Recently, bistable viral infection systems have attracted increased attention. In this paper, we study bistability and robustness for virus infection models with nonmonotonic immune responses in viral infection systems. The results show that the existing transcritical bifurcation undergoes backward or forward bifurcation in viral infection models with nonmonotonic immune responses. Our investigation demonstrates that the backward bifurcation threshold is the elite control threshold. When the immune intensity is greater than the elite control threshold, the virus will be under elite control; when the immune intensity is less than the elite control threshold, the virus may rebound. We also give a new definition of robustness to characterize bistable systems.

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 an SEIRE epidemic model to describe the spread of the epidemic with reinfection. We obtain the important thresholds [Formula: see text] (the basic reproduction number) and [Formula: see text] (a threshold less than one). Our investigations show that when [Formula: see text], the system has an endemic equilibrium, which is globally asymptotically stable. When [Formula: see text], 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 [Formula: see text] is less than [Formula: see text]. 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.