Turing instability and bistable epidemic spread on complex networks: Effects of topology and initial seeding.

Coevolution of epidemic dynamics and network topology driven by disease fatality and waning immunity

Epidemics on complex networks have been shown to exhibit dynamics that are strongly influenced by the topology of the network. However, it remains unclear how the topology of the network is influenced by epidemic properties, such as waning immunity and disease fatality, coupled with demographic changes. To explore the interplay between epidemic dynamics and network topology, we develop an agent-based model of a fatal infectious disease with an imperfect immunity on an initially scale-free network that evolves via demographic and epidemic-induced changes. We show that this model undergoes an epidemic transition between a phase in which the disease eventually dies out and a phase in which it becomes endemic. Moreover, the network may lose its scale-free property as a result of the epidemic spreading, giving rise to a topological transition from a power-law to a non-power-law degree distribution. We validate our findings using heterogeneous mean-field approximations of the agent-based model. These results highlight the importance of accounting for network evolution in epidemic models and advance our understanding of coevolving dynamical systems, in which disease dynamics and network topology continuously shape one another.

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.

Extreme outbreaks in non-Markovian epidemics on complex networks

Extreme epidemic risk is controlled by the right tail of the outbreak-size distribution, but this distribution is generally unknown for non-Markovian spreading on networks. Here we determine this distribution by mapping non-Markovian SIR dynamics to an effective Markovian description. We show that arbitrary infection and recovery time statistics can be incorporated through a single edge transmissibility, yielding an effective Markovian process that reproduces the full outbreak-size statistics. For weakly heterogeneous networks, the reduction yields a universal well-mixed semiclassical theory governed by the bond-percolation reproductive number. Outbreak statistics across diverse waiting-time distributions and topologies collapse onto one predictive curve. For highly heterogeneous and empirical networks, the corresponding effective Markovian dynamics on the network captures the complete distribution. Our results provide a direct route from measured waiting-time distributions to quantitative predictions of network-level extreme-outbreak risk.

Applied Mathematical Modelling

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.

Fear-driven higher-order epidemic dynamics on heterogeneous hypergraphs