Codimension-Two Bifurcation Analysis and Global Dynamics of a Discrete Epidemic Model

In this paper, we study the global dynamics, boundedness, existence of invariant intervals, and identification of codimension-two bifurcation sets with detailed bifurcation analysis at the epidemic fixed point of a discrete epidemic model. More precisely, under definite parametric conditions, it is proved that every positive solution of the discrete epidemic model is bounded, and furthermore, we have also constructed the invariant interval. By the linear stability theory, we have derived the sufficient condition, as well as the necessary and sufficient condition(s) under which fixed points obey certain local dynamical characteristics. We also gave the global analysis at fixed points and proved that both disease-free and epidemic fixed points become globally stable under certain conditions and parameters. Next, in order to study the two-parameter bifurcations of the discrete epidemic model at the epidemic fixed point, we first identified the two-parameter bifurcation sets, and then a detailed two-parameter bifurcation analysis is given by the bifurcation theory and affine transformations. Furthermore, we have given the biological interpretations of the theoretical findings. Finally, numerical simulation validated the theoretical results.