Asymptotic behavior and threshold of a stochastic SIQS epidemic model with vertical transmission and Beddington–DeAngelis incidence

This paper investigates a deterministic and stochastic SIQS epidemic model with vertical transmission and Beddington–DeAngelis incidence. Firstly, for the corresponding deterministic system, the global asymptotic stability of disease-free equilibrium and the endemic equilibrium is proved through the stability theory. Secondly, for the stochastic system, the threshold conditions which decide the extinction or permanence of the disease are derived. By constructing suitable Lyapunov functions, we investigate the oscillation behavior of the stochastic system solution near the endemic equilibrium. The results of this paper show that there exists a great difference between the deterministic and stochastic systems, which implies that the large stochastic noise contributes to inhibiting the spread of disease. Finally, in order to validate the theoretical results, a series of numerical simulations are presented.