EpiEstim (version 1.1-0)

DiscrSI: Discretized Generation Time Distribution Assuming A Shifted Gamma Distribution

Description

DiscrSI computes the discrete distribution of the serial interval, assuming that the serial interval is shifted Gamma distributed, with shift 1.

Usage

DiscrSI(k, mu, sigma)

Arguments

k

positive integer for which the discrete distribution is desired.

mu

a positive real giving the mean of the Gamma distribution.

sigma

a non-negative real giving the standard deviation of the Gamma distribution.

Value

• DiscrSI(k, mu, sigma) gives the discrete probability $w_k$ that the serial interval is equal to $k$.

encoding

UTF-8

Details

Assuming that the serial interval is shifted Gamma distributed with mean $\mu$, standard deviation $\sigma$ and shift $1$, the discrete probability $w_k$ that the serial interval is equal to $k$ is: $w_k = kF_{{\mu-1,\sigma}}(k)+(k-2)F_{{\mu-1,\sigma}}(k-2)-2(k-1)F_{{\mu-1,\sigma}}(k-1)\+(\mu-1)(2F_{{\mu-1+\frac{\sigma^2}{\mu-1},\sigma\sqrt{1+\frac{\sigma^2}{\mu-1}}}}(k-1)-F_{{\mu-1+\frac{\sigma^2}{\mu-1},\sigma\sqrt{1+\frac{\sigma^2}{\mu-1}}}}(k-2)-F_{{\mu-1+\frac{\sigma^2}{\mu-1},\sigma\sqrt{1+\frac{\sigma^2}{\mu-1}}}}(k))$ where $F_{{\mu,\sigma}}$ is the cumulative density function of a Gamma distribution with mean $\mu$ and standard deviation $\sigma$.

References

Cori, A. et al. A new framework and software to estimate time-varying reproduction numbers during epidemics. (submitted)

See Also

OverallInfectivity, EstimateR, WT

Examples

## Computing the discrete serial interval of influenza
MeanFluSI <- 2.6
SdFluSI <- 1.5
DicreteSIDistr <- vector()
for(i in 0:20)
{
    DicreteSIDistr[i+1] <- DiscrSI(i, MeanFluSI, SdFluSI)
}
plot(0:20, DicreteSIDistr, type="h", lwd=10, lend=1, xlab="time (days)", ylab="frequency")
title(main="Discrete distribution of the serial interval of influenza")