Dynamics of stochastic SEIS epidemic model with varying population size

• Stochasticity is introduced into an SEIS model with varying population size.
• Some simple conditions on extinction and persistent in the mean of the disease with probability one are shown.
• When the noises are small, there is a stationary distribution to stochastic model.

We introduce the stochasticity into a deterministic model which has state variables susceptible–exposed–infected with varying population size in this paper. The infected individuals could return into susceptible compartment after recovering. We show that the stochastic model possesses a unique global solution under building up a suitable Lyapunov function and using generalized Itô’s formula. The densities of the exposed and infected tend to extinction when some conditions are being valid. Moreover, the conditions of persistence to a global solution are derived when the parameters are subject to some simple criteria. The stochastic model admits a stationary distribution around the endemic equilibrium, which means that the disease will prevail. To check the validity of the main results, numerical simulations are demonstrated as end of this contribution.