Forces of Infection Allowing for Backward Bifurcation in an Epidemic Model with Vaccination and Treatment

We consider an epidemic model for the dynamics of a vaccine-preventable disease, which incorporates the treatment and an imperfect vaccine given to susceptible individuals. We show that in spite of the simple structure of the model, a backward bifurcation may always occur if the treatment rate is above a threshold value. This occurs regardless of the specific form of the force of infection, which is only required to be infinitesimal of the same order of the size of the infectious compartment I, as I -> 0. This includes many commonly used functionals, as the linear, the monotone saturating Michaelis-Menten and the non-monotone force of infection used to represent the 'psychological effect'.