Weighted and unweighted Epidemic Threshold $$q$$ of a graph.

epidemic_threshold(x, beta = 1)

## Arguments

x an igraph object 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
#>