In this paper, we propose a stochastic SIR epidemic model with vertical transmission and varying total population size. Firstly, we prove the existence and uniqueness of the global positive solution for the stochastic model. Secondly, we establish three thresholds λ1,\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _{1},$$\end{document}λ2\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _{2} $$\end{document} and λ3\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _{3}$$\end{document} of the model. The disease will die out when λ1<0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _{1}<0 $$\end{document} and λ2<0,\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _{2}<0,$$\end{document} or λ1>0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _{1}>0$$\end{document} and λ3<0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _{3}<0$$\end{document}, but the disease will persist when λ1<0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _{1}<0$$\end{document} and λ2>0,\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _{2}>0,$$\end{document} or λ1>0\documentclass[12pt]{minimal}
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\begin{document}$$ \lambda _{1}>0$$\end{document} and λ3>0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _{3}>0$$\end{document} and the law of the solution converge to a unique invariant measure. Moreover, we find that when λ1<0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _{1}<0$$\end{document} some stochastic perturbations can increase the threshold λ2\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _{2}$$\end{document}, while others can decrease the threshold λ2\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _{2}$$\end{document}. That is, some stochastic perturbations enhance the spread of the disease, but others are just the opposite. On the other hand, when λ1>0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _{1}>0$$\end{document}, some stochastic perturbations increase or decrease the threshold λ3\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _{3}$$\end{document} with different parameter sets. Finally, we give some numerical examples to illustrate obtained results.
