Dynamics of an SEIRV endemic model: effects of reinfection

The threshold interpretation of $R_0$ 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 $R_0<1$. We derive the disease-free equilibrium, $R_0$, 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 ${\beta }_2^{\ast }$ determining the local direction of bifurcation at $R_0=1$. When ${\beta }_2>{\beta }_2^{\ast }$, we prove a lower critical threshold $R_0^C<1$ separating elimination from subcritical endemic persistence. We also give an efficient-vaccine approximation and general numerical algorithm for computing $R_0^C$. Simulations using an illustrative Omicron-era parameterisation show how these thresholds organise the model dynamics.