In this paper, we present a method for solving multi-dimensional fractional optimal control problems. Firstly, we derive the Bernstein polynomials operational matrix for the fractional derivative in the Caputo sense, which has not been done before. The main characteristic behind the approach using this technique is that it reduces the problems to those of solving a system of algebraic equations, thus greatly simplifying the problem. The results obtained are in good agreement with the existing ones in the open literature and it is shown that the solutions converge as the number of approximating terms increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach 1.

In this paper, we extend the SIR model with vaccination into a fractional-order model by using a system of fractional ordinary differential equations in the sense of the Caputo derivative of order α ∈ ( 0 , 1 ] $\alpha\in(0,1]$ . By applying fractional calculus, we give a detailed analysis of the equilibrium points of the model. In particular, we analytically obtain a certain threshold value of the basic reproduction number R 0 $R_{0}$ and describe the existence conditions of multiple equilibrium points. Moreover, it is shown that the stability region of the equilibrium points increases by choosing an appropriate value of the fractional order α. Finally, the analytical results are confirmed by some numerical simulations for real data related to pertussis disease.