Backward bifurcations in discrete-time epidemic models with vaccination

We develop a discrete-time Susceptible-Infectious-Susceptible model with Vaccination (SISV) in which vaccination reduces but does not necessarily eliminate the risk of infection. After establishing the stability of the disease-free equilibrium (DFE), we apply an extension of the Fundamental Bifurcation Theorem to study the endemic dynamics arising when the DFE loses stability at the threshold quantity $R_0=1$ , where $R_0$ denotes the basic reproduction number. We find that, as with its continuous-time counterpart, a backward bifurcation may occur in this model leading to bistable dynamics provided that the vaccine does not provide complete protection from infection and infected individuals are able to recover from the disease. For comparison purposes, we also present a second model formulation based on a different ordering of events, an issue that does not appear in continuous-time models. Specifically, we swap the order that vaccination and disease transmission occur within a unit of time. In contrast to the first model, we show that this alternative formulation may produce a backward bifurcation even when vaccination is completely effective.