Regulating Brucellosis Network Dynamics via Clustering: Multistability, Protective Intervals and Transient Oscillations.

The transmission of brucellosis exhibits prominent spatial clustering, yet a clear theoretical understanding of how this clustering quantitatively regulates epidemic spread remains lacking. This paper takes clustering as the core variable and constructs a susceptible-infected-environment network dynamics model with a clustering coefficient using the pair approximation method, aiming to systematically reveal the underlying regulatory mechanisms of clustering on brucellosis transmission. The findings indicate that clustering does not simply intensify transmission but fundamentally alters the dynamical structure of the system. When clustering is present, the system can exhibit multiple complex transmission patterns, including bistability and periodic outbreaks, and can even induce Hopf bifurcation and Bogdanov-Takens bifurcation, leading to unpredictable fluctuations in infection scale. More importantly, clustering exhibits three special regulatory effects: it can create multi-stable transmission regimes, act as a "protective barrier" within specific intervals to suppress the formation of endemic diseases, and induce transient oscillations through unstable limit cycles, thereby creating an early intervention "window period"-with higher clustering levels rendering the epidemic more sensitive to the initial number of infected cases. These findings suggest that clustering is not merely a catalyst for transmission but also a key adjustable lever, offering new perspectives for the precise prevention and control of brucellosis and the optimization of breeding structures.