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

epidemic_threshold(x, beta = 1)

Arguments

x

an igraph object

beta

numeric, between 0 and 1. Probability of transmission.

Value

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.

Details

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.

References

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

Examples

g <- igraph::graph_from_literal(A --+ B --+C, A --+C, B --+D) epidemic_threshold(g)
#> $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 #> #> $weighted #> NULL #>
## weighted graph igraph::E(g)$weight <- c(10, 1, 2, 5) epidemic_threshold(g)
#> $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 #>