Instability-Driven Pattern Formation in a Network Sir Model With Indirect Transmission and Quasi-Laplacian Diffusion

This study investigates how indirect transmission and diffusion asymmetry shape epidemic dynamics in a network-organized SIR model. Using linear stability analysis and eigenmode decomposition, we derive explicit conditions for Hopf bifurcation, Turing instability, and their interaction. The results show that indirect transmission significantly shifts epidemic thresholds, while asymmetric diffusion across network nodes promotes the activation of additional eigenmodes and the emergence of spatially heterogeneous infection patterns. Numerical simulations on random and quasi-Laplacian networks reveal transitions among stable equilibria, periodic outbreaks, and mixed Hopf-Turing regimes, with the specific pattern determined jointly by biological parameters and network topology. To validate the theory, the model was calibrated using real influenza surveillance data from 44 countries. The observed periodicity and spatial clustering closely match the model predictions, demonstrating that instability-driven mechanisms can explain real-world influenza oscillations and heterogeneity. These findings provide a unified theoretical and data-supported framework for understanding epidemic pattern formation and designing interventions that target indirect transmission and mobility-induced spatial instabilities.