Design, Analysis and Comparison of a Nonstandard Computational Method for the Solution of a General Stochastic Fractional Epidemic Model

Malaria is a deadly human disease that is still a major cause of casualties worldwide. In this work, we consider the fractional-order system of malaria pestilence. Further, the essential traits of the model are investigated carefully. To this end, the stability of the model at equilibrium points is investigated by applying the Jacobian matrix technique. The contribution of the basic reproduction number, R0, in the infection dynamics and stability analysis is elucidated. The results indicate that the given system is locally asymptotically stable at the disease-free steady-state solution when R0<1. A similar result is obtained for the endemic equilibrium when R0>1. The underlying system shows global stability at both steady states. The fractional-order system is converted into a stochastic model. For a more realistic study of the disease dynamics, the non-parametric perturbation version of the stochastic epidemic model is developed and studied numerically. The general stochastic fractional Euler method, Runge–Kutta method, and a proposed numerical method are applied to solve the model. The standard techniques fail to preserve the positivity property of the continuous system. Meanwhile, the proposed stochastic fractional nonstandard finite-difference method preserves the positivity. For the boundedness of the nonstandard finite-difference scheme, a result is established. All the analytical results are verified by numerical simulations. A comparison of the numerical techniques is carried out graphically. The conclusions of the study are discussed as a closing note.

The right expression of the equivalent integral equation and non-uniqueness of solution of impulsive fractional order system

The fractional derivatives are not equal for different expressions of the same piecewise function, which caused that the equivalent integral equations of impulsive fractional order system (IFrOS) proposed in existing papers are incorrect. Thus we reconsider two generalized IFrOSs that both have the corresponding impulsive Caputo fractional order system and the corresponding impulsive Riemann-Liouville fractional order system as their special cases, and discover that their equivalent integral equations are two integral equations with some arbitrary constants, which reveal the non-uniqueness of solution of the two generalized IFrOSs. Finally, two numerical examples are offered for explaining the non-uniqueness of solution to the two generalized IFrOSs.