Global dynamics of an epidemic model with a two-threshold policy

This work proposes a non-smooth epidemic model with a two-threshold control strategy. Under the two-threshold policy, social distancing is suggested when the infection level exceeds a certain threshold, and contact tracing is implemented and isolation rate is improved when the infection level exceeds a higher threshold. The global dynamics of the non-smooth system is investigated and it reveals that the disease-free equilibrium, or one of the three endemic equilibria of the subsystems, or one of the two pseudo-equilibria may be globally asymptotically stable, which depends on the two threshold levels and the intensity of control measures. The global stability of the pseudo-equilibrium indicates that the disease could be maintained at a previously given level by choosing the two thresholds properly. It is also obtained that at low thresholds and high intensity of control measures, the peak value is low and peak time is late for the unisolated infections and daily new infections, and the number of cumulative infections is small. However, the effects of control measures are very limited both on the final trend of the epidemic and the cumulative infections if threshold levels are too high.