NB: This vignette is work-in-progress and not yet complete.

This vignette describes how births, deaths and movements can be
incorporated into a model as scheduled events at predefined time-points.
Events can, for example, be used to simulate disese spread among
multiple subpopulations (e.g., farms) when individuals can move between
the subpopulations and thus transfer infection, see Figure 1. In SimInf,
we use `node`

to denote a subpopulation.

\(~\)

Let us define the **6** movement events in Figure 1 to
include them in an SIR model. Below is a `data.frame`

, that
contains the movements. Interpret it as follows:

- In time step
**1**we move**9**individuals from node**3**to node**2** - In time step
**1**we move**2**individuals from node**3**to node**4** - In time step
**2**we move**8**individuals from node**1**to node**3** - In time step
**2**we move**3**individuals from node**4**to node**3** - In time step
**3**we move**5**individuals from node**3**to node**2** - In time step
**3**we move**4**individuals from node**4**to node**2**

```
events <- data.frame(
event = rep("extTrans", 6), ## Event "extTrans" is
## a movement between nodes
time = c(1, 1, 2, 2, 3, 3), ## The time that the event happens
node = c(3, 3, 1, 4, 3, 4), ## In which node does the event occur
dest = c(4, 2, 3, 3, 2, 2), ## Which node is the destination node
n = c(9, 2, 8, 3, 5, 4), ## How many individuals are moved
proportion = c(0, 0, 0, 0, 0, 0), ## This is not used when n > 0
select = c(4, 4, 4, 4, 4, 4), ## Use the 4th column in
## the model select matrix
shift = c(0, 0, 0, 0, 0, 0)) ## Not used in this example
```

and have a look at the `data.frame`

```
## event time node dest n proportion select shift
## 1 extTrans 1 3 4 9 0 4 0
## 2 extTrans 1 3 2 2 0 4 0
## 3 extTrans 2 1 3 8 0 4 0
## 4 extTrans 2 4 3 3 0 4 0
## 5 extTrans 3 3 2 5 0 4 0
## 6 extTrans 3 4 2 4 0 4 0
```

Now, create an SIR model where we turn off the disease dynamics (beta=0, gamma=0) to focus on the scheduled events. Let us start with different number of individuals in each node.

```
library(SimInf)
model <- SIR(u0 = data.frame(S = c(10, 15, 20, 25),
I = c(5, 0, 0, 0),
R = c(0, 0, 0, 0)),
tspan = 0:3,
beta = 0,
gamma = 0,
events = events)
```

The compartments that an event operates on, is controlled by the
select value specified for each event together with the model select
matrix (E). Each row in E corresponds to one compartment in the model,
and the non-zero entries in a column indicate which compartments to
sample individuals from when processing an event. Which column to use in
E for an event is determined by the event select value. In this example,
we use the 4^{th} column which means that all compartments can
be sampled in each movement event (see below).

```
## 3 x 4 sparse Matrix of class "dgCMatrix"
## 1 2 3 4
## S 1 . . 1
## I . 1 . 1
## R . . 1 1
```

In another case you might be interested in only targeting the
susceptibles, which means for this model that we select the first
column. Now, let us run the model and generate data from it. For
reproducibility, we first call the `set.seed()`

function and
specify the number of threads to use since there is random sampling
involved when picking inviduals from the compartments.

And plot (Figure 2) the number of individuals in each node.

\(~\)

Or use the `trajectory()`

function to more easily inspect
the outcome in each node in detail.

```
## node time S I R
## 1 1 0 10 5 0
## 2 2 0 15 0 0
## 3 3 0 20 0 0
## 4 4 0 25 0 0
## 5 1 1 10 5 0
## 6 2 1 17 0 0
## 7 3 1 9 0 0
## 8 4 1 34 0 0
## 9 1 2 6 1 0
## 10 2 2 17 0 0
## 11 3 2 16 4 0
## 12 4 2 31 0 0
## 13 1 3 6 1 0
## 14 2 3 25 1 0
## 15 3 3 12 3 0
## 16 4 3 27 0 0
```

It is possible to assign different probabillities for the compartments that an event sample individuals from. If the weights in the select matrix \(E\) are non-identical, individuals are sampled from a biased urn. To illustrate this, let us create movement events between two nodes for the built-in SIR model, where we start with 300 individuals (\(S = 100\), \(I = 100\), \(R = 100\)) in the first node and then move them, one by one, to the second node.

```
events <- data.frame(
event = rep("extTrans", 300), ## Event "extTrans" is a movement between nodes
time = 1:300, ## The time that the event happens
node = 1, ## In which node does the event occur
dest = 2, ## Which node is the destination node
n = 1, ## How many individuals are moved
proportion = 0, ## This is not used when n > 0
select = 4, ## Use the 4th column in the model select matrix
shift = 0) ## Not used in this example
```

Now, create the model. Then run it, and plot the number of individuals in the second node.

\(~\)

The probability to sample an individual from each compartment is

\[ p_S = \frac{w_S * S}{w_S * S + w_I * I + w_R * R} \]

\[ p_I = \frac{w_I * I}{w_S * S + w_I * I + w_R * R} \]

\[ p_R = \frac{w_R * R}{w_S * S + w_I * I + w_R * R} \]

Where \(w_S\), \(w_I\) and \(w_R\) are the weights in E. These probabilities are applied sequentially, that is the probability of choosing the next item is proportional to the weights amongst the remaining items. Let us now double the weight to sample individuals from the \(I\) compartment and then run the model again.

\(~\)

And a much larger weight to sample individuals from the \(I\) compartment.

\(~\)

Increase the weight for the \(R\) compartment and run the model again.