This practical simulates the early assessment and reconstruction of an Ebola Virus Disease (EVD) outbreak. It introduces various aspects of analysis of the early stage of an outbreak, including contact tracing data, epicurves, growth rate estimation from log-linear models, and more refined estimates of transmissibility. A follow-up practical will provide an introduction to transmission chain reconstruction using outbreaker2.

A novel EVD outbreak in Ankh, Republic of Morporkia

A new EVD outbreak has been notified in the small city of Ankh, located in the Northern, rural district of the Republic of Morporkia. Public Health Morporkia (PHM) is in charge of coordinating the outbreak response, and have contracted you as a consultant in epidemics analysis to inform the response in real time.

Required packages

The following packages, available on CRAN, are needed for this case study:

• xlsx to read .xlsx files
• outbreaks for some other outbreak data
• incidence for epicurves
• epicontacts for contact data visualisation
• distcrete to obtain discretised delay distributions
• epitrix to fit discretised Gamma distributions
• earlyR to estimate R₀
• projections for short term forecasting

To install these packages, use install.packages, e.g.:

install.packages("xlsx")
install.packages("outbreaks")
install.packages("incidence")
install.packages("epicontacts")
install.packages("distcrete")
install.packages("epitrix")
install.packages("earlyR")
install.packages("projections")

Early data

While a new data update is pending, you have been given the following linelist and contact data, from the early stages of the outbreak:

• PHM-EVD-linelist-2017-10-27.xlsx: a linelist containing case information up to the 27th October 2017

• PHM-EVD-contacts-2017-10-27.xlsx: a list of contacts reported between cases up to the 27th October 2017, where from indicates a potential source of infection, and to the recipient of the contact.

To read into R, download these files and use the function read.xlsx() from the xlsx package to import the data. Each import will create a data.frame. Call the first one linelist, and the second one contacts. For instance, you first command line could look like:

linelist <- xlsx::read.xlsx("PHM-EVD-linelist-2017-10-27.xlsx", sheetIndex = 1)

Once imported, the data should look like:

## linelist: one line per case
linelist
##        id      onset    sex age
## 1  39e9dc 2017-10-10 female  62
## 2  664549 2017-10-16   male  28
## 3  b4d8aa 2017-10-17   male  54
## 4  51883d 2017-10-18   male  57
## 5  947e40 2017-10-20 female  23
## 6  9aa197 2017-10-20 female  66
## 7  e4b0a2 2017-10-21 female  13
## 8  af0ac0 2017-10-21   male  10
## 9  185911 2017-10-21 female  34
## 10 601d2e 2017-10-22   male  11
## 11 605322 2017-10-22 female  23
## 12 e399b1 2017-10-23 female  23

## contacts: pairs of cases with reported contacts
contacts
##      from     to
## 1  51883d 185911
## 2  b4d8aa e4b0a2
## 3  39e9dc b4d8aa
## 4  39e9dc 601d2e
## 5  51883d 9aa197
## 6  39e9dc 51883d
## 7  39e9dc e399b1
## 8  b4d8aa af0ac0
## 9  39e9dc 947e40
## 10 39e9dc 664549
## 11 39e9dc 605322

Descriptive analyses

A first look at contacts

Contact tracing is at the centre of an Ebola outbreak response. Using the function make_epicontacts in the epicontacts package, create a new epicontacts object called x. The result should look like:

x
## 
## /// Epidemiological Contacts //
## 
##   // class: epicontacts
##   // 12 cases in linelist; 11 contacts;  directed 
## 
##   // linelist
## 
## # A tibble: 12 x 4
##    id     onset      sex      age
##  * <chr>  <date>     <fct>  <dbl>
##  1 39e9dc 2017-10-10 female   62.
##  2 664549 2017-10-16 male     28.
##  3 b4d8aa 2017-10-17 male     54.
##  4 51883d 2017-10-18 male     57.
##  5 947e40 2017-10-20 female   23.
##  6 9aa197 2017-10-20 female   66.
##  7 e4b0a2 2017-10-21 female   13.
##  8 af0ac0 2017-10-21 male     10.
##  9 185911 2017-10-21 female   34.
## 10 601d2e 2017-10-22 male     11.
## 11 605322 2017-10-22 female   23.
## 12 e399b1 2017-10-23 female   23.
## 
##   // contacts
## 
## # A tibble: 11 x 2
##    from   to    
##    <chr>  <chr> 
##  1 51883d 185911
##  2 b4d8aa e4b0a2
##  3 39e9dc b4d8aa
##  4 39e9dc 601d2e
##  5 51883d 9aa197
##  6 39e9dc 51883d
##  7 39e9dc e399b1
##  8 b4d8aa af0ac0
##  9 39e9dc 947e40
## 10 39e9dc 664549
## 11 39e9dc 605322

You can easily plot these contacts, but with a little bit of tweaking (see ?vis_epicontacts) you can customise shapes by gender:

plot(x, node_shape = "sex", shapes = c(male = "male", female = "female"), selector = FALSE)

What can you say about these contacts?

Looking at incidence curves

The first question PHM asks you is simply: how bad is it?. Given that this is a terrible disease, with a mortality rate nearing 70%, there is a lot of concern about this outbreak getting out of control. The first step of the analysis lies in drawing an epicurve, i.e. an plot of incidence over time.

Using the package incidence, compute daily incidence based on the dates of symptom onset. Store the result in an object called i; the result should look like:

i
## <incidence object>
## [12 cases from days 2017-10-10 to 2017-10-23]
## 
## $counts: matrix with 14 rows and 1 columns
## $n: 12 cases in total
## $dates: 14 dates marking the left-side of bins
## $interval: 1 day
## $timespan: 14 days
plot(i)

If you pay close attention to the dates on the x-axis, you may notice that something is missing. Indeed, the graph stops right after the last case, while the data should be complete until the 27th October 2017. You can remedy this using the argument last_date in the incidence function:

i <- incidence(linelist$onset, last_date = as.Date("2017-10-27"))
i
## <incidence object>
## [12 cases from days 2017-10-10 to 2017-10-27]
## 
## $counts: matrix with 18 rows and 1 columns
## $n: 12 cases in total
## $dates: 18 dates marking the left-side of bins
## $interval: 1 day
## $timespan: 18 days
plot(i)

Statistical analyses

Log-linear model

The simplest model of incidence is probably the log-linear model, i.e. a linear regression on log-transformed incidences. In the incidence package, the function fit will estimate the parameters of this model from an incidence object (here, i). Apply it to the data and store the result in a new object called f. You can print f to derive estimates of the growth rate r and the doubling time, and add the corresponding model to the incidence plot:

f <- fit(i)
## Warning in fit(i): 10 dates with incidence of 0 ignored for fitting
f
## <incidence_fit object>
## 
## $lm: regression of log-incidence over time
## 
## $info: list containing the following items:
##   $r (daily growth rate):
## [1] 0.05352107
## 
##   $r.conf (confidence interval):
##           2.5 %    97.5 %
## [1,] -0.0390633 0.1461054
## 
##   $doubling (doubling time in days):
## [1] 12.95092
## 
##   $doubling.conf (confidence interval):
##         2.5 %    97.5 %
## [1,] 4.744158 -17.74421
## 
##   $pred: data.frame of incidence predictions (8 rows, 5 columns)
plot(i, fit = f)

How would you interpret this result?What criticism would you make on this model?

Estimation of transmissibility (R)

Branching process model

The transmissibility of the disease can be assessed through the estimation of the reproduction number R, defined as the number of expected secondary cases per infected case. In the early stages of an outbreak, and assuming no immunity in the population, this quantity is also the basic reproduction number R₀, i.e. R in a fully susceptible population.

The package earlyR implements a simple maximum-likelihood estimation of R, using dates of onset of symptoms and information on the serial interval distribution. It is a simpler but less flexible version of the model by Cori et al (2013, AJE 178: 1505–1512) implemented in EpiEstim.

Briefly, earlyR uses a simple model describing incidence on a given day as a Poisson process determined by a global force of infection on that day: x_t ∼ 𝒫(λ_t)

where x_t is the incidence (based on symptom onset) on day t and λ_t is the force of infection. Noting R the reproduction number and w() the discrete serial interval distribution, we have:

$$ \lambda_t = R * \sum_{s=1}^t x_s w(t - s) $$

The likelihood (probability of observing the data given the model and parameters) is defined as a function of R:

$$ \mathcal{L}(x) = p(x | R) = \prod_{t=1}^T F_{\mathcal{P}}(x_t, \lambda_t) $$

where F_𝒫 is the Poisson probability mass function.

Estimating the serial interval from older data

As current data are insufficient to estimate the serial interval distribution, some colleague recommends using data from a past outbreak stored in the outbreaks package, as the dataset ebola_sim_clean. Load this dataset, and create a new epicontacts object as before, without plotting it (it is a much larger dataset). Store the new object as old_evd; the output should look like:

old_evd
## 
## /// Epidemiological Contacts //
## 
##   // class: epicontacts
##   // 5,829 cases in linelist; 3,800 contacts;  directed 
## 
##   // linelist
## 
## # A tibble: 5,829 x 9
##    id     generation date_of_infection date_of_onset date_of_hospitalisat~
##  * <chr>       <int> <date>            <date>        <date>               
##  1 d1fafd          0 NA                2014-04-07    2014-04-17           
##  2 53371b          1 2014-04-09        2014-04-15    2014-04-20           
##  3 f5c3d8          1 2014-04-18        2014-04-21    2014-04-25           
##  4 6c286a          2 NA                2014-04-27    2014-04-27           
##  5 0f58c4          2 2014-04-22        2014-04-26    2014-04-29           
##  6 49731d          0 2014-03-19        2014-04-25    2014-05-02           
##  7 f9149b          3 NA                2014-05-03    2014-05-04           
##  8 881bd4          3 2014-04-26        2014-05-01    2014-05-05           
##  9 e66fa4          2 NA                2014-04-21    2014-05-06           
## 10 20b688          3 NA                2014-05-05    2014-05-06           
## # ... with 5,819 more rows, and 4 more variables: date_of_outcome <date>,
## #   outcome <fct>, gender <fct>, hospital <fct>
## 
##   // contacts
## 
## # A tibble: 3,800 x 3
##    from   to     source 
##  * <chr>  <chr>  <fct>  
##  1 d1fafd 53371b other  
##  2 cac51e f5c3d8 funeral
##  3 f5c3d8 0f58c4 other  
##  4 0f58c4 881bd4 other  
##  5 8508df 40ae5f other  
##  6 127d83 f547d6 funeral
##  7 f5c3d8 d58402 other  
##  8 20b688 d8a13d other  
##  9 2ae019 a3c8b8 other  
## 10 20b688 974bc1 other  
## # ... with 3,790 more rows

The function get_pairwise can be used to extract pairwise features of contacts based on attributes of the linelist. For instance, it could be used to test for assortativity, but also to compute delays between connected cases. Here, we use it to extract the serial interval:

old_si <- get_pairwise(old_evd, "date_of_onset")
summary(old_si)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##    1.00    5.00    9.00   11.06   15.00   99.00    1684
old_si <- na.omit(old_si)
summary(old_si)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    1.00    5.00    9.00   11.06   15.00   99.00
hist(old_si, xlab = "Days after symptom onset", ylab = "Frequency",
     main = "Serial interval (empirical distribution)",
     col = "grey", border = "white")

What do you think of this distribution? Make the adjustments you deem necessary, and then use the function fit_disc_gamma from the package epitrix to fit a discretised Gamma distribution to these data. Your results should approximately look like:

si_fit <- fit_disc_gamma(old_si)
si_fit
## $mu
## [1] 11.48373
## 
## $cv
## [1] 0.6429561
## 
## $sd
## [1] 7.383534
## 
## $ll
## [1] -6905.588
## 
## $converged
## [1] TRUE
## 
## $distribution
## A discrete distribution
##   name: gamma
##   parameters:
##     shape: 2.41900859135364
##     scale: 4.74728799288241

Now that you know how to parametrise a discretised Gamma distribution matching previous data on the serial interval, create this distribution using the function distcrete from the similarly named package. Specify make sure you use daily intervals for the discretisation, and use w = 0 to indicate dates should be floored to the day. Store the ouput in an object called si. Your results should look like:

## Warning: package 'distcrete' was built under R version 3.4.4
si
## A discrete distribution
##   name: gamma
##   parameters:
##     shape: 2.41900859135364
##     scale: 4.74728799288241

## look at the fit
hist(old_si, xlab = "Days after symptom onset", ylab = "Frequency",
     main = "Serial interval: fit to data", col = "grey", border = "white",
     nclass = 30, ylim = c(0, 0.07), prob = TRUE)
points(0:60, si$d(0:60), col = "#9933ff", pch = 20)
points(0:60, si$d(0:60), col = "#9933ff", type = "l", lty = 2)

Would you trust this estimation of the generation time? How would you compare it to actual estimates from the West African EVD outbreak (WHO Ebola Response Team (2014) NEJM 371:1481–1495) with a mean of 15.3 days and a standard deviation 9.3 days?

Estimation of R₀

Using the estimates of the mean and standard deviation of the serial interval you just obtained, use the function get_R to estimate the reproduction number, specifying a maximum R of 10 (see ?get_R) and store the result in a new object R:

You can visualise the results as follows:

R
## 
## /// Early estimate of reproduction number (R) //
##  // class: earlyR, list
## 
##  // Maximum-Likelihood estimate of R ($R_ml):
## [1] 1.811812
## 
## 
##  // $lambda:
##   0.04088711 0.05343783 0.06188455 0.06668495 0.06850099 0.0680082...
## 
##  // $dates:
## [1] "2017-10-11" "2017-10-12" "2017-10-13" "2017-10-14" "2017-10-15"
## [6] "2017-10-16"
## ...
## 
##  // $si (serial interval):
## A discrete distribution
##   name: gamma
##   parameters:
##     shape: 2.41900859135364
##     scale: 4.74728799288241
plot(R)

plot(R, "lambdas")
abline(v = linelist$onset, lty = 2)

The first figure shows the distribution of likely values of R, and the Maximum-Likelihood (ML) estimation. The second figure shows the global force of infection over time, with dashed bars indicating dates of onset of the cases.

Interpret these results: what do you make of the reproduction number?What does it reflect? Based on the last part of the epicurve, some colleagues suggest that incidence is going down and the outbreak may be under control. What is your opinion on this?

Short-term forecasting

The function project from the package projections can be used to simulate plausible epidemic trajectories by simulating daily incidence using the same branching process as the one used to estimate R₀ in earlyR. All that is needed is one or several values of R₀ and a serial interval distribution, stored as a distcrete object.

Here, we illustrate how we can simulate 5 random trajectories using a fixed value of $R_0 = $1.81, the ML estimate of R₀:

library(projections)
## Warning: package 'projections' was built under R version 3.4.4
project(i, R = R$R_ml, si = si, n_sim = 5, n_days = 10, R_fix_within = TRUE)
## 
## /// Incidence projections //
## 
##   // class: projections, matrix
##   // 10 dates (rows); 5 simulations (columns)
## 
##  // first rows/columns:
##            [,1] [,2] [,3] [,4] [,5]
## 2017-10-28    0    2    0    0    0
## 2017-10-29    1    0    1    1    0
## 2017-10-30    1    2    0    4    1
## 2017-10-31    0    1    0    0    0
##  .
##  .
##  .
## 
##  // dates:
##  [1] "2017-10-28" "2017-10-29" "2017-10-30" "2017-10-31" "2017-11-01"
##  [6] "2017-11-02" "2017-11-03" "2017-11-04" "2017-11-05" "2017-11-06"

Using the same principle, generate 1,000 trajectories for the next 2 weeks, using a range of plausible values of R₀. Note that you can use sample_R to obtain these values from your earlyR object. Store your results in an object called proj. Plotting the results should give something akin to:

plot(proj, c(.1,.9))

Interpret the following summary:

apply(proj, 1, summary)
##         2017-10-28 2017-10-29 2017-10-30 2017-10-31 2017-11-01 2017-11-02
## Min.         0.000      0.000      0.000      0.000      0.000      0.000
## 1st Qu.      0.000      0.000      0.000      0.000      1.000      1.000
## Median       1.000      1.000      1.000      1.000      1.000      1.000
## Mean         1.403      1.457      1.518      1.494      1.651      1.771
## 3rd Qu.      2.000      2.000      2.000      2.000      2.000      3.000
## Max.         7.000      6.000      8.000     10.000     10.000     13.000
##         2017-11-03 2017-11-04 2017-11-05 2017-11-06 2017-11-07 2017-11-08
## Min.         0.000      0.000      0.000       0.00      0.000      0.000
## 1st Qu.      1.000      1.000      1.000       1.00      1.000      1.000
## Median       2.000      2.000      2.000       2.00      2.000      2.000
## Mean         1.905      2.046      2.313       2.42      2.663      2.942
## 3rd Qu.      3.000      3.000      3.000       3.00      4.000      4.000
## Max.        12.000     11.000     25.000      26.00     27.000     20.000
##         2017-11-09 2017-11-10
## Min.         0.000      0.000
## 1st Qu.      1.000      1.000
## Median       2.000      3.000
## Mean         3.144      3.606
## 3rd Qu.      4.000      5.000
## Max.        25.000     27.000
apply(proj, 1, function(x) mean(x>0))
## 2017-10-28 2017-10-29 2017-10-30 2017-10-31 2017-11-01 2017-11-02 
##      0.742      0.742      0.741      0.738      0.781      0.771 
## 2017-11-03 2017-11-04 2017-11-05 2017-11-06 2017-11-07 2017-11-08 
##      0.808      0.808      0.809      0.816      0.824      0.850 
## 2017-11-09 2017-11-10 
##      0.859      0.877

According to these results, what are the chances that more cases will appear in the near future?Is this outbreak being brought under control? Would you recommend scaling up / down the response?

Follow-up…

For a follow-up on this outbreak, have a look at the second part of this simulated response, which includes a data update, genetic sequences, and the use of outbreak reconstruction tools.

About this document

Contributors

• Thibaut Jombart: initial version

Contributions are welcome via pull requests. The source file is hosted on github.

Legal stuff

License: CC-BY Copyright: Thibaut Jombart, 2017