In this paper, we examine the long-term dynamics of obesity under conditions of treatment response and relapse using a four-compartment dynamical model motivated by the growing use of GLP-1 receptor agonist therapy. The model is partially calibrated to Georgia Behavioral Risk Factor Surveillance System (BRFSS) data from 2011 to 2024 and reproduces the observed trends closely. We establish the existence of a unique endemic equilibrium, linking model parameters directly to long-run prevalence. Simulations show that increasing the effective treatment-transition rate on its own produces only modest reductions in prevalence. Sensitivity analysis shows that, under the baseline setting, equilibrium obesity prevalence is most strongly influenced by obesity progression and partial recovery. Under the enhanced treatment-response setting, the effective treatment-transition rate becomes the strongest negative driver, while obesity progression remains the strongest positive driver and the influence of relapse increases substantially. Improving access to GLP-1-based treatment may increase the effective treatment-transition rate, but access alone is not sufficient. Lasting reductions in prevalence also require managing relapse, so policy design needs to address adherence and long-term retention, not access alone.
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