Multiplex epidemiological networks with delays: mechanisms of Turing instability and spatiotemporal oscillation

Abstract

Understanding and controlling the spatial-temporal spread of infectious diseases are vital challenges, especially in light of increased population mobility and complex human interactions. Traditional epidemic models often neglect the heterogeneous, multiplexed, and time-delay nature of real-world transmission. This work proposes a comprehensive epidemic model on multiplex networks, explicitly incorporating both incubation and infectious period delays. Using local stability and bifurcation theory, we derive precise mathematical criteria for Turing instability and periodic oscillations in these systems, revealing that the self-recovery rate induces subcritical bifurcations, while time delays can trigger supercritical bifurcations and persistent oscillations. Our analysis demonstrates that multiplex network topology and delays jointly regulate the emergence of spatiotemporal patterns—specifically, clustered outbreaks and wave-like, periodic fluctuations. As network connectivity or delays increase, both spatial aggregation and the amplitude of epidemic cycles intensify. Numerical simulations reflect these findings and are quantitatively validated using real COVID-19 data from Brazil, South Africa, and 35 countries in the Americas, confirming our theoretical predictions. These insights advance the mechanistic understanding of epidemic pattern formation and highlight the importance of considering network structure and time delays in early warning and targeted intervention strategies.