Network topology and double delays in turing instability and pattern formation

Investigating Turing patterns in complex networks presents a significant challenge, particularly in understanding the transition from simple to complex systems. We examine the network-organized SIR model, incorporating the Matthew effect and double delays, to demonstrate how network structures directly impact critical delay values, providing insights into historical patterns of disease spread. The study reveals that both susceptible and infected individuals experience a latent period due to interactions between the Matthew effect and incubation, mirroring historical patterns observed in seasonal flu outbreaks. The emergence of chaotic states is observed when two delays intersect critical curves, highlighting the complex dynamics that can arise in historical epidemic models. A novel approach is introduced, utilizing eigenvalue ratios from minimum/maximum Laplacian matrices (excluding 0) and critical delay values, to identify stable regions within network-organized systems, providing a new tool for historical epidemiological analysis. The paper further explores dynamic and biological mechanisms, discussing how these findings can inform historical and contemporary strategies for managing infectious disease outbreaks.