The impact of the Caputo fractional difference operator on the dynamical behavior of a discrete-time SIR model for influenza A virus

Since there are few studies that deal with the fractional-order discrete-time epidemic models, thispaper presents a new fractional-order discrete-time SIR epidemic model that is constructed based onthe Caputo fractional difference operator. The effect of the fractional orders on the global dynamics ofthe SIR model is analyzed. In particular, the existence and stability of equilibrium points of the modelare presented. Furthermore, we investigate the qualitative dynamical properties of the SIR model forboth commensurate and incommensurate fractional orders using powerful nonlinear tools such asphase attractors, bifurcation diagrams, maximum Lyapunov exponent, chaos diagrams, and 0-1 test.In addition, the complexity of the discrete model is measured via the spectral entropy complexityalgorithm. Further, an active controller is designed to stabilize the chaotic dynamics of the fractional-order SIR model. Finally, the suggested model isfitted with real data to show the accuracy of thecurrent stability study. Our goal was achieved by confirming that the proposed SIR model can displaya variety of epidiomologically observed states, including stable, periodic, and chaotic behaviors. Thefindings suggest that any change in parameter values or fractional orders could lead to unpredictablebehavior. As a result, there is a need for additional research on this topic.