Vaccination threshold size and backward bifurcation of SIR model with state-dependent pulse control

Depending on the potential susceptible human size, we consider the state-dependent integrated infectious disease control strategies including vaccination, isolation and treatment. Correspondingly, we propose a state-dependent pulse SIR model, in which whether the control measures implemented or not depends on the threshold size of susceptible population. By defining the Poincaré map, we first investigate the existence and global stability of the semi-trivial (or disease free) periodic solution, and the threshold condition is proposed. Further, by employing bifurcation theories of the one-parameter family of maps related to the Poincaré map, we then focus on the bifurcation with respect to the key parameters. The main results reveal that backward bifurcation via transcritical bifurcation or pitchfork bifurcation can occur for all the interesting parameters including isolation rate, vaccination rate, threshold susceptible population size and birth rate. The complex relationships between the basic reproduction number of classical SIR model and the threshold condition of the model with state-dependent pulse control depict that the control strategies related to the four parameters should be carefully designed, otherwise the paradoxical effects could occur and the gains cannot make up for losses. For example, too small vaccination rate will result in an increasing of threshold condition and the number of infected population. Therefore, our results suggest that when the state-dependent feedback control strategy is implemented for infectious disease control, the effective and optimal control program should take the population dynamics, the threshold susceptible population size, vaccination and isolation or treatment rate into consideration.