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]. 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 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.

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By using maximum principle approach, the existence, uniqueness and stability of a coupled fractional partial differential equations is studied. A new fractional characteristic finite difference scheme is given for solving the coupled system. This method is based on shifted Grünwald approximation and Diethelm's algorithm. We obtain the optimal convergence rate for this scheme and drive the stability estimates. The results are justified by implementing an example of the fractional order time and space dependent in concept of the complex Lévy motion. Also, the numerical results are examined for disinfection and sterilization of tetanus.