Persistence and extinction for an age-structured stochastic SVIR epidemic model with generalized nonlinear incidence rate

We formulate an epidemic model with age of vaccination and generalized nonlinear incidence rate, where the total population consists of the susceptible, the vaccinated, the infected and the removed. We then reach a stochastic SVIR model when the fluctuation is introduced into the transmission rate. By using Itô's formula and Lyapunov methods, we first show that the stochastic epidemic model admits a unique global positive solution with the positive initial value. We then obtain the sufficient conditions of the stochastic epidemic model. Moreover, the threshold tells the disease spreads or not is derived. If the intensity of the white noise is small enough and R˜0<1, then the disease eventually becomes extinct with negative exponential rate. If R˜0>1, then the disease is weakly permanent. The persistence in the mean of the infected is also obtained when the indicator Rˆ0>1, which means the disease will prevail in a long run. As a consequence, several illustrative examples are separately carried out with numerical simulations to support the main results of this paper.