Author: James M. Azam and Sebastian Funk

# Getting started with epichains

#### James M. Azam and Sebastian Funk

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

- To understand the theoretical background of the branching processes methods used here, refer to the theory vignette `vignette("theoretical_background")`.
- To understand the software development design decisions and implementation details of the package, refer to the design vignette `vignette("design-principles")`.

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).

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).

    library("epichains")
    library("epicontacts")

## 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`.

    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
    # 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`.

    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

    # 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

    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`.

    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:

    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 \(3\):

    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 \(0.9\).

    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.

    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

    # 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`.

    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.

    # 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

We can plot individual chains in an `<epichains>` object using the {epicontacts} package.

To do this, we first need to create an `<epicontacts>` object using the `epicontacts::make_epicontacts()` function, which requires a linelist and a contacts data.frame (See `?epicontacts::make_epicontacts()`).

### Some notes about interoperability between `<epicontacts>` and `<epichains>` objects

- For the `epicontacts::make_epicontacts()` function to work:

  - the `linelist` and `contacts` data.frames must have a column named “id” that uniquely identifies each case.
  - The `linelist` and `contacts` objects can be the same.

- An `<epichains>` object contains multiple independent chains with their own unique infectee ids, so this does not work out of the box with `make_epicontacts()` unless you use a subset chain.

- For now, we will have to select and visualise individual chains from the `<epichains>` object.

- In future versions of this package, we will make this interaction seamless by providing user-friendly functionality to convert an `<epichains>` object to an `<epicontacts>` in a way that allows us to plot all chains at once.

Let’s subset one of the chains with the biggest size from the `sim_chains` object created above.

    # Get the biggest chain
    longest_chain <- sim_chains[sim_chains$chain == which.max(
      unname(table(sim_chains$chain))
    ), ]

    # Convert to data.frame to view the whole data
    as.data.frame(longest_chain)
    #>    chain infector infectee generation time
    #> 6      6       NA        1          1    0
    #> 16     6        1        2          2    3
    #> 17     6        1        3          2    3
    #> 18     6        1        4          2    3
    #> 27     6        2        5          3    6
    #> 28     6        3        6          3    6
    #> 29     6        3        7          3    6
    #> 30     6        4        8          3    6
    #> 42     6        5        9          4    9
    #> 43     6        6       10          4    9
    #> 44     6        7       11          4    9
    #> 45     6        8       12          4    9

Now, we can plot the chain using the `{epicontacts}` package.

    # Create an `<epicontacts>` object
    epc <- make_epicontacts(
      linelist = longest_chain,
      contacts = longest_chain,
      id = "infectee",
      from = "infector",
      to = "infectee",
      directed = TRUE
    )

    # Plot the chain
    plot(epc)

Aggregated `<epichains_tree>` objects can also 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.

    # 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.