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In this article, we propose the mathematical modeling of monkeypox, a viral zoonotic disease, to study its near outbreaks in the United States. We use integer and fractional derivatives to simulate the transmission model dynamics effectively. The model parameter values are estimated using the Levenberg–Marquardt algorithm with the lsqcurvefit function in MATLAB. We derive the numerical solution of the integer-order model by using a neural network approach. For solving the fractional-order model, we use the finite-difference predictor–corrector method in the sense of the Caputo derivative. We perform several graphical simulations of future disease outbreaks. Implementing the Caputo fractional derivatives is motivated by its nonlocal nature, which includes memory effects in the model. The new data simulation and various solution approaches are the novelty of this research. This work aims to explore the transmission pattern and duration of a monkeypox outbreak in the United States. [ABSTRACT FROM AUTHOR]