  The main purpose of this paper is to study the local dynamics and
bifurcations of a discrete-time SIR epidemiological model. The existence and
stability of disease-free and endemic fixed points are investigated along with
a fairly complete classification of the systems bifurcations, in particular, a
complete analysis on local stability and codimension 1 bifurcations in the
parameter space. Sufficient conditions for positive trajectories are given. The
existence of a 3-cycle is shown, which implies the existence of cycles of
arbitrary length by the celebrated Sharkovskii's theorem. Generacity of some
bifurcations is examined both analytically and through numerical computations.
Bifurcation diagrams along with numerical simulations are presented. The system
turns out to have both rich and interesting dynamics.
