Unification of the different fractional time derivatives: An application to the epidemic-antivirus dynamical system in computer networks

Abstract

Different versions of the fractional derivative have been proposed in the literature. One of the objectives of the present work is to unify these versions. Which is done by reducing a fractional derivative to nonautonomous ordinary ones. Thus, fractional ODEs are transformed to non-autonomous ODEs. A second objective is to find the exact solutions of the fractional model equations of the dynamics between the epidemic and antivirus in computer networks. The model considered here, is an extension to the SIR model by accounting for the antivirus dynamics that results to the antidotal state (A), which is abbreviated SIRA. The study is carried here by reducing the fractional SIRA model to non-autonomous ordinary SIRA equations. A novel approach is proposed for solving the reduced system by implementing the extended unified method. Numerical evaluation of the exact solutions of susceptible, infected, recovered and antidotal species, are carried. The cases of Caputo and Caputo–Fabrizio-fractional derivatives are considered. It is observed that, in view of the model considered, the numbers of the suspected and infected computers decrease, with time, to zero in the two case. In both two cases, the number of recovered computers increases rapidly to an asymptotic state, while the variation of the antidotal, against time, is not significant. It is also, remarked that the highest values of SIR correspond to the smallest fractional order in both two cases. We think that the results, obtained here, are consistent with those expected in studying an epidemic-antivirus NLDS.