The Dynamics of a Discrete Fractional-Order Logistic Growth Model with Infectious Disease

In this paper, we study the dynamics of a discrete fractional-order logistic growth model with infectious disease. We obtain the discrete model by applying the piecewise constant arguments to the fractional-order model. This model contains three fixed points namely the origin point, the disease-free point, and the endemic point. We confirm that the origin point is always exists and unstable, the disease-free point is always exists and conditionally stable, and the endemic point is conditionally exists and stable. We also investigate the existence of forward, period-doubling, and Neimark-Sacker bifurcation. The numerical simulations are also presented to confirm the analytical results. We also show numerically the existence of period-3 solution which leads to the occurrence of chaotic behavior.