This research presents a new fractional-order discrete-time susceptible-infected-recovered (SIR) epidemic model with vaccination. The dynamical behavior of the suggested model is examined analytically and numerically. Through using phase attractors, bifurcation diagrams, maximum Lyapunov exponent and the 0−1 test, it is verified that the newly introduced fractional discrete SIR epidemic model vaccination with both commensurate and incommensurate fractional orders has chaotic behavior. The discrete fractional model gives more complex dynamics for incommensurate fractional orders compared to commensurate fractional orders. The reasonable range of commensurate fractional orders is between $\gamma$ = 0.8712 and $\gamma$ = 1, while the reasonable range of incommensurate fractional orders is between ${\gamma }_2$ = 0.77 and ${\gamma }_2$ = 1. Furthermore, the complexity analysis is performed using approximate entropy ($ApEn$) and $C_0$ complexity to confirm the existence of chaos. Finally, simulations were carried out on MATLAB to verify the efficacy of the given findings.

For a ∈ (0, 1/2], r ∈ (0, 1), let