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Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay
1.  School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China 
2.  School of Mathematics and Statistics, Guangxi Colleges and Universities Key Laboratory, of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, Guangxi 537000, China 
3.  School of Mathematics and Statistics, Key Laboratory of Applied Statistics of MOE, Northeast, Normal University, Changchun, Jilin 130024, China 
4.  Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 121589, Saudi Arabia 
5.  College of Science, China University of Petroleum (East China), Qingdao 266580, China 
6.  School of Mathematics and Statistics, Key Laboratory of Applied Statistics of MOE, Northeast, Normal University, Changchun, Jilin 130024, China 
7.  Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 121589, Saudi Arabia 
8.  Department of Mathematics, QuaidiAzam University 45320, Islamabad 44000, Pakistan 
9.  Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 121589, Saudi Arabia 
In this paper, we consider two SEIR epidemic models with distributed delay in random environments. First of all, by constructing a suitable stochastic Lyapunov function, we obtain the existence of stationarity of the positive solution to the stochastic autonomous system. Then we establish sufficient conditions for extinction of the disease. Finally, by using Khasminskii's theory of periodic solutions, we prove that the stochastic nonautonomous epidemic model admits at least one nontrivial positive Tperiodic solution under a simple condition.
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J. Artalejo, A. Economou and M. LopezHerrero, The stochastic SEIR model before extinction: Computational approaches, Appl. Math. Comput., 265 (2015), 10261043. Google Scholar 
[3] 
Z. Bai and Y. Zhou, Existence of two periodic solutions for a nonautonomous SIR epidemic model, Appl. Math. Model., 35 (2011), 382391. Google Scholar 
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E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model, Mathematical Ecology, World Scientific, Teaneck, NJ, (1988), 317342. Google Scholar 
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Z. Feng, W. Huang and C. C. Castillo, Global behavior of a multigroup SIS epidemic model with age structure, J. Diff. Equ., 218 (2005), 292324. Google Scholar 
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R. Z. Has'minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1980. Google Scholar 
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W. Huang, K. L. Cooke and C. C. Castillo, Stability and bifurcation for a multiplegroup model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math., 52 (1992), 835854. Google Scholar 
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L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Diff. Equ., 217 (2005), 2653. Google Scholar 
[10] 
C. Ji, D. Jiang, Q. Yang and N. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica, 48 (2012), 121131. Google Scholar 
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D. Jiang, J. Yu, C. Ji and N. Shi, Asymptotic behavior of global positive solution to a stochastic SIR model, Math. Comput. Model., 54 (2011), 221232. Google Scholar 
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L. Jódar, R. J. Villanueva and A. Arenas, Modeling the spread of seasonal epidemiological diseases: theory and applications, Math. Comput. Model., 48 (2008), 548557. Google Scholar 
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R. Khasminskii, Stochastic Stability of Differential Equations, 2^{nd} edition, SpringerVerlag, Berlin, Heidelberg, 2012. Google Scholar 
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C. Koide and H. Seno, Sex ratio features of twogroup SIR model for asymmetric transmission of heterosexual disease, Math. Comput. Model., 23 (1996), 6791. Google Scholar 
[15] 
A. Lahrouz and A. Settati, Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation, Appl. Math. Comput., 233 (2014), 1019. Google Scholar 
[16] 
A. Lahrouz, L. Omari, D. Kiouach and A. Belmaâti, Complete global stability for an SIRS epidemic model with generalized nonlinear incidence and vaccination, Appl. Math. Comput., 218 (2012), 65196525. Google Scholar 
[17] 
D. Li and D. Xu, Periodic solutions of stochastic delay differential equations and applications to Logistic equation and neural networks, J. Korean Math. Soc., 50 (2013), 11651181. Google Scholar 
[18] 
M. Y. Li, Z. Shuai and C. Wang, Global stability of multigroup epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 3847. Google Scholar 
[19] 
Y. Lin, D. Jiang and S. Wang, Stationary distribution of a stochastic SIS epidemic model with vaccination, Physica A, 394 (2014), 187197. Google Scholar 
[20] 
Y. Lin, D. Jiang and T. Liu, Nontrivial periodic solution of a stochastic epidemic model with seasonal variation, Appl. Math. Lett., 45 (2015), 103107. Google Scholar 
[21] 
Q. Liu and Q. Chen, Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence, Physica A, 428 (2015), 140153. Google Scholar 
[22] 
Q. Liu, D. Jiang, N. Shi, T. Hayat and A. Alsaedi, Nontrivial periodic solution of a stochastic nonautonomous SISV epidemic model, Physica A, 462 (2016), 837845. Google Scholar 
[23] 
Z. Liu, Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates, Nonlinear Anal. Real World Appl., 14 (2013), 12861299. Google Scholar 
[24] 
X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, 2008. Google Scholar 
[25] 
P. Witbooi, Stability of an SEIR epidemic model with independent stochastic perturbations, Physica A, 392 (2013), 49284936. Google Scholar 
[26] 
Q. Yang, D. Jiang, N. Shi and C. Ji, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. Math. Anal. Appl., 388 (2012), 248271. Google Scholar 
[27] 
Q. Yang and X. Mao, Extinction and recurrence of multigroup SEIR epidemic models with stochastic perturbations, Nonlinear Anal. Real World Appl., 14 (2013), 14341456. Google Scholar 
[28] 
C. Yuan, D. Jiang, D. O'Regan and R. Agarwal, Stochastically asymptotically stability of the multigroup SEIR and SIR models with random perturbation, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 25012516. Google Scholar 
[29] 
Y. Zhao and D. Jiang, The asymptotic behavior and ergodicity of stochastically perturbed SVIR epidemic model, Int. J. Biomath., 9 (2016), 1650042 (14 pages). Google Scholar 
[30] 
Y. Zhao and D. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Appl. Math. Comput., 243 (2014), 718727. Google Scholar 
[31] 
Y. Zhou, W. Zhang and S. Yuan, Survival and stationary distribution of a SIR epidemic model with stochastic perturbations, Appl. Math. Comput., 244 (2014), 118131. Google Scholar 
[32] 
C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 11551179. Google Scholar 
show all references
References:
[1] 
L. Arnold, W. Horsthemke and J. Stucki, The influence of external real and white noise on the LotkaVolterra model, J. Biomedical, 21 (1979), 451471. Google Scholar 
[2] 
J. Artalejo, A. Economou and M. LopezHerrero, The stochastic SEIR model before extinction: Computational approaches, Appl. Math. Comput., 265 (2015), 10261043. Google Scholar 
[3] 
Z. Bai and Y. Zhou, Existence of two periodic solutions for a nonautonomous SIR epidemic model, Appl. Math. Model., 35 (2011), 382391. Google Scholar 
[4] 
E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model, Mathematical Ecology, World Scientific, Teaneck, NJ, (1988), 317342. Google Scholar 
[5] 
R. Durrett, Stochastic Calculus, CRC Press, 1996. Google Scholar 
[6] 
Z. Feng, W. Huang and C. C. Castillo, Global behavior of a multigroup SIS epidemic model with age structure, J. Diff. Equ., 218 (2005), 292324. Google Scholar 
[7] 
R. Z. Has'minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1980. Google Scholar 
[8] 
W. Huang, K. L. Cooke and C. C. Castillo, Stability and bifurcation for a multiplegroup model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math., 52 (1992), 835854. Google Scholar 
[9] 
L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Diff. Equ., 217 (2005), 2653. Google Scholar 
[10] 
C. Ji, D. Jiang, Q. Yang and N. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica, 48 (2012), 121131. Google Scholar 
[11] 
D. Jiang, J. Yu, C. Ji and N. Shi, Asymptotic behavior of global positive solution to a stochastic SIR model, Math. Comput. Model., 54 (2011), 221232. Google Scholar 
[12] 
L. Jódar, R. J. Villanueva and A. Arenas, Modeling the spread of seasonal epidemiological diseases: theory and applications, Math. Comput. Model., 48 (2008), 548557. Google Scholar 
[13] 
R. Khasminskii, Stochastic Stability of Differential Equations, 2^{nd} edition, SpringerVerlag, Berlin, Heidelberg, 2012. Google Scholar 
[14] 
C. Koide and H. Seno, Sex ratio features of twogroup SIR model for asymmetric transmission of heterosexual disease, Math. Comput. Model., 23 (1996), 6791. Google Scholar 
[15] 
A. Lahrouz and A. Settati, Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation, Appl. Math. Comput., 233 (2014), 1019. Google Scholar 
[16] 
A. Lahrouz, L. Omari, D. Kiouach and A. Belmaâti, Complete global stability for an SIRS epidemic model with generalized nonlinear incidence and vaccination, Appl. Math. Comput., 218 (2012), 65196525. Google Scholar 
[17] 
D. Li and D. Xu, Periodic solutions of stochastic delay differential equations and applications to Logistic equation and neural networks, J. Korean Math. Soc., 50 (2013), 11651181. Google Scholar 
[18] 
M. Y. Li, Z. Shuai and C. Wang, Global stability of multigroup epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 3847. Google Scholar 
[19] 
Y. Lin, D. Jiang and S. Wang, Stationary distribution of a stochastic SIS epidemic model with vaccination, Physica A, 394 (2014), 187197. Google Scholar 
[20] 
Y. Lin, D. Jiang and T. Liu, Nontrivial periodic solution of a stochastic epidemic model with seasonal variation, Appl. Math. Lett., 45 (2015), 103107. Google Scholar 
[21] 
Q. Liu and Q. Chen, Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence, Physica A, 428 (2015), 140153. Google Scholar 
[22] 
Q. Liu, D. Jiang, N. Shi, T. Hayat and A. Alsaedi, Nontrivial periodic solution of a stochastic nonautonomous SISV epidemic model, Physica A, 462 (2016), 837845. Google Scholar 
[23] 
Z. Liu, Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates, Nonlinear Anal. Real World Appl., 14 (2013), 12861299. Google Scholar 
[24] 
X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, 2008. Google Scholar 
[25] 
P. Witbooi, Stability of an SEIR epidemic model with independent stochastic perturbations, Physica A, 392 (2013), 49284936. Google Scholar 
[26] 
Q. Yang, D. Jiang, N. Shi and C. Ji, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. Math. Anal. Appl., 388 (2012), 248271. Google Scholar 
[27] 
Q. Yang and X. Mao, Extinction and recurrence of multigroup SEIR epidemic models with stochastic perturbations, Nonlinear Anal. Real World Appl., 14 (2013), 14341456. Google Scholar 
[28] 
C. Yuan, D. Jiang, D. O'Regan and R. Agarwal, Stochastically asymptotically stability of the multigroup SEIR and SIR models with random perturbation, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 25012516. Google Scholar 
[29] 
Y. Zhao and D. Jiang, The asymptotic behavior and ergodicity of stochastically perturbed SVIR epidemic model, Int. J. Biomath., 9 (2016), 1650042 (14 pages). Google Scholar 
[30] 
Y. Zhao and D. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Appl. Math. Comput., 243 (2014), 718727. Google Scholar 
[31] 
Y. Zhou, W. Zhang and S. Yuan, Survival and stationary distribution of a SIR epidemic model with stochastic perturbations, Appl. Math. Comput., 244 (2014), 118131. Google Scholar 
[32] 
C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 11551179. Google Scholar 
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