A state-dependent impulsive system with ratio-dependent action threshold for investigating SIR model

In general, there is an imperative to amalgamate timely interventions and comprehensive measures for the efficacious control of infectious diseases. The deployment of such measures is intricately tied to the system's state and its transmission rate, presenting formidable challenges for stability and bifurcation analyses. In our pursuit of devising qualitative techniques for infectious disease analysis, we introduced a model that incorporates state-dependent transmission interventions. Through the introduction of state-dependent control, characterized by a non-linear action threshold contingent upon the combination of susceptible population density and its rate of change, we employ analytical methods to scrutinize various facets of the model. This encompasses addressing the existence, stability, and bifurcation phenomena concerning disease-free periodic solutions (DFPS). The analysis of the established Poincaré map leads us to the conclusion that DFPS indeed exists and maintains stability under specific conditions. Significantly, we have formulated a distinctive single-parameter family of discrete mappings, leveraging the bifurcation theorems of discrete maps to dissect the transcritical bifurcations around DFPS with respect to parameters such as $ET$ and ${\eta }_1$. Under particular conditions, these phenomena may give rise to effects like backward bifurcation and bistability. Through the analytical methodologies developed in this study, our objective is to unveil a more comprehensive understanding of infectious disease models and their potential relevance across diverse domains.