This study integrates advanced machine learning techniques, namely Artificial Neural Networks, Long Short-Term Memory, and Gated Recurrent Unit models, to forecast monkeypox outbreaks in Canada, Spain, the USA, and Portugal. The research focuses on the effectiveness of these models in predicting the spread and severity of cases using data from June 3 to December 31, 2022, and evaluates them against test data from January 1 to February 7, 2023. The study highlights the potential of neural networks in epidemiology, especially concerning recent monkeypox outbreaks. It provides a comparative analysis of the models, emphasizing their capabilities in public health strategies. The research identifies optimal model configurations and underscores the efficiency of the Levenberg-Marquardt algorithm in training. The findings suggest that ANN models, particularly those with optimized Root Mean Squared Error, Mean Absolute Percentage Error, and the Coefficient of Determination values, are effective in infectious disease forecasting and can significantly enhance public health responses.