seir.auxiliary {fastbeta}R Documentation

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.


[Package fastbeta version 0.4.0 Index]