Temporal Heterogeneities Increase the Prevalence of Epidemics on Evolving Networks

Empirical studies suggest that contact patterns follow heterogeneous inter-event times, meaning that intervals of high activity are followed by periods of inactivity. Combined with birth and death of individuals, these temporal constraints affect the spread of infections in a non-trivial way and are dependent on the particular contact dynamics. We propose a stochastic model to generate temporal networks where vertices make instantaneous contacts following heterogeneous inter-event times, and leave and enter the system at fixed rates. We study how these temporal properties affect the prevalence of an infection and estimate R0, the number of secondary infections, by modeling simulated infections (SIR, SI and SIS) co-evolving with the network structure. We find that heterogeneous contact patterns cause earlier and larger epidemics on the SIR model in comparison to homogeneous scenarios. In case of SI and SIS, the epidemics is faster in the early stages (up to 90% of prevalence) followed by a slowdown in the asymptotic limit in case of heterogeneous patterns. In the presence of birth and death, heterogeneous patterns always cause higher prevalence in comparison to homogeneous scenarios with same average inter-event times. Our results suggest that R0 may be underestimated if temporal heterogeneities are not taken into account in the modeling of epidemics.

💡 Research Summary

The paper investigates how temporal heterogeneities—specifically bursty inter‑event time distributions—affect epidemic spread on networks that also experience birth and death of individuals. The authors construct a minimal stochastic model of a temporal network: each node alternates between “on” (active) and “off” (inactive) states, with the waiting time τ between successive active periods drawn from either a power‑law with exponential cutoff (heterogeneous case, P_het(τ)∝τ^{–α}e^{–τ/τ_c}) or an exponential (homogeneous case, P_hom(τ)=λe^{–λτ}) having the same mean. When a node becomes active it selects uniformly another active node, forms a one‑step contact, and then returns to the off state. This yields a rapidly mixing network without degree heterogeneity, but with strong temporal fluctuations.

Node lifetimes follow a Poisson process with death rate μ=1/τ_death; each death is immediately compensated by a newborn node, keeping the total population size N constant. On top of this evolving substrate the authors simulate three classic compartmental models: SIR (susceptible‑infected‑recovered), SI (susceptible‑infected), and SIS (susceptible‑infected‑susceptible). Infection probability per contact is set to β=1, while the infectious period τ_I is varied. Simulations start with a single infected node; newcomers are always susceptible.

Key findings for the SIR model: (1) When τ_I is short and the power‑law exponent α is low (strong burstiness), the heterogeneous timing yields a substantially higher peak prevalence—up to 54 % above the homogeneous baseline—and the peak occurs 20–40 time steps earlier. (2) For larger τ_I or larger α (weaker burstiness), the homogeneous case overtakes, producing higher peaks. Introducing birth/death reduces peaks in both scenarios, but the reduction is more pronounced for homogeneous timing, so heterogeneous timing consistently yields higher overall prevalence in the presence of turnover. Moderate turnover (τ_death≈5–6) creates a secondary, smaller wave driven by susceptible newcomers; this secondary wave is more evident under heterogeneous timing.

For the SI model, heterogeneous timing accelerates the early phase, infecting ~85 % of the population faster than the homogeneous case, but then the approach to full saturation slows dramatically, resulting in a longer tail. The homogeneous case, while slower initially, reaches full infection earlier overall. Adding death eliminates the long tail for the heterogeneous case and maintains higher average prevalence across all death rates; the homogeneous case can even produce near‑zero average outbreaks when death is frequent.

The SIS model mirrors the SI behavior: an early surge in prevalence under heterogeneous timing is followed by a lower steady‑state infection level after a brief oscillatory transient, whereas the homogeneous case stabilizes at a higher endemic level.

These results imply that the basic reproduction number R₀, often estimated from early growth rates, can be severely underestimated if temporal burstiness is ignored. Burstiness amplifies early transmission but can hinder long‑term persistence, a dual effect that standard static‑network or homogeneous‑temporal models miss. The authors argue that realistic epidemic forecasting and control policies must incorporate the full temporal structure of contacts, especially in settings where bursty interactions and demographic turnover coexist (e.g., sexual networks, hospital staff, tourism). The study provides a clean, analytically tractable framework that isolates the impact of inter‑event time distributions, offering a valuable baseline for future work that may add further realism such as degree heterogeneity, spatial constraints, or adaptive behavior.