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A discrete-time SIS epidemic model with vaccination is presented and studied. The model includes deaths due to disease and the total population size is variable. First, existence and positivity of the solutions are discussed and equilibria of the model and basic reproduction number are obtained. Next, the stability of the equilibria is studied and conditions of stability are obtained in terms of the basic reproduction number R 0. Also, occurrence of the fold bifurcation, the flip bifurcation, and the Neimark-Sacker bifurcation is investigated at equilibria. In addition, obtained results are numerically discussed and some diagrams for bifurcations, Lyapunov exponents, and solutions of the model are presented.

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A discrete-time SIS epidemic model with vaccination is introduced and formulated by a system of difference equations. Some necessary and sufficient conditions for asymptotic stability of the equilibria are obtained. Furthermore, a sufficient condition is also presented. Next, bifurcations of the model including transcritical bifurcation, period-doubling bifurcation, and the Neimark-Sacker bifurcation are considered. In addition, these issues will be studied for the corresponding model with constant population size. Dynamics of the model are also studied and compared in detail with those found theoretically by using bifurcation diagrams, analysis of eigenvalues of the Jacobian matrix, Lyapunov exponents and solutions of the models in some examples.