Stability and Bifurcations of a Two-Patch SIS Model with Delayed Surveillance Feedback

In this paper, we propose a delayed two-patch SIS epidemic model with surveillancemediated infection and dispersal rates. The well-posedness of the model is established using functional differential equation theory, and the basic reproduction number R 0 is derived via the next-generation matrix method. Analytical thresholds related to dispersal behavior are obtained, and the existence and stability of boundary and endemic equilibria are investigated using stability and Hopf bifurcation theory. Numerical simulations validate the theoretical results and highlight the impact of time delay, infection rates, and surveillance-based dispersal rates on disease dynamics. Our results reveal that the dispersal of susceptible individuals alone can initiate transmission in otherwise disease-free patches. We also find that R 0 and key thresholds are independent of time delay, while the transient dynamics and stability of equilibria are significantly influenced by heterogeneous parameters such as self-protection level and attention to past surveillance. Dispersal rates exhibit bidirectional effects (either inhibiting or enhancing) on the basic reproduction number R 0 in heterogeneous environments, and it exhibits high sensitivity to this effect. The synergistic interactions among self-protection intensity, time delay and surveillance-mediated parameters significantly influence the bifurcation dynamics of endemic equilibria. Therefore, for the effective disease control strategies, we must account for not only the management of population mobility, but also the multifaceted influences of infection rates, media influence, and regional heterogeneity.