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Volume 8, Number 4, 2018, Pages 1061-1084                                                                DOI:10.11948/2018.1061
The dynamic behavior of deterministic and stochastic delayed SIQS model
Xiaobing Zhang,Haifeng Huo,Hong Xiang and Dungang Li
Keywords:Random perturbations, It\^{o}$^{\prime} $s formula, the threshold, time delay.
Abstract:
      In this paper, we present the deterministic and stochastic delayed SIQS epidemic models. For the deterministic model, the basic reproductive number $R_{0}$ is given. Moreover, when $R_{0}<1$, the disease-free equilibrium is globally asymptotical stable. When $R_{0}>1$ and additional conditions hold, the endemic equilibrium is globally asymptotical stable. For the stochastic model, a sharp threshold $\overset{\wedge }{R}_{0}$ which determines the extinction or persistence in the mean of the disease is presented. Sufficient conditions for extinction and persistence in the mean of the epidemic are established. Numerical simulations are also conducted in the analytic results.
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