Abstract
We state and study a discrete SIR epidemic model with bilinear incidence rate and constant recovery. We obtain conditions for the existence of the disease-free equilibrium and endemic equilibria. Theoretic analysis shows that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less than unity, and the numerical simulations illustrate that it is asymptotically stable when the number is greater than unity. We also obtain conditions for the stability of the endemic equilibria. More attention is paid to the existence of the saddle-node bifurcation, the flip bifurcation, the 1:1 resonance, and the Neimark–Sacker bifurcation. We obtain sufficient conditions for those bifurcations. Our numerical simulations demonstrate our theoretical results and the complexity of the model.
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