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. |
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
|
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.