Epidemic modelling requires knowledge of the social network

TL;DR

The paper investigates how social-network structure affects epidemic forecasts when using compartmental models. It uses a simple ABM SIR across Erdős–Rényi and scale-free networks to show that degree heterogeneity and super-spreading events can dominate epidemic dynamics, biasing $R_0$ estimation and lowering the effective herd-immunity threshold below $1/R_0$. It also demonstrates that time-varying networks can induce multiple waves even with persistent population immunity, challenging random-mixing predictions. The work highlights the need to incorporate network topology into standard models and to collect network data to improve forecasts and guide interventions, especially those targeting super-spreading events.

Abstract

Compartmental models of epidemics are widely used to forecast the effects of communicable diseases such as COVID-19 and to guide policy. Although it has long been known that such processes take place on social networks, the assumption of random mixing is usually made, which ignores network structure. However, super-spreading events have been found to be power-law distributed, suggesting that the underlying networks may be scale free or at least highly heterogeneous. The random-mixing assumption would then produce an overestimation of the herd-immunity threshold for given $R_0$; and a (more significant) overestimation of $R_0$ itself. These two errors compound each other, and can lead to forecasts greatly overestimating the number of infections. Moreover, if networks are heterogeneous and change in time, multiple waves of infection can occur, which are not predicted by random mixing. A simple SIR model simulated on both Erdős-Rényi and scale-free networks shows that details of the network structure can be more important than the intrinsic transmissibility of a disease. It is therefore crucial to incorporate network information into standard models of epidemics.

Figures (4)

• Figure 1: Time series for the proportions of agents in the Infectious (panel A) and Recovered ( B) states, from the SIR model described in the main text, in three scenarios: Erdős-Rényi (ER) random graphs and probability of infection $\beta=0.48$ (dark blue circles); scale-free (SF) networks with exponent $\alpha=2.5$ and $\beta=0.48$ (light blue triangles); and SF networks with $\alpha=2.5$ and $\beta=0.12$ (red diamonds). Number of vertices $N=10^4$, mean degree $\langle k\rangle = 5$, averages over $100$ networks in each case, bars represent one standard deviation. At time $t=0$ all agents are Susceptible, except for 150 randomly chosen agents set to Infectious, for the SF network with $\beta=0.12$; or 50 randomly chosen agents for the other two cases (the discrepancy is to showcase the overlapping curves better). Lines (splines) are a guide for the eye.
• Figure 2: Proportion of agents ever infected, $\rho$, against probability of infection, $\beta$, for SF networks with $\alpha=2$ (dark blue circles) and $\alpha=3$ (light blue triangles), and for ER random graphs (red diamonds) (Panel A). Estimated value of basic reproduction number, $R_0^e$, from Eq. (\ref{['eq_R0e']}) against $\beta$ on the same networks ( B). Proportion infected, $\rho$, against SF exponent $\alpha$ for infection probability $\beta=0.6$ (dark blue circles), $\beta=0.3$ (light blue triangles) and $\beta=0.1$ (red diamonds) ( C). And $R_0^e$ against $\alpha$ for the same values of $\beta$ ( D). All agents are initially Susceptible except for 50 randomly chosen to be set to Infectious. All other parameters as in Fig. \ref{['fig_1']}.
• Figure 3: Proportion of agents ever infected, $\rho$, against estimated basic reproduction number, $R_0^e$, from Eq. (\ref{['eq_R0e']}) for SF networks with exponent $\alpha=2$ (dark blue circles) and $\alpha=3$ (light blue triangles), and for ER random graphs (red diamonds). Different values for the same network correspond to the different values of $\beta$ used in Fig. \ref{['fig_hetero']}A and B. All other parameters as in Fig. \ref{['fig_hetero']}.
• Figure 4: Time series for proportions of agents in the Infectious (panels A and C) and Recovered ( B and D) states for ER random graphs (blue circles) and SF networks with exponent $\alpha=2.2$ (red diamonds). At time $t=0$ all agents are Susceptible, except for 50 randomly chosen agents set to Infectious. At times $t=15$ and $t=30$, the networks are replaced with new ones, randomly generated with the same network parameters; and 50 randomly chosen Susceptible agents are set to Infectious. Panels A and B: Transmissibility is constant at $\beta=0.4$ in the ER case and $\beta=0.1$ in the SF case. Panels C and D: Transmissibility is increased at times $t=15$ and $t=30$. In the ER case, $\beta=0.4$ until $t=15$, $\beta=0.6$ until $t=30$, and $\beta=1$ thereafter. In the SF case, $\beta=0.1$ until $t=15$, $\beta=0.2$ until $t=30$, and $\beta=0.4$ thereafter. All other parameters as in Fig. \ref{['fig_1']}.