Backward bifurcation in a discrete SIS model with vaccination

A simple discrete SIS model with vaccination is proposed. Its dynamics depend on a lumped parameter R_vac. The model exhibits the classical threshold behavior when vaccination is totally ineffective. When vaccination is partially effective, a backward transcritical bifurcation may occur at R _vac = 1. In this case, the model also undergoes a saddle-node bifurcation at certain parameter values when R_vac < 1. The disease can persist for R_vac > 1 and can be eradicated for R _vac < 1 if a forward transcritical bifurcation occurs at R _vac = 1. However, the disease may persist even when R_vac < 1 if a backward bifurcation occurs at R_vac = 1.