Risk Behavior, Backward Bifurcation, and Vaccine Effectiveness in Disease Dynamics.

Vaccination is a cornerstone of infectious disease control, yet vaccines are not fully protective, leaving a fraction of the vaccinated population susceptible to infection. This partial protection can alter behavior, as individuals who perceive themselves as immune may reduce adherence to preventive measures. Motivated by this, we investigate how behavioral changes among non-immune vaccinated individuals influence the dynamics of a directly transmitted disease and the basic reproduction number. We propose a model that incorporates vaccine failure through three facets (take, degree, and duration) alongside a behavioral parameter that modifies contact rates according to compliance with mitigation measures. Our analysis highlights the critical role of the behavioral index in key phenomena, including backward bifurcation and overall disease dynamics. We identify two thresholds. The first specifies the values of the behavioral index for which backward bifurcation does not arise, thereby indicating the conditions under which the disease may persist. The second establishes a relationship between the behavioral index and vaccine efficacy, which allows us to compare the transmission dynamics of our model with those of the classical vaccination model.