seir.auxiliary: Auxiliary Functions for the SEIR Model without Forcing

In davidearn/fastbeta: Fast Approximation of Time-Varying Infectious Disease Transmission Rates

Auxiliary Functions for the SEIR Model without Forcing

Description

Calculate the basic reproduction number, endemic equilibrium, and Jacobian matrix of the SEIR model without forcing.

Usage

seir.R0      (beta, nu = 0, mu = 0, sigma = 1, gamma = 1, delta = 0,
              m = 1L, n = 1L, N = 1)
seir.ee      (beta, nu = 0, mu = 0, sigma = 1, gamma = 1, delta = 0,
              m = 1L, n = 1L, N = 1)
seir.jacobian(beta, nu = 0, mu = 0, sigma = 1, gamma = 1, delta = 0,
              m = 1L, n = 1L)

Arguments

beta, nu, mu, sigma, gamma, delta	non-negative numbers. beta, nu, and mu are the rates of transmission, birth, and natural death. m*sigma, n*gamma, and delta are the rates of removal from each latent, infectious, and recovered compartment.
m	a non-negative integer indicating a number of latent stages.
n	a positive integer indicating a number of infectious stages.
N	a non-negative number indicating a population size for the (nu == 0 && mu == 0) case.

Details

If \mu, \nu = 0, then the basic reproduction number is computed as

\mathcal{R}_{0} = N \beta / \gamma

and the endemic equilibrium is computed as

\begin{bmatrix} S^{\hphantom{1}} \\ E^{i} \\ I^{j} \\ R^{\hphantom{1}} \end{bmatrix} = \begin{bmatrix} \gamma / \beta \\ w \delta / (m \sigma) \\ w \delta / (n \gamma) \\ w \end{bmatrix}

where w is chosen so that the sum is N.

If \mu, \nu > 0, then the basic reproduction number is computed as

\mathcal{R}_{0} = \nu \beta a^{-m} (1 - b^{-n}) / \mu^{2}

and the endemic equilibrium is computed as

\begin{bmatrix} S^{\hphantom{1}} \\ E^{i} \\ I^{j} \\ R^{\hphantom{1}} \end{bmatrix} = \begin{bmatrix} \mu a^{m} / (\beta (1 - b^{-n})) \\ w a^{m - i} b^{n} (\delta + \mu) / (m \sigma) \\ w b^{n - j} (\delta + \mu) / (n \gamma) \\ w \end{bmatrix}

where w is chosen so that the sum is \nu / \mu, the population size at equilibrium, and a = 1 + \mu / (m \sigma) and b = 1 + \mu / (n \gamma).

Currently, none of the functions documented here are vectorized. Arguments must have length 1.

Value

seir.R0 returns a numeric vector of length 1. seir.ee returns a numeric vector of length 1+m+n+1. seir.jacobian returns a function of one argument x (which must be a numeric vector of length 1+m+n+1) whose return value is a square numeric matrix of size length(x).

See Also

seir, for the system of ordinary differential equations on which these computations are predicated.

davidearn/fastbeta documentation built on July 4, 2025, 6:28 p.m.