In this study, we extend the traditional compartmental “Susceptible, Infected, Recovered” (SIR) model to a susceptible-asymptomatic-infected-recovered (SAIR) framework with logistic growth in the susceptible population, aiming to capture the impact of asymptomatic transmission and population dynamics on disease spread. The inclusion of an asymptomatic compartment allows for a more accurate representation of diseases where undetected carriers contribute significantly to transmission, as seen in infections like COVID-19. We derive stability conditions for the disease-free equilibrium (DFE) and the endemic equilibrium (EE) using Routh-Hurwitz criteria, examining how the basic reproduction number \( R_0 \) and population carrying capacity \( K \) influence disease persistence. Our analysis reveals that for low values of \( K \) , the DFE is stable, suggesting that the disease will eventually be eliminated. However, as \( K \) increases, the system shifts to an EE, where infection persists within the population. At higher \( K \) values, a Hopf bifurcation occurs, leading to stable limit cycle oscillations and periodic outbreaks, rather than a steady endemic state. These findings highlight critical thresholds, including the branch point (BP) and Hopf bifurcation (HB), where qualitative changes in stability occur, impacting long-term disease dynamics.