Stability and codimension 2 bifurcations of a discrete time SIR model

Abstract

Dynamic behaviours of an epidemic model of discrete time SIR type, have been discussed. The existence and stability conditions of fixed points and some of the codim-1 bifurcations of this model are investigated in [34], but we make bifurcation analysis more general than their work. It is shown that SIR model undergoes codimension 1 (codim 1) bifurcations such as transcritical, flip (period doubling), Neimark-Sacker, and codimension 2 (codim 2) bifurcations including resonance 1:2, resonance 1:3 and resonance 1:4. For each bifurcation, normal form coefficients along with its scenario are investigated thoroughly. Bifurcation curves of fixed points are drawn with the aid of numerical continuation. Besides, using numerical simulation, in addition to confirming the results of our analysis, more behavior is extracted from the model, such as the bifurcations of higher iterations like the fourth, the eight, etc. It is observed that the discrete epidemic model has richer dynamic behaviours compared to the continuous one.