d_reproductiondistn {modelSSE}

The distribution of individual reproduction number

Description

This function (i.e., d_reproductiondistn()) is the probability density function (PDF) of individual reproduction number that was modelled as a shifted gamma distribution.

Usage

d_reproductiondistn(
  x = 1,
  epi.para = list(mean = 1, disp = 0.5, shift = 0.2),
  offspring.type = "D",
  is.log = FALSE
)

Arguments

x

A scalar, or a vector of non-negative integer.

epi.para

A list (list) of pre-defined epidemiological parameters for offspring distribution, in the format of list(mean = ?, disp = ?, shift = ?), where the three parameters accept non-negative values. Each parameter can be either a scalar, or a vector. For the parameters being assigned with values of vectors, the vectors should be of the same length. For Delaporte distribution, the value of mean should be larger than the value of shift.

offspring.type

A character label (character) indicating the type of distribution used to describe the offspring distribution. It only accepts one of the following values:

• "D" indicates the Delaporte distribution for offspring cases, where reproduction number follows a shifted gamma distribution;

• "NB" indicates the negative binomial distribution for offspring cases, where reproduction number follows a (non-shifted, or standard) gamma distribution;

• "G" indicates the geometric distribution for offspring cases, where reproduction number follows an exponential distribution; or

• "P" indicates the Poisson distribution for offspring cases, where reproduction number follows a Dirac delta distribution.

By default, offspring.type = 'D'.

is.log

A logical variable, under which probability would be taken natural logarithm, if is.log = TRUE. By default, is.log = FALSE.

Value

d_reproductiondistn() is the probability density function (PDF), and it returns value of probability density (non-negative value).

Note

Only the PDF of individual reproduction number (i.e., d_reproductiondistn()) was created here (without cumulative distribution, quantile, or random variable generating functions). The function d_reproductiondistn() was used mainly for data visualization purpose (rather than using for analysis), because the distribution of individual reproduction number was not explicitly used in model fitting to the disease contact tracing data.

When offspring.type = "P", individual reproduction number follows a Dirac delta distribution, which is difficult to return any value, or visualize the PDF, because of the nature of this pulse function.

References

Lloyd-Smith JO, Schreiber SJ, Kopp PE, Getz WM. Superspreading and the effect of individual variation on disease emergence. Nature. 2005;438(7066):355-359. doi:10.1038/nature04153

Zhao S, Chong MK, Ryu S, Guo Z, He M, Chen B, Musa SS, Wang J, Wu Y, He D, Wang MH. Characterizing superspreading potential of infectious disease: Decomposition of individual transmissibility. PLoS Computational Biology. 2022;18(6):e1010281. doi:10.1371/journal.pcbi.1010281

Examples

## an example to visualize individual reproduction number is as follows.
plot(seq(0.01,9.99, length.out = 1001), d_reproductiondistn(
  x = seq(0,10, length.out = 1001),
  epi.para = list(mean = 2, disp = 1.5, shift = 0.5),
  offspring.type = "D",
  is.log = FALSE
), type = 'l', xlab = 'individual reproduction number', ylab = 'density')