Multiple Disease-Free Equilibria: Toward a Vector Structure of R 0.

Traditional epidemiological models rely on the basic reproductive number, R 0 , as a single threshold quantity to determine whether a disease will persist or die out. However, increased structural complexity in models can lead to the emergence of multiple disease-free equilibria (DFEs). To illustrate this dynamic diversity, we analyze a compartmental model based on the Health Belief Model, where risk perception, social imitation, and behavioral fatigue among susceptible individuals generate the coexistence of multiple DFEs. This case study exposes the limitations of a single scalar threshold and motivates the introduction of a methodological approach: a vector-valued reproductive number and three-dimensional bifurcation diagrams. The analysis reveals complex scenarios including multistability, regions with specific dynamic behaviors, and outcomes conditioned by the initial population distribution. This heuristic approach suggests an extension of the classical concept of R 0 and offers a new analytical perspective that lays the groundwork for studying disease dynamics in populations with complex structure.