We consider a fractional-order model of glucose and insulin interaction based on the intra-venous glucose tolerance test (IVGTT). We show the existence of the model's solution, uniqueness, non-negativity, and boundadness. In addition, for the proposed fractional-order model, we establish sufficient conditions for stability or instability. Some conditions for bifurcation in the proposed model are presented using bifurcation theory. Further, in the case of first order the model is discretized by applying the forward Euler scheme. We investigate how small the time step size must be chosen to guarantee that the steady state solution is an attractive fixed point of the discretized model. Numerical simulations that we provided support the analytical results.