Turing instability induced by crossing curves in network-organized system

Several factors significantly contribute to the onset of infectious diseases, including direct and indirect transmissions and their respective impacts on incubation periods. The intricate interplay of these factors within social networks remains a puzzle yet to be unraveled. In this study, we conduct a stability analysis within a network-organized SIR model incorporating dual delays to explore the influence of direct and indirect incubation periods on disease spread. Additionally, we investigate how compound networks affect the critical incubation period. Our findings reveal several vital insights. First, by examining crossing curves and the dispersion equation, we establish the conditions for Turing instability and delineate the stable regions associated with dual delays. Second, we ascertain that the critical incubation value exhibits an inverse relationship with a network’s eigenvalues, indicating that the Laplacian matrix does not solely dictate periodic behavior in the context of delays. Furthermore, our study elucidates the impact of delays and networks on pattern formation, revealing distinct pattern types across different regions. Specifically, our observations suggest that effectively curtailing the spread of infectious diseases during an outbreak is more achievable when the incubation period for indirect contact is shorter and for direct contact is longer. Namely, our network framework enables regulation of the optimal combination of $(\tau _{1},\tau _{2})$ to mitigate the risk of infectious diseases. In summary, our results offer valuable theoretical insights that can inform strategies for preventing and managing infectious diseases.