Abstract:

Authors discussed the stochastic SIQS epidemic system with environmental white noise. When the basic reproduction number was not more than 1, we gave the asymptotic behavior of the stochastic system around the diseasefree equilibrium point of the deterministic model by stochastic Lyapunov analysis method. The result shows that the disease will die out when the white noises are small. When the basic reproduction number is more than 1, it is shown that
 there is a stationary distribution based on the Hasminskii’s ergodic theory and it is ergodic, which reveals that the disease will prevail under some conditions.

Key words: stochastic differential equation, extinction, ergodicity, Lyapunov function, stationary distribution