In order to effectively manage infectious diseases, it is crucial to understand the interplay between disease dynamics and human conduct. Various factors can impact the control of an epidemic, including social interventions, adherence to health protocols, mask-wearing, and vaccination. This article presents the development of an innovative hybrid model, known as the Combined Dynamic-Learning Model, that integrates classical recurrent dynamic models with four different learning methods. The model is composed of two approaches: The first approach introduces a traditional dynamic model that focuses on analyzing the impact of vaccination on the occurrence of an epidemic, and the second approach employs various learning methods to forecast the potential outcomes of an epidemic. Furthermore, our numerical results offer an interesting comparison between the traditional approach and modern learning techniques. Our classic dynamic model is a compartmental model that aims to analyze and forecast the diffusion of epidemics. The model we propose has a recurrent structure with piecewise constant parameters and includes compartments for susceptible, exposed, vaccinated, infected, and recovered individuals. This model can accurately mirror the dynamics of infectious diseases, which enables us to evaluate the impact of restrictive measures on the spread of diseases. We conduct a comprehensive dynamic analysis of our model. Additionally, we suggest an optimal numerical design to determine the parameters of the system. Also, we use regression tree learning, bidirectional long short-term memory, gated recurrent unit, and a combined deep learning method for training and evaluation of an epidemic. In the final section of our paper, we apply these methods to recently published data on COVID-19 in Austria, Brazil, and China from 26 February 2021 to 4 August 2021, which is when vaccination efforts began. To evaluate the numerical results, we utilized various metrics such as RMSE and R-squared. Our findings suggest that the dynamic model is ideal for long-term analysis, data fitting, and identifying parameters that impact epidemics. However, it is not as effective as the supervised learning method for making long-term forecasts. On the other hand, supervised learning techniques, compared to dynamic models, are more effective for predicting the spread of diseases, but not for analyzing the behavior of epidemics.

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