The global spread of SARS-CoV-2 (COVID-19) continues to strain public health systems, emphasizing the need for reliable analytical and forecasting tools. This study develops an \(\textbf{SIHRS}\) epidemic model with a nonlinear incidence rate that incorporates crowding effects, government-imposed restrictions, behavioral changes, and public compliance. The model is calibrated using COVID-19 case data from the United Kingdom through the Delayed Rejection Adaptive Metropolis (DRAM) MCMC algorithm. Key epidemiological drivers and parameter uncertainties are assessed using Partial Rank Correlation Coefficient (PRCC) based sensitivity analysis, while the basic reproduction number serves as a threshold indicator determining the long-term disease transmission potential. The analytical results determine the criteria under which the zero-infection and endemic equilibrium states remain stable, complemented by bifurcation analysis revealing transcritical, Hopf, and saddle-node bifurcations that portray how small changes in key parameters can cause major shifts in epidemic behavior. In order to enhance the predictive capability, we develop epidemiological-machine learning integrated models by embedding transmission dynamics into ARIMA and ARNN frameworks. These hybrid models are used to forecast short-term COVID-19 incidence in the UK, demonstrating improved predictive accuracy compared to ML-only approaches. Overall, our findings emphasize the value of integrating mechanistic modeling with data-driven methods to enhance efficient epidemic forecasting and strengthen preparedness for potential resurgence of COVID-19 and other infectious diseases.
