Existence and stability of equilibria in infectious disease dynamics with behavioral feedback

Mathematical models have provided a general framework for understanding the dynamics and control of infectious disease. Many compartmental models are limited in that they do not account for the range of behavioral feedbacks that have been observed in the response to emerging infections. Here we expand on the SIR compartmental model framework by introducing a general class of behavioral feedbacks that encompasses both individual responses and nonpharmaceutical interventions. By linking transmission dynamics and behavior, this class of models can capture the interplay of disease incidence, behavioral response, and controls such as vaccination. We prove mathematically the existence of two endemic equilibria depending on the vaccination rate: one in the presence of low vaccination but with reduced societal activity (the "new normal"), and one with return to normal activity but with vaccination rate below that required for disease elimination. Establishing the existence and stability of these equilibria is a precursor to designing control strategies that may exploit them.