# Computes reproduction numbers from fitted models

### Description

The S3 generic function `R0`

defined in package surveillance is intended to
compute reproduction numbers from fitted epidemic models.
The package currently defines a method for the `"twinstim"`

class, which
computes expected numbers of infections caused by infected individuals depending on the event type
and marks attached to the individual, which contribute to the infection pressure
in the epidemic predictor of that class.
There is also a method for simulated `"epidataCS"`

(just a wrapper for the `"twinstim"`

-method).

### Usage

1 2 3 4 5 6 7 8 9 |

### Arguments

`object` |
A fitted epidemic model object for which an |

`newevents` |
an optional For the |

`trimmed` |
logical indicating if the individual reproduction numbers should be
calculated by integrating the epidemic intensities over the
observation period and region only ( |

`newcoef` |
the model parameters to use when calculating reproduction numbers.
The default ( |

`...` |
additional arguments passed to methods.
Currently unused for the |

`eta` |
a value for the epidemic linear predictor, see details. |

`eps.s,eps.t` |
the spatial/temporal radius of interaction.
If |

### Details

For the `"twinstim"`

class, the individual-specific expected
number *μ_j* of infections caused by individual (event) *j*
inside its theoretical (untrimmed) spatio-temporal range of interaction
given by its `eps.t`

(*ε*) and `eps.s`

(*δ*) values is defined as follows (cf. Meyer et al, 2012):

*μ_j = e^{η_j} \cdot
\int_{b(\bold{0},δ)} f(\bold{s}) d\bold{s} \cdot
\int_0^ε g(t) dt .*

Here, *b(\bold{0},δ)* denotes the disc centred at (0,0)' with
radius *δ*, *η_j* is the epidemic linear predictor,
*g(t)* is the temporal interaction function, and *f(\bold{s})*
is the spatial interaction function. For a type-specific
`twinstim`

, there is an additional factor for the number of event
types which can be infected by the type of event *j* and the
interaction functions may be type-specific as well.

Alternatively to the equation above,
the `trimmed`

(observed) reproduction numbers
are obtain by integrating over the observed infectious domains of the
individuals, i.e. integrate *f* over the intersection of the
influence region with the observation region `W`

(i.e. over *\{ W \cap b(\bold{s}_j,δ) \} - \bold{s}_j*)
and *g* over the intersection of the observed infectious period with
the observation period *(t_0;T]* (i.e. over
*(0; \min(T-t_j,ε)]*).

The function `simpleR0`

computes

*\exp(η) \cdot
\int_{b(\bold{0},δ)} f(\bold{s}) d\bold{s} \cdot
\int_0^{ε} g(t) dt ,*

where *η* defaults to *γ_0* disregarding any epidemic
effects of types and marks. It is thus only
suitable for simple epidemic `twinstim`

models with
`epidemic = ~1`

, a diagonal (or secondary diagonal) `qmatrix`

,
and type-invariant interaction functions.
`simpleR0`

mainly exists for use by `epitest`

.

(Numerical) Integration is performed exactly as during the fitting of
`object`

, for instance `object$control.siaf`

is queried if
necessary.

### Value

For the `R0`

methods,
a numeric vector of estimated reproduction numbers from the fitted
model `object`

corresponding to the rows of `newevents`

(if
supplied) or the original fitted events including events of the prehistory.

For `simpleR0`

, a single number (see details).

### Author(s)

Sebastian Meyer

### References

Meyer, S., Elias, J. and Höhle, M. (2012):
A space-time conditional intensity model for invasive meningococcal
disease occurrence. *Biometrics*, **68**, 607-616.
\Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.1111/j.1541-0420.2011.01684.x")}

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 | ```
## load the 'imdepi' data and a model fit
data("imdepi")
data("imdepifit")
## calculate individual and type-specific reproduction numbers
R0s <- R0(imdepifit)
tapply(R0s, imdepi$events@data[names(R0s), "type"], summary)
## untrimmed R0 for a specific event
R0(imdepifit, newevents=marks(imdepi)[1,], trimmed=FALSE)
### compute a Monte Carlo confidence interval
## use a simpler model with constant 'siaf' for speed
simplefit <- update(imdepifit, epidemic=~type, siaf=NULL, subset = NULL)
## we'd like to compute the mean R0's by event type
meanR0ByType <- function (newcoef) {
R0events <- R0(simplefit, newcoef=newcoef)
tapply(R0events, imdepi$events@data[names(R0events),"type"], mean)
}
(meansMLE <- meanR0ByType(newcoef=NULL))
## sample B times from asymptotic multivariate normal of the MLE
B <- 5 # CAVE: toy example! In practice this has to be much larger
set.seed(123)
parsamples <- MASS::mvrnorm(B, mu=coef(simplefit), Sigma=vcov(simplefit))
## for each sample compute the 'meanR0ByType'
meansMC <- apply(parsamples, 1, meanR0ByType)
## get the quantiles and print the result
cisMC <- apply(cbind(meansMLE, meansMC), 1, quantile, probs=c(0.025,0.975))
print(rbind(MLE=meansMLE, cisMC))
### R0 for a simple epidemic model (homogeneous individuals)
mepi1 <- update(simplefit, epidemic = ~1, data = imdepi,
subset = type == "B", model = TRUE, verbose = FALSE)
## using the default spatial and temporal ranges of interaction
(R0B <- simpleR0(mepi1)) # eps.s=200, eps.t=30
stopifnot(identical(R0B, R0(mepi1, trimmed = FALSE)[[1]]))
## assuming less interaction
simpleR0(mepi1, eps.s = 50, eps.t = 15)
``` |