Dynamic analysis of an infectious disease model with media coverage and saturated treatment

In the context of infectious disease outbreaks, media coverage and healthcare resources are critical in shaping transmission dynamics. To capture the coupled effects of disease spread, media influence, behavioral adaptation, and healthcare burden, we develop a novel five-dimensional compartmental model. The susceptible population is divided into unaware and aware classes, and a media compartment is introduced to capture media-induced behavioral heterogeneity. Media effects on the contact rate are modeled by an exponential-decay term, while limited healthcare resources are represented by a saturating recovery function. We investigate the model both analytically and numerically. We first establish the existence, uniqueness, non-negativity, and boundedness of solutions, thereby ensuring biological feasibility. Second, by deriving the basic reproduction number, we examined the existence and stability of equilibria. A normalized forward sensitivity analysis indicates that the recruitment rate Λ, effective contact rate β, and natural death rate μ are the key parameters governing $R_{0}$. Bifurcation analysis further demonstrates the occurrence of forward and backward bifurcations, as well as Hopf bifurcations. Numerical continuation, time-series, and phase-space simulations confirm the theoretical findings and delineate parameter regimes associated with stable equilibria, periodic oscillations, and instability. This study reveals how media coverage, behavioral adaptation, and healthcare resource constraints jointly drive bifurcations and oscillations, explaining the emergence of epidemic steady states and periodic outbreaks. It also shows that controlling population inflow, reducing contact rates, and optimizing healthcare resource allocation can effectively mitigate periodic outbreaks. The results advance nonlinear epidemic theory and provide quantitative guidance for predicting and controlling complex epidemics.