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First, we collect and analyze statistical data for epidemiological surveillance of HIV\/AIDS prevalence in Yemen from 2000 to 2022. Then, we employ the statistical analysis software EViews and apply ARIMA models to predict the number of HIV\/AIDS cases from 2023 to 2024. The results of the selected model, free of standard problems, predicted a future increase in HIV\/AIDS cases in Yemen. Next, relying on the well\u2010known fixed\u2010point theorem and a set of other associated results, we prove the existence and uniqueness results of the fractional model. Moreover, we use the Adams\u2013Bashforth method to approximate the solutions of this system numerically. 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