Estimate what proportion of cases cause a certain proportion of transmission

Calculates the proportion of cases that cause a certain percentage of transmission.

It is commonly estimated what proportion of cases cause 80% of transmission (i.e. secondary cases). This can be calculated using proportion_transmission() at varying values of \(R\) and for different values of percentage transmission.

There are two methods for calculating the proportion of transmission, \(p_{80}\) (default) and \(t_{20}\), see method argument or details for more information.

Usage

proportion_transmission(
  R,
  k,
  prop_transmission,
  method = c("p_80", "t_20"),
  simulate = FALSE,
  ...,
  offspring_dist,
  format_prop = TRUE
)

Arguments

R: A number specifying the \(R\) parameter (i.e. average secondary cases per infectious individual).
k: A number specifying the \(k\) parameter (i.e. dispersion in offspring distribution from fitted negative binomial).
prop_transmission: A number of the proportion of transmission for which a proportion of cases has produced. Must be between 0 and 1 (exclusive, \((0, 1)\)).
method: A character string defining which method is used to calculate the proportion of transmission. Options are "p_80" (default) or "t_20". See details for more information on each of these methods.
simulate: A logical whether the calculation should be done numerically (i.e. simulate secondary contacts) or analytically. Default is FALSE which uses the analytical calculation.
...: dots not used, extra arguments supplied will cause a warning.
offspring_dist: An <epiparameter> object. An S3 class for working with epidemiological parameters/distributions, see epiparameter::epiparameter().
format_prop: A logical determining whether the proportion column of the <data.frame> returned by the function is formatted as a string with a percentage sign (%), (TRUE, default), or as a numeric (FALSE).

Value

A <data.frame> with the value for the proportion of cases for a given value of R and k.

Details

Calculates the expected proportion of transmission from a given proportion of infectious cases. There are two methods to calculate this with distinct formulations, \(p_{80}\) and \(t_{20}\) these can be specified by the method argument.

method = p_80 calculates relative transmission heterogeneity from the offspring distribution of secondary cases, \(Z\), where the upper proportion of the distribution comprise \(x\%\) of total number of cases given R0 and k, where \(x\) is typically defined as 0.8 or 80%. e.g. 80% of all transmissions are generated by the upper 20% of cases, or p_80 = 0.2, per the 80/20 rule. In this formulation, changes in R can have a significant effect on the estimate of \(p_{80}\) even when \(k\) is constant. Importantly, this formulation does not allow for true homogeneity when k = Inf i.e. \(p_{80} = 0.8\).

method = t_20 calculates a similar ratio, instead in terms of the theoretical individual reproductive number and infectiousness given \(R_0\) and \(k\). The individual reproductive number, \(\nu\), is described in Lloyd-Smith et al. (2005), "as a random variable representing the expected number of secondary cases caused by a particular infected individual. Values for \(\nu\) are drawn from a continuous gamma probability distribution with population mean \(R_0\) and dispersion parameter \(k\), which encodes all variation in infectious histories of individuals, including properties of the host and pathogen and environmental circumstances." The value of \(k\) corresponds to the shape parameters of the gamma distribution which encodes the variation in the gamma-Poisson mixture a.k.a. the negative binomial.

For method = t_20, we define the upper proportion of infectiousness, which is typically 0.2 i.e. the upper 20% most infectious cases, again per the 80/20 rule. e.g. the most infectious 20% of cases, are expected to produce 80% of all infections, or t_20 = 0.8. Unlike method = p_80, changes in R have no effect on the estimate of t_80 when k is constant, but R is still required for the underlying calculation. This formulation does allow for true homogeneity when k = Inf i.e. t_20 = 0.2, or t_80 = 0.8.

Multiple values of R and k can be supplied and a <data.frame> of every combination of these will be returned.

The numerical calculation for method = p_80 uses random number generation to simulate secondary contacts so the answers may minimally vary between calls. The number of simulation replicates is fixed to 10⁵.

References

The analytical calculation is from:

Endo, A., Abbott, S., Kucharski, A. J., & Funk, S. (2020) Estimating the overdispersion in COVID-19 transmission using outbreak sizes outside China. Wellcome Open Research, 5. doi:10.12688/wellcomeopenres.15842.3

The \(t_{20}\) method follows the formula defined in section 2.2.5 of the supplementary material for:

Lloyd-Smith, J. O., Schreiber, S. J., Kopp, P. E., & Getz, W. M. (2005) Superspreading and the effect of individual variation on disease emergence. Nature. Nov;438(7066):355–9. doi:10.1038/nature04153

The original code for the \(t_{20}\) method is from ongoing work originating from https://github.com/dcadam/kt and:

Adam, D., Gostic, K., Tsang, T., Wu, P., Lim, W. W., Yeung, A., Wong, J., Lau, E., Du, Z., Chen, D., Ho, L.-M., Martín-Sánchez, M., Cauchemez, S., Cobey, S., Leung, G., & Cowling, B. (2022) Time-varying transmission heterogeneity of SARS and COVID-19 in Hong Kong. doi:10.21203/rs.3.rs-1407962/v1

Examples

# example of single values of R and k
prop_transmission <- 0.8 # 80% of transmission
R <- 2
k <- 0.5
proportion_transmission(
  R = R,
  k = k,
  prop_transmission = prop_transmission
)
#>   R   k prop_80
#> 1 2 0.5   26.4%

# example with multiple values of k
k <- c(0.1, 0.2, 0.3, 0.4, 0.5, 1)
proportion_transmission(
  R = R,
  k = k,
  prop_transmission = prop_transmission
)
#>   R   k prop_80
#> 1 2 0.1   9.21%
#> 2 2 0.2   15.6%
#> 3 2 0.3   20.3%
#> 4 2 0.4   23.8%
#> 5 2 0.5   26.4%
#> 6 2 1.0   35.6%

# example with vectors of R and k
R <- c(1, 2, 3)
proportion_transmission(
  R = R,
  k = k,
  prop_transmission = prop_transmission
)
#>    R   k prop_80
#> 1  1 0.1   8.69%
#> 2  2 0.1   9.21%
#> 3  3 0.1   9.39%
#> 4  1 0.2   14.3%
#> 5  2 0.2   15.6%
#> 6  3 0.2     16%
#> 7  1 0.3   18.2%
#> 8  2 0.3   20.3%
#> 9  3 0.3     21%
#> 10 1 0.4   20.8%
#> 11 2 0.4   23.8%
#> 12 3 0.4     25%
#> 13 1 0.5   22.6%
#> 14 2 0.5   26.4%
#> 15 3 0.5     28%
#> 16 1 1.0     30%
#> 17 2 1.0   35.6%
#> 18 3 1.0   37.8%