Ratio-Dependent Impulsive Control and Threshold Dynamics for an SIRI Model Incorporating Overexposure and Reinfection

To dynamically target interventions across populations of varying sizes during epidemics, we propose an SIRI epidemic model with ratio-dependent threshold control. The model captures the intermittent nature of integrated interventions and their impulsive effects on disease transmission dynamics. We first analyze the continuous subsystem, establishing the existence of multiple equilibria and bistability induced by backward bifurcation. Using Poincaré mapping and qualitative analytical techniques, we then investigate the threshold dynamics of the impulsive system by reducing the phase space to a positively invariant set. The effect of ratio-dependent threshold control on dynamics reveals that the solution behavior is jointly determined by the basic reproduction number, the predefined ratio threshold, and initial conditions. Numerical simulations validate the theoretical results, highlighting the emergence of periodic attractors of order-1 and order-2 solutions.