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\title{Asymptotic behaviors of stochastic epidemic models with jump-diffusion}
\author{ Nguyen  Thanh Dieu \thanks{ Department of Mathematics, Vinh University,
182 Le Duan, Vinh city, Nghe An province, Vietnam, dieunguyen2008@gmail.com. 
%This research is funded in part by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.03-2017.23.
}\and  Takasu Fugo\thanks{Dept. Environmental Science,
Nara Women's University,
Kita-Uoya Nishimachi, Nara 630-8506, Japan, takasu@es.nara-wu.ac.jp.} \and
 Nguyen Huu Du,\thanks{Corresponding author: Department of Mathematics, Mechanics and
Informatics, Hanoi National University,
 334 Nguyen Trai, Thanh Xuan, Hanoi Vietnam, dunh@vnu.edu.vn.
}  
}
\maketitle
\begin{abstract} In this talk, we classify the asymptotic behavior for a class of stochastic epidemic models represented by
 stochastic differential  systems  where the Brownian motions and L\'{e}vy jumps perturb to the linear terms of each equation.  We  construct a threshold value between permanence and extinction  and develop the ergodicity of the underlying system. It is
shown that  the transition probabilities  converge in total variation norm  to the invariant measure.  Our results can be considered  as a significant contribution in studying the long term behavior of stochastic differential models  because there are no restrictions imposed to the parameters of  models. Techniques used in proving results of this paper are new and   suitable to deal with other stochastic models  in biology where the noises may perturb to nonlinear terms of  equations or with delay equations.
%of the results given by Y. Zhou et al. \cite{ZZ}.
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\end{abstract}
\begin{thebibliography}{}






\bibitem{DDDY}N. T. Dieu, D. H. Nguyen,  N. H. Du, G.  Yin, Classification of asymptotic behavior in a stochastic SIR model. {SIAM J. Appl. Dyn. Syst.} {15} (2016)  no. 2  1062-1084.

\bibitem{DFD}N. T. Dieu, T. Fugo, N. H. Du, Asymptotic behaviors of stochastic epidemic models with jump-diffusion. Appl. Math. Model. 86 (2020), 259--270.

\bibitem{LJHA}Q. Liu,  D. Jiang, N. Shi, T. Hayat,  Dynamics of a stochastic delayed SIR epidemic model with vaccination and double diseases driven by L\'evy jumps,  {Phys. A}  {492} (2018)  2010-2018.


\bibitem{ZZ} Y. Zhou, W. Zhang,  Threshold of a stochastic SIR epidemic model with L\'{e}vy jumps,  {Phys. A}  {446} (2016)  204-216.




\bibitem{D2018} N. T. Dieu,  Asymptotic properties of a stochastic SIR epidemic model with Beddington-DeAngelis incidence rate. J. Dynam. Differential Equations 30 (2018), no. 1, 93--106.




\bibitem{DD} N. H. Du, N. T.  Dieu,  Long-time behavior of an SIR model with perturbed disease transmission coefficient. \emph{Discrete Contin. Dyn. Syst. Ser. B} \textbf{21} (2016), no. 10, 3429-3440.











\end{thebibliography}
	
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