Nonlinear Dynamics and Behavioral Effects in an Extended S 1 S 2 V IR Epidemic Model

In the present paper, we construct an extended S 1 S 2 V IR-type epidemiological model which includes behavioral adaptation, nonlinear incidence, and vaccination with leaky efficacy. The susceptible population is divided into two behavioral classes, whereby the model can depict various dynamic responses to perceived infection risk. We formulate the complete system of differential equations, find all diseasefree and endemic equilibria, and carry out their local and global stability analyses by using Lyapunov methods and Routh-Hurwitz criteria. The analysis confirms a transcritical bifurcation at the threshold for effective reproduction R v = 1 and a saddle-node bifurcation in both infected and susceptible populations due to behavioral feedback. Numerical simulations and data-driven calibration against COVID-19 data will show that the model replicates the observed epidemic patterns and Quantify the combined effects of vaccination and behavioral transitions. The presented framework gives a complete mathematical background for infectious diseases in which human behavior and imperfect vaccination play an important role.