Measures of network diffusion

These functions allow measurement of various features of a diffusion process at the network level:

net_transmissibility() measures the average transmissibility observed in a diffusion simulation, or the number of new infections over the number of susceptible nodes.
net_recovery() measures the average number of time steps nodes remain infected once they become infected.
net_reproduction() measures the observed reproductive number in a diffusion simulation as the network's transmissibility over the network's average infection length.
net_immunity() measures the proportion of nodes that would need to be protected through vaccination, isolation, or recovery for herd immunity to be reached.

net_transmissibility(.data)

net_recovery(.data, censor = TRUE)

net_reproduction(.data)

net_immunity(.data, normalized = TRUE)

Arguments

.data: Network data with nodal changes, as created by play_diffusion(), or a valid network diffusion model, as created by as_diffusion().
censor: Where some nodes have not yet recovered by the end of the simulation, right censored values can be replaced by the number of steps. By default TRUE. Note that this will likely still underestimate recovery.
normalized: Logical scalar, whether the centrality scores are normalized. Different denominators are used depending on whether the object is one-mode or two-mode, the type of centrality, and other arguments.

Transmissibility

net_transmissibility() measures how many directly susceptible nodes each infected node will infect in each time period, on average. That is: $$T = \frac{1}{n}\sum_{j=1}^n \frac{i_{j}}{s_{j}}$$ where $i$ is the number of new infections in each time period, $j \in n$, and $s$ is the number of nodes that could have been infected in that time period (note that $s \neq S$, or the number of nodes that are susceptible in the population). $T$ can be interpreted as the proportion of susceptible nodes that are infected at each time period.

Recovery time

net_recovery() measures the average number of time steps that nodes in a network remain infected. Note that in a diffusion model without recovery, average infection length will be infinite. This will also be the case where there is right censoring. The longer nodes remain infected, the longer they can infect others.

Reproduction number

net_reproduction() measures a given diffusion's reproductive number. Here it is calculated as: $$R = \min\left(\frac{T}{1/L}, \bar{k}\right)$$ where $T$ is the observed transmissibility in a diffusion and $L$ is the observed recovery length in a diffusion. Since $L$ can be infinite where there is no recovery or there is right censoring, and since network structure places an upper limit on how many nodes each node may further infect (their degree), this function returns the minimum of $R_0$ and the network's average degree.

Interpretation of the reproduction number is oriented around R = 1. Where $R > 1$, the 'disease' will 'infect' more and more nodes in the network. Where $R < 1$, the 'disease' will not sustain itself and eventually die out. Where $R = 1$, the 'disease' will continue as endemic, if conditions allow.

Herd immunity

net_immunity() estimates the proportion of a network that need to be protected from infection for herd immunity to be achieved. This is known as the Herd Immunity Threshold or HIT: $$1 - \frac{1}{R}$$ where $R$ is the reproduction number from net_reproduction(). The HIT indicates the threshold at which the reduction of susceptible members of the network means that infections will no longer keep increasing. Note that there may still be more infections after this threshold has been reached, but there should be fewer and fewer. These excess infections are called the overshoot. This function does not take into account the structure of the network, instead using the average degree.

Interpretation is quite straightforward. A HIT or immunity score of 0.75 would mean that 75% of the nodes in the network would need to be vaccinated or otherwise protected to achieve herd immunity. To identify how many nodes this would be, multiply this proportion with the number of nodes in the network.

References

On epidemiological models

Kermack, William O., and Anderson Gray McKendrick. 1927. "A contribution to the mathematical theory of epidemics". Proc. R. Soc. London A 115: 700-721. doi:10.1098/rspa.1927.0118

On the basic reproduction number

Diekmann, Odo, Hans J.A.P. Heesterbeek, and Hans J.A.J. Metz. 1990. "On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations". Journal of Mathematical Biology, 28(4): 365–82. doi:10.1007/BF00178324

Kenah, Eben, and James M. Robins. 2007. "Second look at the spread of epidemics on networks". Physical Review E, 76(3 Pt 2): 036113. doi:10.1103/PhysRevE.76.036113

On herd immunity

Garnett, G.P. 2005. "Role of herd immunity in determining the effect of vaccines against sexually transmitted disease". The Journal of Infectious Diseases, 191(1): S97-106. doi:10.1086/425271

Examples

  smeg <- generate_smallworld(15, 0.025)
  smeg_diff <- play_diffusion(smeg, recovery = 0.2)
  plot(smeg_diff)

  # To calculate the average transmissibility for a given diffusion model
  net_transmissibility(smeg_diff)
#> [1] 2.2
  # To calculate the average infection length for a given diffusion model
  net_recovery(smeg_diff)
#> [1] 5
  # To calculate the reproduction number for a given diffusion model
  net_reproduction(smeg_diff)
#> [1] 4
  # Calculating the proportion required to achieve herd immunity
  net_immunity(smeg_diff)
#> [1] 0.75
  # To find the number of nodes to be vaccinated
  net_immunity(smeg_diff, normalized = FALSE)
#> [1] 12