Analysis of a hybrid SIR model combining the fixed-moments pulse interventions with susceptibles-triggered threshold policy

In our study, a novel hybrid mathematical model is proposed to describe the susceptibles-triggered vaccination and isolation strategies at fixed monitoring moments, which is more in line with the general rule of the implementation of control measures of infectious diseases in practice. We theoretically investigate the complex dynamic behaviors with the appearance of (k+m)T-periodic solutions. Firstly, we investigate the existence and stability of the order-1 (k+1)T-disease-free periodic solution with nonnegative integer k and the order-m (1+m)T-disease-free periodic solution with m being any positive integer. Furthermore, we study the existence and stability of (3+2)T-disease-free periodic solution. These theoretical results indicate that new kinds of disease-free periodic solutions can be studied in depth. We also show other kinds of the high-order disease-free periodic solutions through the numerical simulations. In addition, through studying the bifurcation near the disease-free periodic solution, we discuss the existence and stability of the endemic periodic solutions. Numerical simulations show the complex dynamics, such as the bistability of the endemic equilibrium and an endemic periodic solution or a disease-free periodic solution, bistability of two endemic periodic solutions, and tri-stability of three endemic periodic solutions. Our main results indicate that there exists a critical monitoring period or vaccination rate such that the proposed intervention strategy can successfully control and eliminate infectious diseases. Furthermore, the selection of threshold value is highly maneuverable.