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Getting started with epichains

James M. Azam and Sebastian Funk

Source: vignettes/epichains.Rmd
epichains.Rmd

This vignette demonstrates how get started with using the epichains package for simulating transmission chains or estimating the likelihood of observing a transmission chain.

Infectious disease epidemics spread through populations when a chain of infected individuals transmit the infection to others. Branching processes can be used to model this process. A branching process is a stochastic process where each infectious individual in a generation gives rise to a random number of individuals in the next generation, starting with the index case in generation 1. The distribution of the number of offspring is called the offspring distribution.

The size of the transmission chain is the total number of individuals infected by a single case, and the length of the transmission chain is the number of generations from the first case to the last case they produced before the chain ended. The size calculation includes the first case and the length calculation includes the first generation when the first case starts the chain (See figure below).

The figure shows an example of a transmission chain starting with a single case C1. The complete chain is depicted as blue circles linked by orange directed arrows showing the cases produced in each generation. Each generation is marked by a brown rectangle and labelled Gen 1, Gen 2, and Gen 3. The chain grows through generations Gen 1, Gen 2, and Gen 3. Case C1 starts in Gen 1 and produces cases C2, and C3 in Gen 2. In Gen 3, C2 produces C4 and C5, and C3 produces C6. The chain ends in Gen 3. The chain size of C1 is 6, which includes C1 (that is the sum of all the blue circles, representing cases) and the length is 3, which includes Gen 1 (maximum number of generations reached by C1's chain).

Figure 1: An example of a transmission chain starting with a single case C1. Cases are represented by blue circles and arrows indicate who infected whom. The chain grows through generations Gen 1, Gen 2, and Gen 3, producing cases C2, C3, C4, C5, and C6. The chain ends at generation Gen 3 with cases C4, C5, and C6. The size of C1’s chain is 6, including C1 (that is, the sum of all blue circles) and the length is 3, which includes Gen 1 (maximum number of generations reached by C1’s chain).

epichains provides methods to analyse and simulate the size and length of branching processes with an arbitrary offspring distribution. These can be used, for example, to analyse the distribution of chain sizes or length of infectious disease outbreaks, as discussed in Farrington, Kanaan, and Gay (2003) and Blumberg and Lloyd-Smith (2013).

Code

Transmission chains likelihoods

Use case

Suppose we have observed a number of transmission chains, each arising from a separate index case. What are the likely transmission properties (reproduction number and/or superspreading coefficient) that generated these chains (assuming these parameters are the same across all the chains)?

The first step in answering this question is to calculate the likelihood of observing the observed chain summaries given the transmission properties. This is where the likelihood() function comes in. The returned estimate can then be used to infer the transmission properties using estimation frameworks such as maximum likelihood or Bayesian inference.

epichains does not provide these estimation frameworks.

What we have

  1. Data on observed transmission chains summaries (sizes or lengths).

What we assume

  1. An offspring distribution that governs the number of secondary cases produced by each case.
  2. Parameters of the offspring distribution (for example, mean and overdispersion if using a negative binomial; which can then be interpreted as reproduction number and superspreading coefficient, respectively)
  3. (optional) An observation process/probability that generates the observed chain summaries.
  4. (optional) A threshold of chain summaries that are considered too large to be consistent with a reproduction number of \(R<1\) (e.g., \(N\) cases or \(T\) generations in total).

likelihood()

This function calculates the likelihood/loglikelihood of observing a vector of outbreak summaries obtained from transmission chains. “Outbreak summaries” here refer to transmission chain sizes or lengths/durations.

likelihood() requires a vector of chain summaries (sizes or lengths), chains, the corresponding statistic to use, statistic, the offspring distribution, offspring_dist and its associated parameters.

offspring_dist is specified by the R function that is used to generate random numbers, i.e. rpois for Poisson, rnbinom for negative binomial, or a custom function.

By default, the result is a log-likelihood but if log = FALSE, then likelihoods are returned (See ?likelihood for more details).

To understand how likelihood() works see the section on How likelihood() works.

Let’s look at the following example where we estimate the log-likelihood of observing chain_sizes.

Code
set.seed(121)
# example of observed chain sizes
# randomly generate 20 chains of size between 1 to 10
chain_sizes <- sample(1:10, 20, replace = TRUE)
chain_sizes
#>  [1]  4  7  4  9  1  3  7  5  1  8 10  2  1  2  4  5 10  6  8  7
Code
# estimate loglikelihood of the observed chain sizes for given lambda
likelihood_eg <- likelihood(
  chains = chain_sizes,
  statistic = "size",
  offspring_dist = rpois,
  nsim_obs = 100,
  lambda = 0.5
)
# Print the estimate
likelihood_eg
#> [1] -67.82879

Joint and individual log-likelihoods

likelihood(), by default, returns the joint log-likelihood, given by the sum of log-likelihoods of each observed chain. If instead the individual log-likelihoods are desired (for example for calculating Watanabe–Akaike information criterion values, then the individual argument must be set to TRUE. To return likelihoods instead, set log = FALSE.

Code
set.seed(121)
# example of observed chain sizes
# randomly generate 20 chains of size between 1 to 10
chain_sizes <- sample(1:10, 20, replace = TRUE)
chain_sizes
#>  [1]  4  7  4  9  1  3  7  5  1  8 10  2  1  2  4  5 10  6  8  7
Code

# estimate loglikelihood of the observed chain sizes
likelihood_ind_eg <- likelihood(
  chains = chain_sizes,
  statistic = "size",
  offspring_dist = rpois,
  nsim_obs = 100,
  lambda = 0.5,
  individual = TRUE
)
# Print the estimate
likelihood_ind_eg
#> [[1]]
#>  [1] -3.098612 -4.508584 -3.098612 -5.269208 -0.500000 -2.480829 -4.508584
#>  [8] -3.622329 -0.500000 -4.900542 -5.619471 -1.693147 -0.500000 -1.693147
#> [15] -3.098612 -3.622329 -5.619471 -4.086190 -4.900542 -4.508584

Observation probability

likelihood() uses the argument obs_prob to model the observation probability.

By default, it assumes perfect observation, where obs_prob = 1 (See ?likelihood), meaning that all transmission events are observed and recorded in the data.

If observations are imperfect, the obs_prob must be less than 1. In the case of imperfect observation, “true” chain sizes or lengths are simulated nsim_obs times, and the likelihood calculated for each of the simulations.

For example, if the probability of observing each case is obs_prob = 0.30, we use

Code
set.seed(121)
# example of observed chain sizes; randomly generate 20 chains of size 1 to 10
chain_sizes <- sample(1:10, 20, replace = TRUE)
# get their likelihood
liks <- likelihood(
  chains = chain_sizes,
  statistic = "size",
  offspring_dist = rpois,
  obs_prob = 0.3,
  lambda = 0.5,
  nsim_obs = 10
)
liks
#>  [1] -183.8670 -162.1382 -166.6720 -170.1077 -152.9557 -157.9814 -166.0587
#>  [8] -174.8488 -156.4197 -160.0229

This returns 10 likelihood values (because nsim_obs = 10), which can be averaged to come up with an overall likelihood estimate.

How likelihood() works

likelihood() first checks if an analytical solution of the likelihood exists for the given offspring distribution and chain statistic specified. If there’s none, simulations are used to estimate the likelihoods.

The epichains package includes closed-form (analytical) solutions for calculating the likelihoods associated with certain summaries of transmission chains (“size” or “length”) for specific offspring distributions.

For the size distributions, the package provides the Poisson, negative binomial, and gamma-borel mixture.

To provide the gamma-Borel size likelihood, the Borel distribution is needed. However, base R does not provide this distribution natively like poisson and gamma, so epichains provides its density and random generator.

For the length distribution, there’s the Poisson and geometric distributions. These can be used with likelihood() based on what is specified for offspring_dist and statistic.

If an analytical solution does not exist, an internal simulation function, .offspring_ll() is employed. It uses simulations to approximate the probability distributions (using a linear approximation to the cumulative distribution for unobserved sizes/lengths). If simulation is to be used, an extra argument nsim_offspring must be passed to likelihood() to specify the number of simulations to be used for this approximation. Approximate values of the likelihood will vary with every call to the simulation (because the simulations used for estimation vary), and it may be worth calling likelihood() multiple times in this case to see the error this may introduce.

For example, let’s look at an example where chain_sizes is observed and we want to calculate the likelihood of this being drawn from a binomial distribution with probability prob = 0.9.

Code
set.seed(121)
# example of observed chain sizes; randomly generate 20 chains of size 1 to 10
chain_sizes <- sample(1:10, 20, replace = TRUE)
# get their likelihood
liks <- likelihood(
  chains = chain_sizes,
  offspring_dist = rbinom,
  statistic = "size",
  size = 1,
  prob = 0.9,
  nsim_offspring = 250
)
liks
#> [1] -56.76805

Transmission chain simulation

Use case

We want to simulate a scenario where a number of chains are each produced by a separate index case. We want to simulate the chains of transmission that result from these cases, given a specific offspring distribution and a reproduction number.

What we have

  1. An initial number of cases that seed the outbreak.

What we assume

  1. An offspring distribution that governs the number of secondary cases produced by each case.
  2. Parameters of the offspring distribution (for example, mean and overdispersion if using a negative binomial; which can then be interpreted as reproduction number and superspreading coefficient, respectively)
  3. (optional) A threshold of chain summaries that are considered too large to be consistent with a reproduction number of \(R<1\) (e.g., \(N\) cases or \(T\) generations in total).
  4. (optional) A generation time distribution that governs the time between successive cases.

There are two simulation functions, herein referred to collectively as the simulate_*() functions that can help us achieve this.

simulate_chains()

simulate_chains() simulates an outbreak from a given number of infections and an offspring distribution. By default, it assumes the population is infinite with no pre-existing immunity.

The function tracks and returns information on infectors (ancestors), infectees, the generation of infection, and the time, if a generation time function is specified.

Let’s look at an example where we simulate a transmission tree for \(10\) index cases. We assume a poisson offspring distribution with mean, \(\text{lambda} = 0.9\), and a generation time of \(3\) days:

Code
set.seed(32)
# Define generation time
generation_time_fn <- function(n) {
  gt <- rep(3, n)
  return(gt)
}

sim_chains <- simulate_chains(
  n_chains = 10,
  statistic = "size",
  offspring_dist = rpois,
  stat_threshold = 10,
  generation_time = generation_time_fn,
  lambda = 0.9
)

head(sim_chains)
#>    chain infector infectee generation time
#> 11     1        1        2          2    3
#> 12     2        1        2          2    3
#> 13     3        1        2          2    3
#> 14     3        1        3          2    3
#> 15     4        1        2          2    3
#> 16     6        1        2          2    3

By default, simulate_chains() assumes an infinite population but can account for susceptible depletion when a finite pop is specified.

Pre-existing immunity levels can also be specified, which will be applied to pop before the simulation is initialised.

Here is a quick example where we simulate an outbreak in a population of size \(1000\) with \(10\) index cases and \(10/%\) pre-existing immunity. We assume individuals have a poisson offspring distribution with mean, \(\text{lambda} = 1\), and fixed generation time of \(3\):

Code
set.seed(32)
# Define generation time
generation_time_fn <- function(n) {
  gt <- rep(3, n)
  return(gt)
}

sim_chains_with_pop <- simulate_chains(
  pop = 1000,
  n_chains = 10,
  percent_immune = 0.1,
  offspring_dist = rpois,
  statistic = "size",
  lambda = 1,
  generation_time = generation_time_fn
)

head(sim_chains_with_pop)
#>    chain infector infectee generation time
#> 11     1        1        2          2    3
#> 12     2        1        2          2    3
#> 13     3        1        2          2    3
#> 14     3        1        3          2    3
#> 15     4        1        2          2    3
#> 16     6        1        2          2    3

simulate_chain_stats()

simulate_chain_stats() is a performant version of simulate_chains() that does not track information on each infector and infectee. It returns the eventual size or length/duration of each transmission chain. This function is especially useful for calculating numerical likelihoods in likelihood().

Here is an example to simulate the previous examples without intervention, returning the size of each of the \(10\) chains. It assumes a Poisson offspring distribution distribution with mean of \(0.9\).

Code
set.seed(123)

simulate_chain_stats_eg <- simulate_chain_stats(
  n_chains = 10,
  statistic = "size",
  offspring_dist = rpois,
  stat_threshold = 10,
  lambda = 0.9
)

# Print the results
simulate_chain_stats_eg
#> `epichains_summary` object 
#> 
#>  [1]   1 Inf   4   4 Inf   1   2 Inf   5   3
#> 
#>  Simulated sizes: 
#> 
#> Max: >=10
#> Min: 1

S3 Methods

Summarising

You can run summary() on the object returned by simulate_chains() to obtain the chain summaries per index case.

Code
set.seed(32)
# Define generation time
generation_time_fn <- function(n) {
  gt <- rep(3, n)
  return(gt)
}

sim_chains <- simulate_chains(
  n_chains = 10,
  statistic = "size",
  offspring_dist = rpois,
  stat_threshold = 10,
  generation_time = generation_time_fn,
  lambda = 0.9
)

summary(sim_chains)
#> `epichains_summary` object 
#> 
#>  [1] Inf   2 Inf   2   1 Inf   3 Inf   9   1
#> 
#>  Simulated sizes: 
#> 
#> Max: >=10
#> Min: 1
Code

# Example with simulate_chain_stats()
set.seed(32)

simulate_chain_stats_eg <- simulate_chain_stats(
  n_chains = 10,
  statistic = "size",
  offspring_dist = rpois,
  stat_threshold = 10,
  lambda = 0.9
)

# Get summaries
summary(simulate_chain_stats_eg)
#> $n_chains
#> [1] 10
#> 
#> $max_stat
#> [1] Inf
#> 
#> $min_stat
#> [1] 1

This summary is the same as the output of simulate_chain_stats() if the same inputs are used. simulate_chain_stats() is a more performant version of simulate_chains(), hence, you can use it instead, if you are only interested in the summary of the simulated chains without details about the infection tree. :::

We can confirm if the two outputs are the same using base::setequal(), which checks if two objects are equal and returns TRUE/FALSE.

Code
setequal(
  # summary of output from simulate_chains()
  summary(sim_chains),
  # results from simulate_chain_stats()
  simulate_chain_stats_eg
)
#> [1] TRUE

Aggregating

You can aggregate <epichains_tree> objects returned by simulate_chains() a <data.frame> with columns “cases” and either “generation” or “time”, depending on the value of by.

To aggregate over “time”, you must have specified a generation time distribution (generation_time) in the simulation step.

Code
# Example with simulate_chains()
set.seed(32)

# Define generation time
generation_time_fn <- function(n) {
  gt <- rep(3, n)
  return(gt)
}

sim_chains <- simulate_chains(
  n_chains = 10,
  statistic = "size",
  offspring_dist = rpois,
  stat_threshold = 10,
  generation_time = generation_time_fn,
  lambda = 0.9
)

aggregate(sim_chains, by = "time")
#>   time cases
#> 1    0    10
#> 2    3    12
#> 3    6    14
#> 4    9    17
#> 5   12     3
#> 6   15     4
#> 7   18     3

Plotting

Aggregated <epichains_tree> objects can easily be plotted using base R or ggplot2 with little to no data manipulation.

Here is an end-to-end example from simulation through aggregation to plotting.

Code
# Run simulation with simulate_chains()
set.seed(32)
# Define generation time
generation_time_fn <- function(n) {
  gt <- rep(3, n)
  return(gt)
}

sim_chains <- simulate_chains(
  n_chains = 10,
  statistic = "size",
  offspring_dist = rpois,
  stat_threshold = 1000,
  generation_time = generation_time_fn,
  lambda = 2
)

# Aggregate cases over time
sim_chains_aggreg <- aggregate(sim_chains, by = "time")

# Plot cases over time
plot(
  sim_chains_aggreg,
  type = "b",
  col = "red",
  lwd = 2,
  xlab = "Time",
  ylab = "Cases"
)

References

Blumberg, S., and J. O. Lloyd-Smith. 2013. “Comparing Methods for Estimating R0 from the Size Distribution of Subcritical Transmission Chains.” Epidemics 5 (3): 131–45. https://doi.org/10.1016/j.epidem.2013.05.002.
Farrington, C. P., M. N. Kanaan, and N. J. Gay. 2003. “Branching Process Models for Surveillance of Infectious Diseases Controlled by Mass Vaccination.” Biostatistics (Oxford, England) 4 (2): 279–95. https://doi.org/10.1093/biostatistics/4.2.279.