Dynamics of an SEIRV endemic model: effects of reinfection.

The threshold interpretation of R0 can fail to describe elimination in epidemic models with imperfect vaccination and reinfection. We study an SEIRV model in which vaccinated individuals remain partially susceptible and recovered individuals may be reinfected. The main novelty is an explicit threshold account of how reinfection can sustain endemic equilibria even when the vaccinated-population reproduction number satisfies R0<1. We derive the disease-free equilibrium, R0, and a sufficient condition for convergence to the disease-free state, showing why local disease-free stability and global elimination are distinct. We reduce the endemic-equilibrium problem to a scalar equation and identify a reinfection threshold β2∗ determining the local direction of bifurcation at R0=1. When β2>β2∗, we prove a lower critical threshold R0C<1 separating elimination from subcritical endemic persistence. We also give an efficient-vaccine approximation and general numerical algorithm for computing R0C. Simulations using an illustrative Omicron-era parameterisation show how these thresholds organise the model dynamics.