On pulse vaccine strategy in a periodic stochastic SIR epidemic model

A periodic stochastic SIR epidemic model with pulse vaccination is studied. The system has global positive solutions and under some conditions it admits a unique positive periodic disease-free solution, which is globally exponentially stable in mean square. The mathematical expectation and variance of the positive periodic solution are obtained. Two threshold parameters R1 and R2 (R1>R2) are identified; if R1<1, the susceptible will be persistent in the mean and the disease will go to extinction; if R2>1, the susceptible and the disease will be weakly persistent in the mean. We show that by repeatedly vaccinating the susceptible population in series of pulses, it is possible to eradicate the infective from the entire model population in the random environment.