An Epidemic Threshold Set for Networks

The Structure of Spreading on Temporal Networks

It is shown that spreading dynamics on temporal networks can analogously be mapped to reachability in temporal event graphs, which provides a theoretical and computational framework for a class of processes, such as variants of the susceptible-infected-susceptible model.

Applied Mathematical Modelling

Multi-scale local network structure critically impacts epidemic spread and interventions

Network epidemic simulation enables fine-grained understanding of epidemic behavior. However, empirical samples of interaction networks display properties that are challenging to capture with popular synthetic models of networks. Our empirical results show that epidemic spread behavior is sensitive to a form of multi-scale local structure that is absent in common baseline models, (e.g., Erdős–Rényi, Chung-Lu, etc). This structure critically impacts the effect of local quarantining and stops epidemic spread in samples of interaction networks, even when it cannot be halted in simple synthetic models of those networks. Insights from our analysis include how epidemics on networks with widespread multi-scale local structure are easier to mitigate, as well as characterizing which nodes are ultimately not likely to be infected. We demonstrate that this structure results from more than just local triangle structure in the network, and we illustrate processes based on homophily or social influence and random walks that suggest how this multi-scale local structure arises and use it to cleanly isolate intervention sensitivity to multi-scale local structure.

Dynamics of SIRS epidemic model in heterogeneous hypernetworks.

In this paper, the complex epidemic behavior of higher-order interactions is investigated by considering heterogeneous hypernetworks with collective contagion mechanisms, and a susceptible-infected-recovered-susceptible epidemic model is established in hypernetworks. First, we propose a mean-field model in heterogeneous hypernetworks based on hyperdegree and conduct analyses of this model. For general heterogeneous hypernetworks, we determine the epidemic threshold for disease spread and the threshold for the emergence of backward bifurcation, performing comprehensive dynamical analyses. Specifically, in a heterogeneous hypernetwork where the hyperedge size does not exceed 4, the existence of endemic equilibria is examined, and conditions for the system possessing a varying number of endemic equilibria are provided. By analyzing the model, we identify three types of bifurcations and two distinct bistable phenomena. Finally, theoretical predictions are confirmed through numerical simulations.

Dynamics of pairwise-based SIS epidemic models on clustered networks with higher-order interactions

This work investigates the susceptible-infected-susceptible epidemic model on clustered regular networks using the pair mean-field approach, where higher-order effects are explicitly considered in dynamic analysis and proves the effectiveness of the pair mean-field method for analyzing higher-order network dynamics.

Traffic-driven sequential spread of two interacting epidemics on complex networks.

While the coupling between traffic dynamics and epidemic spreading on complex networks has been widely studied, existing research has predominantly focused on single-pathogen transmission. In this paper, we investigate the sequential spreading dynamics of two interacting epidemics driven by traffic flow. The model incorporates a key mechanism whereby prior infection with the first disease alters a node's susceptibility to the second disease. We develop a heterogeneous mean-field framework for the coupled spreading process and derive analytical expressions for the epidemic thresholds. Our results show that the interaction parameter α and the infection rate of the first epidemic β1 jointly determine the outbreak threshold and stationary prevalence of the second epidemic. The parameter α regulates how prior infection influences susceptibility to the second epidemic, ranging from suppressive interaction (α < 1) through neutral interaction (α = 1) to synergistic interaction (α > 1). A pronounced nonlinear threshold response emerges: in the suppressive interaction regime, increasing β1 below its critical point substantially raises the epidemic threshold of the second disease, whereas in the synergistic interaction regime, it lowers the threshold. Once the first epidemic exceeds its critical point, both effects gradually saturate. Numerical simulations show good agreement with the theoretical predictions and further demonstrate the robustness of the results across different network sizes and average degrees. These findings reveal how epidemic interactions and network structure jointly shape sequential spreading dynamics in traffic-driven systems, providing new insights into coupled contagion processes on complex networks.