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This paper discusses a randomized non-autonomous logistic equationstandard Brownian motion. In [D.Q. Jiang, N.Z. Shi, A note on non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl. 303 (2005) 164-172], the authors show that E[1/N (t)] has a unique positive T -periodic solution E[1/N p (t)] provided a(t), b(t) and α(t) are continuous T -periodic functions, a(t) > 0, b(t) > 0 and T 0 [a(s) − α 2 (s)] ds > 0. We show that this equation is stochastically permanent and the solution N p (t) is globally attractive provided a(t), b(t) and α(t) are continuous T -periodic functions, a(t) > 0, b(t) > 0 and min t∈[0,T ] a(t) > max t∈[0,T ] α 2 (t). By the way, the similar results of a generalized non-autonomous logistic equation with random perturbation are yielded.

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In this paper, we present a DI SIR epidemic model with two categories stochastic perturbations. The long time behavior of the two stochastic systems is studied. Mainly, we show how the solution goes around the infection-free equilibrium and the endemic equilibrium of deterministic system under different conditions.

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