Stability and bifurcations of a minimal model for the effect of PrEP-related risk compensation in epidemics of sexually transmitted infections

HIV pre-exposure prophylaxis (PrEP) greatly reduces HIV infection risk when taken as prescribed, offering near-complete protection even without condoms. However, this protection may prompt increased risk behaviors (through risk compensation), facilitating the spread of other sexually transmitted infections (STIs). We analyze a minimal model describing how PrEP-related risk compensation affects STI epidemics among high-infection-risk men who have sex with men, focusing on two primary parameters: PrEP uptake (P) and risk awareness (H). These factors shape STI transmission through feedback mechanisms involving risk compensation and testing. We derive the system’s basic reproduction number, R0, and identify a transcritical bifurcation at R0=1, where the disease-free equilibrium becomes unstable, and an asymptotically stable endemic equilibrium appears. We determine critical behavioral and policy thresholds separating these regimes and analyze typical parameter values. Notably, saturation of behavioral response produces a “back-bending” R0=1 curve in the (P,H) plane, where increasing risk awareness may paradoxically hinder containment. We analytically characterize the conditions under which this effect occurs. More broadly, our model provides a general framework for studying nonlinear feedbacks among behavioral adaptation, interventions, and disease dynamics, revealing how feedbacks can drive complex epidemic responses.