0.88 (0.73 -- 1.1) <data.table[1x9]>
## 4:                        Rate of growth -0.012 (-0.028 -- 0.0052) <data.table[1x9]>
## 5:          Doubling/halving time (days)          -60 (130 -- -25) <data.table[1x9]>

For further analyses and custom plotting, you can access the summarised daily estimates via $estimates$summarised. We will convert this from the default data.table to a tibble for ease of use with dplyr.

## extract summary and convert to tibble
estimates <- as_tibble(epinow_res$estimates$summarised)
estimates

As an example, let’s make a plot of the doubling time and R_t. We will only look at the first few months of the outbreak when R_t is well above one, to avoid plotting extremely high doublings times.

We use the formula log(2)/growth_rate to calculate the doubling time from the estimated growth rate.

## make wide df for median plotting
df_wide <- estimates %>%
  filter(
    variable %in% c("growth_rate", "R"),
    date < as.Date("2014-09-01")
  ) %>%
  ## convert growth rates to doubling times
  mutate(
    across(
      c(median, lower_90:upper_90),
      ~ case_when(
        variable == "growth_rate" ~ log(2)/.x,
        TRUE ~ .x
      )
    ),
    ## rename variable to reflect transformation
    variable = replace(variable, variable == "growth_rate", "doubling_time")
  )

## make long df for quantile plotting
df_long <- df_wide %>%
  ## here we match matching quantiles (e.g. lower_90 to upper_90)
  pivot_longer(
    lower_90:upper_90,
    names_to = c(".value", "quantile"),
    names_pattern = "(.+)_(.+)"
  )

## make plot
ggplot() +
  geom_ribbon(
    data = df_long,
    aes(x = date, ymin = lower, ymax = upper, alpha = quantile),
    color = NA
  ) +
  geom_line(
    data = df_wide,
    aes(x = date, y = median)
  ) +
  ## use label_parsed to allow subscript label
  facet_wrap(
    ~ variable,
    ncol = 1,
    scales = "free_y",
    labeller = as_labeller(c(R = "R[t]", doubling_time = "Doubling~time"), label_parsed),
    strip.position = 'left'
  ) +
  ## manually define quantile transparency
  scale_alpha_manual(
    values = c(`20` = 0.7, `50` = 0.4, `90` = 0.2),
    labels = function(x) paste0(x, "%")
  ) +
  labs(
    x = NULL,
    y = NULL,
    alpha = "Credibel\ninterval"
  ) +
  scale_x_date(
    date_breaks = "1 month",
    date_labels = "%b %d\n%Y"
  ) +
  theme_minimal(base_size = 14) +
  theme(
    strip.background = element_blank(),
    strip.placement = 'outside'
  )

EpiEstim

To run EpiEstim, we need to provide data on daily incidence and specify the serial interval (i.e. the distribution of delays between symptom onset of primary and secondary cases).

Incidence data can be provided to EpiEstim as a vector, a data frame, or an incidence object from the original incidence package. You can even distinguish between imports and locally acquired infections; see the documentation at ?estimate_R for further details.

We will create the input using incidence2. See the page on Epidemic curves for more examples with the incidence2 package. Since there have been updates to the incidence2 package that don’t completely align with estimateR()’s expected input, there are some minor additional steps needed. The incidence object consists of a tibble with dates and their respective case counts. We use complete() from tidyr to ensure all dates are included (even those with no cases), and then rename() the columns to align with what is expected by estimate_R() in a later step.

## get incidence from onset date
cases <- incidence2::incidence(linelist, date_index = date_onset) %>% # get case counts by day
  tidyr::complete(date_index = seq.Date(                              # ensure all dates are represented
    from = min(date_index, na.rm = T),
    to = max(date_index, na.rm=T),
    by = "day"),
    fill = list(count = 0)) %>%                                       # convert NA counts to 0
  rename(I = count,                                                   # rename to names expected by estimateR
         dates = date_index)

The package provides several options for specifying the serial interval, the details of which are provided in the documentation at ?estimate_R. We will cover two of them here.

Using serial interval estimates from the literature

Using the option method = "parametric_si", we can manually specify the mean and standard deviation of the serial interval in a config object created using the function make_config. We use a mean and standard deviation of 12.0 and 5.2, respectively, defined in this paper:

## make config
config_lit <- make_config(
  mean_si = 12.0,
  std_si = 5.2
)

We can then estimate R_t with the estimate_R function:

epiestim_res_lit <- estimate_R(
  incid = cases,
  method = "parametric_si",
  config = config_lit
)

## Default config will estimate R on weekly sliding windows.
##     To change this change the t_start and t_end arguments.

and plot a summary of the outputs:

plot(epiestim_res_lit)

Using serial interval estimates from the data

As we have data on dates of symptom onset and transmission links, we can also estimate the serial interval from the linelist by calculating the delay between onset dates of infector-infectee pairs. As we did in the EpiNow2 section, we will use the get_pairwise function from the epicontacts package, which allows us to calculate pairwise differences of linelist properties between transmission pairs. We first create an epicontacts object (see Transmission chains page for further details):

## generate contacts
contacts <- linelist %>%
  transmute(
    from = infector,
    to = case_id
  ) %>%
  drop_na()

## generate epicontacts object
epic <- make_epicontacts(
  linelist = linelist,
  contacts = contacts, 
  directed = TRUE
)

We then fit the difference in onset dates between transmission pairs, calculated using get_pairwise, to a gamma distribution. We use the handy fit_disc_gamma from the epitrix package for this fitting procedure, as we require a discretised distribution.

## estimate gamma serial interval
serial_interval <- fit_disc_gamma(get_pairwise(epic, "date_onset"))

We then pass this information to the config object, run EpiEstim again and plot the results:

## make config
config_emp <- make_config(
  mean_si = serial_interval$mu,
  std_si = serial_interval$sd
)

## run epiestim
epiestim_res_emp <- estimate_R(
  incid = cases,
  method = "parametric_si",
  config = config_emp
)

## Default config will estimate R on weekly sliding windows.
##     To change this change the t_start and t_end arguments.

## plot outputs
plot(epiestim_res_emp)

Specifying estimation time windows

These default options will provide a weekly sliding estimate and might act as a warning that you are estimating R_t too early in the outbreak for a precise estimate. You can change this by setting a later start date for the estimation as shown below. Unfortunately, EpiEstim only provides a very clunky way of specifying these estimations times, in that you have to provide a vector of integers referring to the start and end dates for each time window.

## define a vector of dates starting on June 1st
start_dates <- seq.Date(
  as.Date("2014-06-01"),
  max(cases$dates) - 7,
  by = 1
) %>%
  ## subtract the starting date to convert to numeric
  `-`(min(cases$dates)) %>%
  ## convert to integer
  as.integer()

## add six days for a one week sliding window
end_dates <- start_dates + 6
  
## make config
config_partial <- make_config(
  mean_si = 12.0,
  std_si = 5.2,
  t_start = start_dates,
  t_end = end_dates
)

Now we re-run EpiEstim and can see that the estimates only start from June:

## run epiestim
epiestim_res_partial <- estimate_R(
  incid = cases,
  method = "parametric_si",
  config = config_partial
)

## plot outputs
plot(epiestim_res_partial)

## Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.

Analysing outputs

The main outputs can be accessed via $R. As an example, we will create a plot of R_t and a measure of “transmission potential” given by the product of R_t and the number of cases reported on that day; this represents the expected number of cases in the next generation of infection.

## make wide dataframe for median
df_wide <- epiestim_res_lit$R %>%
  rename_all(clean_labels) %>%
  rename(
    lower_95_r = quantile_0_025_r,
    lower_90_r = quantile_0_05_r,
    lower_50_r = quantile_0_25_r,
    upper_50_r = quantile_0_75_r,
    upper_90_r = quantile_0_95_r,
    upper_95_r = quantile_0_975_r,
    ) %>%
  mutate(
    ## extract the median date from t_start and t_end
    dates = epiestim_res_emp$dates[round(map2_dbl(t_start, t_end, median))],
    var = "R[t]"
  ) %>%
  ## merge in daily incidence data
  left_join(cases, "dates") %>%
  ## calculate risk across all r estimates
  mutate(
    across(
      lower_95_r:upper_95_r,
      ~ .x*I,
      .names = "{str_replace(.col, '_r', '_risk')}"
    )
  ) %>%
  ## seperate r estimates and risk estimates
  pivot_longer(
    contains("median"),
    names_to = c(".value", "variable"),
    names_pattern = "(.+)_(.+)"
  ) %>%
  ## assign factor levels
  mutate(variable = factor(variable, c("risk", "r")))

## make long dataframe from quantiles
df_long <- df_wide %>%
  select(-variable, -median) %>%
  ## seperate r/risk estimates and quantile levels
  pivot_longer(
    contains(c("lower", "upper")),
    names_to = c(".value", "quantile", "variable"),
    names_pattern = "(.+)_(.+)_(.+)"
  ) %>%
  mutate(variable = factor(variable, c("risk", "r")))

## make plot
ggplot() +
  geom_ribbon(
    data = df_long,
    aes(x = dates, ymin = lower, ymax = upper, alpha = quantile),
    color = NA
  ) +
  geom_line(
    data = df_wide,
    aes(x = dates, y = median),
    alpha = 0.2
  ) +
  ## use label_parsed to allow subscript label
  facet_wrap(
    ~ variable,
    ncol = 1,
    scales = "free_y",
    labeller = as_labeller(c(r = "R[t]", risk = "Transmission~potential"), label_parsed),
    strip.position = 'left'
  ) +
  ## manually define quantile transparency
  scale_alpha_manual(
    values = c(`50` = 0.7, `90` = 0.4, `95` = 0.2),
    labels = function(x) paste0(x, "%")
  ) +
  labs(
    x = NULL,
    y = NULL,
    alpha = "Credible\ninterval"
  ) +
  scale_x_date(
    date_breaks = "1 month",
    date_labels = "%b %d\n%Y"
  ) +
  theme_minimal(base_size = 14) +
  theme(
    strip.background = element_blank(),
    strip.placement = 'outside'
  )

24.4 Projecting incidence

EpiNow2

Besides estimating R_t, EpiNow2 also supports forecasting of R_t and projections of case numbers by integration with the EpiSoon package under the hood. All you need to do is specify the horizon argument in your epinow function call, indicating how many days you want to project into the future; see the EpiNow2 section under the “Estimating R_t” for details on how to get EpiNow2 up and running. In this section, we will just plot the outputs from that analysis, stored in the epinow_res object.

## define minimum date for plot
min_date <- as.Date("2015-03-01")

## extract summarised estimates
estimates <-  as_tibble(epinow_res$estimates$summarised)

## extract raw data on case incidence
observations <- as_tibble(epinow_res$estimates$observations) %>%
  filter(date > min_date)

## extract forecasted estimates of case numbers
df_wide <- estimates %>%
  filter(
    variable == "reported_cases",
    type == "forecast",
    date > min_date
  )

## convert to even longer format for quantile plotting
df_long <- df_wide %>%
  ## here we match matching quantiles (e.g. lower_90 to upper_90)
  pivot_longer(
    lower_90:upper_90,
    names_to = c(".value", "quantile"),
    names_pattern = "(.+)_(.+)"
  )

## make plot
ggplot() +
  geom_histogram(
    data = observations,
    aes(x = date, y = confirm),
    stat = 'identity',
    binwidth = 1
  ) +
  geom_ribbon(
    data = df_long,
    aes(x = date, ymin = lower, ymax = upper, alpha = quantile),
    color = NA
  ) +
  geom_line(
    data = df_wide,
    aes(x = date, y = median)
  ) +
  geom_vline(xintercept = min(df_long$date), linetype = 2) +
  ## manually define quantile transparency
  scale_alpha_manual(
    values = c(`20` = 0.7, `50` = 0.4, `90` = 0.2),
    labels = function(x) paste0(x, "%")
  ) +
  labs(
    x = NULL,
    y = "Daily reported cases",
    alpha = "Credible\ninterval"
  ) +
  scale_x_date(
    date_breaks = "1 month",
    date_labels = "%b %d\n%Y"
  ) +
  theme_minimal(base_size = 14)

projections

The projections package developed by RECON makes it very easy to make short term incidence forecasts, requiring only knowledge of the effective reproduction number R_t and the serial interval. Here we will cover how to use serial interval estimates from the literature and how to use our own estimates from the linelist.

Using serial interval estimates from the literature

projections requires a discretised serial interval distribution of the class distcrete from the package distcrete. We will use a gamma distribution with a mean of 12.0 and and standard deviation of 5.2 defined in this paper. To convert these values into the shape and scale parameters required for a gamma distribution, we will use the function gamma_mucv2shapescale from the epitrix package.

## get shape and scale parameters from the mean mu and the coefficient of
## variation (e.g. the ratio of the standard deviation to the mean)
shapescale <- epitrix::gamma_mucv2shapescale(mu = 12.0, cv = 5.2/12)

## make distcrete object
serial_interval_lit <- distcrete::distcrete(
  name = "gamma",
  interval = 1,
  shape = shapescale$shape,
  scale = shapescale$scale
)

Here is a quick check to make sure the serial interval looks correct. We access the density of the gamma distribution we have just defined by $d, which is equivalent to calling dgamma:

## check to make sure the serial interval looks correct
qplot(
  x = 0:50, y = serial_interval_lit$d(0:50), geom = "area",
  xlab = "Serial interval", ylab = "Density"
)

Using serial interval estimates from the data

As we have data on dates of symptom onset and transmission links, we can also estimate the serial interval from the linelist by calculating the delay between onset dates of infector-infectee pairs. As we did in the EpiNow2 section, we will use the get_pairwise function from the epicontacts package, which allows us to calculate pairwise differences of linelist properties between transmission pairs. We first create an epicontacts object (see Transmission chains page for further details):

## generate contacts
contacts <- linelist %>%
  transmute(
    from = infector,
    to = case_id
  ) %>%
  drop_na()

## generate epicontacts object
epic <- make_epicontacts(
  linelist = linelist,
  contacts = contacts, 
  directed = TRUE
)

We then fit the difference in onset dates between transmission pairs, calculated using get_pairwise, to a gamma distribution. We use the handy fit_disc_gamma from the epitrix package for this fitting procedure, as we require a discretised distribution.

## estimate gamma serial interval
serial_interval <- fit_disc_gamma(get_pairwise(epic, "date_onset"))

## inspect estimate
serial_interval[c("mu", "sd")]

## $mu
## [1] 11.51242
## 
## $sd
## [1] 7.700005

Projecting incidence

To project future incidence, we still need to provide historical incidence in the form of an incidence object, as well as a sample of plausible R_t values. We will generate these values using the R_t estimates generated by EpiEstim in the previous section (under “Estimating R_t”) and stored in the epiestim_res_emp object. In the code below, we extract the mean and standard deviation estimates of R_t for the last time window of the outbreak (using the tail function to access the last element in a vector), and simulate 1000 values from a gamma distribution using rgamma. You can also provide your own vector of R_t values that you want to use for forward projections.

## create incidence object from dates of onset
inc <- incidence::incidence(linelist$date_onset)

## 256 missing observations were removed.

## extract plausible r values from most recent estimate
mean_r <- tail(epiestim_res_emp$R$`Mean(R)`, 1)
sd_r <- tail(epiestim_res_emp$R$`Std(R)`, 1)
shapescale <- gamma_mucv2shapescale(mu = mean_r, cv = sd_r/mean_r)
plausible_r <- rgamma(1000, shape = shapescale$shape, scale = shapescale$scale)

## check distribution
qplot(x = plausible_r, geom = "histogram", xlab = expression(R[t]), ylab = "Counts")

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

We then use the project() function to make the actual forecast. We specify how many days we want to project for via the n_days arguments, and specify the number of simulations using the n_sim argument.

## make projection
proj <- project(
  x = inc,
  R = plausible_r,
  si = serial_interval$distribution,
  n_days = 21,
  n_sim = 1000
)

We can then handily plot the incidence and projections using the plot() and add_projections() functions. We can easily subset the incidence object to only show the most recent cases by using the square bracket operator.

## plot incidence and projections
plot(inc[inc$dates > as.Date("2015-03-01")]) %>%
  add_projections(proj)

You can also easily extract the raw estimates of daily case numbers by converting the output to a dataframe.

## convert to data frame for raw data
proj_df <- as.data.frame(proj)
proj_df

24.5 Resources

• Here is the paper describing the methodology implemented in EpiEstim.
• Here is the paper describing the methodology implemented in EpiNow2.
• Here is a paper describing various methodological and practical considerations for estimating R_t.