Abstract:

In this talk, we formulate an epidemic model with age of vaccination and generalized nonlinear incidence rate, where total population size consists of the susceptible, the vaccinated, the infected and the removed. We then reach a stochastic SVIR model when the fluctuation is introduced into transmission rate. By using Ito's formula and Lyapunov methods, we firstly show that stochastic epidemic model admits a unique global positive solution with positive initial values. Then, sufficient conditions to stochastic epidemic model are derived. Moreover, the threshold that tells diseases spread or not is derived. If intensity of white noise is small enough and $\tilde{R}_0<1$, then diseases eventually become extinct with negative exponential rate. If $\tilde{R}_0>1$, then diseases are weak permanent. The persistence in the mean for the infected is also obtained when indicator $\hat{R}_0>1$, which means diseases will prevail in a long run. As a consequence, several illustrative examples are separately carried out with numerical simulations to support the main results.
