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This article presents the mathematical formulation for the monkeypox infection using the Mittag–Leffler kernel. A detailed mathematical formulation of the fractional-order Atangana-Baleanu derivative is given. The existence and uniqueness results of the fractional-order system is established. The local asymptotical stability for the disease-free case, when ℛ0<1{{\mathcal{ {\mathcal R} }}}_{0}\lt 1, is given. The global asymptotical stability is given when ℛ0>1{{\mathcal{ {\mathcal R} }}}_{0}\gt 1. The backward bifurcation analysis for fractional system is shown. The authors give a numerical scheme, solve the model, and present the results graphically. Some graphical results are shown for disease curtailing in the USA.