A fractional-order vaccination model to analyze the dynamics of Mpox Clade I and II with real data.

A fractional vaccination model for monkeypox infection with two strains, based on real data, is proposed. We consider monkeypox case data for the period January 1, 2025, to July 31, 2025. The model is first formulated using an integer-order derivative and then extended to a fractional-order derivative. It is shown that the fractional model exists and has a unique solution. Equilibrium points are obtained, and their stability is analyzed; the model is found to be locally asymptotically stable whenever [Formula: see text]. Multiple equilibria are identified, and under certain conditions, the existence of a positive endemic equilibrium and the possibility of backward bifurcation are demonstrated. A nonlinear least-squares approach is used for the estimation of model parameters, and sensitivity analysis identifies parameters with high potential for disease control. The numerical values estimated for the parameters give [Formula: see text]. A numerical scheme is proposed to accurately solve the two-strain mpox model, and graphical results illustrate how disease elimination can be achieved based on the fractional-order parameter and contact rates.