Bursts of Vertex Activation and Epidemics in Evolving Networks

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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 , 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 is generally higher for heterogeneous patterns, except for sufficiently large infection duration and transmission probability. Networks of sexual contacts and of spatial proximity are of interest for the understanding of epidemics because they define potential pathways by which sexual and airborne infections spread. These networks are not static but vary, with both vertices and links appearing and disappearing at different times. One of the temporal properties observed across systems is that the time lapse between two contacts is irregular, which means that high activity is followed by long intervals of idleness. In this article, by using a theoretical model of a dynamic network co-evolving with a simulated infection, we show that such heterogeneity leads to earlier epidemic outbreaks and increased prevalence of infections for a range of parameters, in comparison to scenarios of regular activity, which is the current modeling paradigm in mathematical epidemiology. We also include a turnover rate to model individuals entering and leaving the system, and we show that if turnover is high, the relative difference in the prevalence of heterogeneous and homogeneous contact patterns increases due to the continuous influx of susceptible individuals. These heterogeneities also increase the expected number of secondary infections produced by a single infected vertex in a completely susceptible population.
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