Persistence and Extinction of a Stochastic Epidemic Model with Delay and Proportional Vaccination

Shu-qi GAN, Feng-ying WEI


A type of susceptible-infection-vaccinated epidemic model with proportional vaccination and generalized nonlinear rate is formulated and investigated in the paper. We show that the stochastic epidemic model admits a unique and global positive solution with probability one when constructing a proper C2 -function therewith. Then the sufficient condition that guarantees the diseases vanish is derived when the indicator 1 0 R  . Further, if 1 0 R  , then we obtain that the solution is weakly permanent with probability one. And we also derived the sufficient conditions of the persistence in the mean for the susceptible and infected under the condition 0 R 1.


Stochastic epidemic model, Time delay, Extinction, Persistence in the mean, Threshold


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