Epidemic spreading on clustered networks with behavioral adaptation.

Understanding disease spread requires coupling behavioral dynamics with contact network structure, yet their interaction is often overlooked or treated separately. We address this by formulating a susceptible-infected-recovered model on clustered networks with stochastic behavioral adaptation. In our model, individuals are characterized by two vigilance states: non-vigilant and vigilant, and update their states upon contact with infected neighbors according to a stochastic transition rule. Using a multitype branching process combined with percolation theory, we establish a theoretical framework for calculating the epidemic invasion probability, epidemic threshold, and final epidemic size and validate them through extensive stochastic simulations. Our results show that behavioral adaptation and transmission probabilities influence the epidemic threshold through distinct mechanisms: behavioral effects act in an approximately linear and additive manner, whereas transmission introduces nonlinear coupling between infection channels. Moreover, network clustering plays a dual role in epidemic invasion: it enhances outbreak probability in low-transmissibility regimes via local reinforcement of infection pathways, but suppresses large-scale spreading under high transmissibility due to local saturation and reduced effective branching. In addition, behavioral adaptation not only modulates the epidemic threshold but also reorganizes outbreak composition: non-vigilant infections dominate the epidemic burden, whereas vigilant infections exhibit a pronounced ridge-like maximum arising from a flux balance between vigilance induction and relaxation. Overall, our results reveal a rich interplay between behavioral adaptation, transmission heterogeneity, and network clustering in shaping both the onset and structure of epidemic outbreaks.