Calculate serial interval mixture density assuming underlying gamma distribution

This function computes the weighted mixture density for serial intervals based on different transmission routes in an outbreak. It implements part of the Vink et al. (2014) method for serial interval estimation, assuming an underlying gamma distribution for the serial interval.

Usage

f_gam(x, w1, w2, w3, mu, sigma)

Arguments

x: quantile or vector of quantiles (time in days since index case symptom onset)
w1: probability weight of being a co-primary case
w2: probability weight of being a primary-secondary case
w3: probability weight of being a primary-tertiary case
mu: mean serial interval in days (must be positive)
sigma: standard deviation of serial interval in days (must be positive)

Value

Vector of weighted density values corresponding to input quantiles x. Returns the sum of densities from all four transmission routes.

Details

The function models four distinct transmission routes:

Co-primary (CP): Cases infected simultaneously from the same source
Primary-secondary (PS): Direct transmission from index case
Primary-tertiary (PT): Transmission through one intermediate case
Primary-quaternary (PQ): Transmission through two intermediate cases

Each route contributes to the overall serial interval distribution with different means and variances. The co-primary component uses a modified gamma distribution to account for simultaneous infections, while subsequent generations follow gamma distributions with progressively longer means and larger variances.

This function is primarily used internally by si_estim when dist = "gamma" is specified, and by plot_si_fit for visualizing fitted distributions.

The weights w1, w2, and w3 must sum to <= 1, with the remaining probability (1 - w1 - w2 - w3) assigned to primary-quaternary cases. The function converts the mean and standard deviation to gamma distribution shape (k) and scale (\theta) parameters using the method of moments: $$k = \mu^2 / \sigma^2$$ $$\theta = \sigma^2 / \mu$$

References

Vink, M. A., Bootsma, M. C. J., & Wallinga, J. (2014). Serial intervals of respiratory infectious diseases: A systematic review and analysis. American Journal of Epidemiology, 180(9), 865-875.

Examples

if (FALSE) { # \dontrun{
x <- seq(0.1, 30, by = 0.1)
density_values <- f_gam(x, w1 = 0.1, w2 = 0.6, w3 = 0.2, mu = 6.5, sigma = 2.8)
plot(x, density_values, type = "l")
} # }