In this paper we consider stochastic stability, namely, asymptotic stability in mean, asymptotic mean-square stability, asymptotic stability a.s., exponential p-stability and stability with stochastic Lyapunov fucntion for vector stochastic differential equations. We apply these results to the stochastic epidemic models induced by bacteriophages in the marine bacteria populations. The novelty of the paper consists of new stability results for vector SDE and their applications to a new stochastic epidemic model.

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We consider an averaging principle for the endemic SIR model in a semi-Markov random media. Under stationary conditions of a semi-Markov media we show that the perturbed endemic SIR model converges to the classic endemic SIR model with averaged coefficients. Numerical toy examples and their interpretations are also presented for two-state Markov and semi-Markov chains. We also discuss two numerical examples involving real data: 1) Dengue Fever Disease (Indonesia and Malaysia (2009)) and 2) Cholera Outbreak in Zimbabwe (2008-2009). Novelty of the paper consists in studying of an endemic SIR model in semi-Markov random media and in implementations and interpretations of the results through numerical toy examples and discussion of numerical examples with real data.

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