The Endemic Trap: Proactive Early Intervention as the Key to Elimination in the Epidemiological Dynamics of Dual Thresholds

Abstract

Discrete system dynamics play a vital role in understanding the spread of infectious diseases, especially during pandemics. In this study, bifurcation analysis is employed on a discrete model incorporating interventions (vaccination and quarantine) and medical resource limitations to reveal qualitative changes in epidemic behavior. We investigate why early policy speed is crucial and how strategies must be adapted when a region faces an endemic state. Through the analysis of dual threshold bifurcation types I and II and numerical simulations, we demonstrate that, in the early phase (type I), complacency can lead to a serious dilemma, pushing the epidemic system (which describes how a disease moves through a population) into a costly, difficult-to-control endemic condition, and if a high endemic state is reached (type II) and medical capacity is exceeded, inadequate interventions risk causing an unsustainable return to high case numbers. Even in a low endemic state, the system can remain highly vulnerable, where even small disturbances can trigger dangerous new surges. The ultimate threat is a dramatic surge in cases from shifts in transmission (e.g., new variants), which can undo all progress. These findings emphasize the importance of dynamic epidemiological thresholds and the need for proactive, consistent, and adaptive policies to mitigate future pandemic impacts.