  Recent evidences show that individuals who recovered from COVID-19 can be
reinfected. However, this phenomenon has rarely been studied using mathematical
models. In this paper, we propose a SEIRE epidemic model to describe the spread
of the epidemic with reinfection. We obtain the important thresholds $R_0$ (the
basic reproduction number) and Rc (a threshold less than one). Our
investigations show that when $R_0 > 1$, the system has an endemic equilibrium,
which is globally asymptotically stable. When $R_c < R_0 < 1$, the epidemic
system exhibits bistable dynamics. That is, the system has backward bifurcation
and the disease cannot be eradicated. In order to eradicate the disease, we
must ensure that the basic reproduction number $R_0$ is less than $R_c$. The
basic reinfection number is obtained to measure the reinfection force, which
turns out to be a new tipping point for disease dynamics. We also give
definition of robustness, a new concept to measure the difficulty of completely
eliminating the disease for a bistable epidemic system. Numerical simulations
are carried out to verify the conclusions.
