`epidemic_threshold.Rd`

Weighted and unweighted Epidemic Threshold \(q\) of a graph.

epidemic_threshold(x, beta = 1)

x | an |
---|---|

beta | numeric, between 0 and 1. Probability of transmission. |

a list the weighted and unweighted Potential for Transmission \(R_0\) and its inverse, the Epidemic Threshold \(q\). As an attribute named "sna", a data.frame with the in/out-degrees of each node and their individual contribution to R0.

If the input graph is unweighted, the weighted component is NULL.

The Epidemic Threshold \(q\) quantifies the minimal expeced
transmission coefficient necessary for diffusing an epidemy in a
network. It is computed as the inverse of the *Potential for
transmission* of the network: a measure of the expected number of
nodes affected by an infectious node, which is a generalisation of
the Basic Reproduction Number \(R_0\) of an epidemy to the
context of a network. It thus quantifies the potential for
transmission of an infection throughout the contact network. It is
computed in terms of the incoming-outgoing rates from the network's
nodes: $$R_0 = \beta \frac{\overline{k_\mathrm{in}\cdot
k_\mathrm{out}}}{\overline{k_\mathrm{in}}},$$
where \(\beta\) is the transmission coefficient among animals,
\(k_\mathrm{in/out}\) are the in/out-degrees of a node
and the \(\overline{\cdot}\) symbol represents the average value
across all nodes in the graph.

The unweighted value computed above is most appropriate for a highly infectious epidemy with high animal-prevalence on nodes, as it assumes that any contact is potentially infectious.

In the weighted formulation, \(k_\mathrm{in/out}\) are the weight values for the incoming/outgoing edges in each node. It is more appropriate for low-prevalence diseases, where the transmission probability is assumed proportional to the number of contacts.

The default value of 1 for the probability of transmission \(\beta\) implies that every infectious contact leads to transmission.

Volkova VV, Howey R, Savill NJ, Woolhouse MEJ (2010) Sheep Movement Networks and the Transmission of Infectious Diseases. PLoS ONE 5(6): e11185. https://doi.org/10.1371/journal.pone.0011185

#> $unweighted #> R0 q #> 0.5 2.0 #> attr(,"sna") #> node indeg outdeg R0k #> 1 A 0 2 0.0 #> 2 B 1 2 0.5 #> 3 C 2 0 0.0 #> 4 D 1 0 0.0 #> #> $weighted #> NULL #>#> $unweighted #> R0 q #> 0.5 2.0 #> attr(,"sna") #> node indeg outdeg R0k #> 1 A 0 2 0.0 #> 2 B 1 2 0.5 #> 3 C 2 0 0.0 #> 4 D 1 0 0.0 #> #> $weighted #> R0 q #> 3.8888889 0.2571429 #> attr(,"sna") #> node in_w out_w R0k_w #> 1 A 0 11 0.000000 #> 2 B 10 7 3.888889 #> 3 C 3 0 0.000000 #> 4 D 5 0 0.000000 #>