Network frailty and the geometry of herd immunity

Abstract

The spread of infectious disease through communities depends fundamentally on the underlying patterns of contacts between individuals. Generally, the more contacts one individual has, the more vulnerable they are to infection during an epidemic. Thus, outbreaks disproportionately impact the most highly connected demographics. Epidemics can then lead, through immunization or removal of individuals, to sparser networks that are more resistant to future transmission of a given disease. Using several classes of contact networks-Poisson, scale-free and small-world-we characterize the structural evolution of a network due to an epidemic in terms of frailty (the degree to which highly connected individuals are more vulnerable to infection) and interference (the extent to which the epidemic cuts off connectivity among the susceptible population that remains following an epidemic). The evolution of the susceptible network over the course of an epidemic differs among the classes of networks; frailty, relative to interference, accounts for an increasing component of network evolution on networks with greater variance in contacts. The result is that immunization due to prior epidemics can provide greater community protection than random vaccination on networks with heterogeneous contact patterns, while the reverse is true for highly structured populations.

Figure 1

Fraction of the immunized population that escaped infection as a function of the proportion immunized for small-world, Poisson and scale-free networks of 1000 nodes. Mean degree is scaled such that R₀=3.9 for the three classes of networks. Grey boxes indicate the central 50% simulation runs for populations immunized by a prior epidemic. White boxes indicate the same for randomly vaccinated populations. Note that boxes are offset for clarity.

Figure 2

Mean original degree and mean residual degree (scaled to 〈k〉=1) of the active epidemic network (susceptible and infectious nodes) for 100 simulated network epidemics and analytical predictions for (a) small-world, (b) Poisson and (c) scale-free networks of 1000 nodes. Each epidemic was simulated on a separate network with β=0.05. The dashed curve gives the mean original degree of nodes across all the networks and the solid curve gives the mean residual degree across time. Points indicate the simulated trajectories. The solid diamond indicates the predicted final mean original degree, 〈k〉_r, and the solid circle indicates the predicted mean residual degree, ${\langle k_{\text{r}}\rangle }_{\text{r}}$. The solid vertical bar indicates the predicted frailty and the dashed bar indicates the predicted interference.

Figure 3

(a) Mean original (scaled), 〈k〉_r, and (b) residual degree (scaled), ${\langle k_{\text{r}}\rangle }_{\text{r}}$, as a function of the epidemic size (proportion of nodes infected by the end of the epidemic) for the three types of networks.