Bursts of vertex activation and epidemics in evolving networks

Abstract

The dynamic nature of contact patterns creates diverse temporal structures. In particular, empirical studies have shown that contact patterns follow heterogeneous inter-event time intervals, meaning that periods of high activity are followed by long periods of inactivity. To investigate the impact of these heterogeneities in the spread of infection from a theoretical perspective, we propose a stochastic model to generate temporal networks where vertices make instantaneous contacts following heterogeneous inter-event intervals, and may leave and enter the system. We study how these properties affect the prevalence of an infection and estimate R(0), the number of secondary infections of an infectious individual in a completely susceptible population, by modeling simulated infections (SI and SIR) that co-evolve with the network structure. We find that heterogeneous contact patterns cause earlier and larger epidemics in the SIR model in comparison to homogeneous scenarios for a vast range of parameter values, while smaller epidemics may happen in some combinations of parameters. In the case of SI and heterogeneous patterns, the epidemics develop faster in the earlier stages followed by a slowdown in the asymptotic limit. For increasing vertex turnover rates, heterogeneous patterns generally cause higher prevalence in comparison to homogeneous scenarios with the same average inter-event interval. We find that [Formula: see text] is generally higher for heterogeneous patterns, except for sufficiently large infection duration and transmission probability.

Figure 1. Prevalence of the infection in SI epidemics.

The prevalence in case of SI epidemics for HET and HOM contact patterns with (blue curves) and (red curves). Each column corresponds to a different , (A) , (B) and (C) . The x-axis is in log-scale.

Figure 2. Prevalence of the infection in SIR epidemics.

Curves correspond to the fraction of infected (i.e. the prevalence – blue) and fraction of susceptible individuals (red). Each panel contains a different configuration: (A) and ; (B) and ; (C) and ; (D) and ; (E) and ; (F) and . The x-axis is in log-scale.

Figure 3. Intensity of the peak prevalence for SIR epidemics.

Difference in the intensity of peak prevalence for (A) deterministic () and for (B) stochastic () SIR dynamics with various infective intervals in case of . F statistics for (C) deterministic () and for (D) stochastic () SIR (red and white mean that HET and HOM peak intensities are statistically different, that is, ), raw p-values are in Text S1; the difference relative to the HET case, that is, for (E) deterministic () and for (F) stochastic () SIR.

Figure 4. Time of the peak prevalence for SIR epidemics.

Difference in the time of peak prevalence for (A) deterministic () and for (B) stochastic () SIR dynamics with various infective intervals in case of ; F statistics for (C) deterministic () and for (D) stochastic () SIR (red and white mean that HET and HOM peak times are statistically different, i.e. ), raw p-values are in Text S1; the difference relative to the HET case, that is, for (E) deterministic () and for (F) stochastic () SIR.

Figure 5. Estimation of for deterministic SIR epidemics.

Numerical estimation of for SIR in case of (A) and (C) (with ), and in case of (B) and (D) (with ). The results are independent of the network size (see Text S1). Dashed lines correspond to . The F statistics is presented above the plots. Dashed lines correspond to ; and are statistically different if .

Figure 6. Estimation of for stochastic SIR epidemics.

Numerical estimation of for HET network () and the difference of between HET and HOM networks, that is, . for HET in case of (A) and (E) ; for (B) and (F) ; F statistics for (C) and (G) (red and white mean that HET and HOM cases are statistically different, that is, ); p-values for (D) and (H) .

Figure 7. Distribution of outbreak sizes by random initial infection seeds.

Fraction of times an epidemic outbreak with size is observed at time . The results correspond to the SIR model with and network configurations with .