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Periodic Solution and Ergodic Stationary Distribution of SEIS Dynamical Systems with Active and Latent Patients

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Abstract

This paper proposes two new stochastic non-autonomous SEIS epidemic dynamical models with latent and active patients. For the non-autonomous periodic system, we first obtain the sufficient conditions for the existence of nontrivial positive periodic solution by constructing some suitable stochastic Lyapunov functions with regime switching. Then we prove the existence of ergodic stationary distribution of the stochastic SEIS epidemic model with Markov conversion by using the stochastic qualitative theory. At last, some rigorous computer numerical simulations illustrate our theoretical results. The results show that the large random disturbance can destroy the periodic solution and the ergodic stationary distribution.

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Acknowledgements

This work was supported by the SDUST Research Fund (2014TDJH102), and the Research Fund for the Taishan Scholar Project of Shandong Province of China.

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Authors and Affiliations

  1. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, People’s Republic of China

    Haokun Qi, Xiaona Leng & Xinzhu Meng

  2. State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao, 266590, People’s Republic of China

    Xinzhu Meng

  3. Department of Mathematics, Swinburne University of Technology, Melbourne, VIC, 3122, Australia

    Tonghua Zhang

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  1. Haokun Qi
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  2. Xiaona Leng
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  3. Xinzhu Meng
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  4. Tonghua Zhang
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Correspondence to Xinzhu Meng.

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Qi, H., Leng, X., Meng, X. et al. Periodic Solution and Ergodic Stationary Distribution of SEIS Dynamical Systems with Active and Latent Patients. Qual. Theory Dyn. Syst. 18, 347–369 (2019). https://doi.org/10.1007/s12346-018-0289-9

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  • DOI: https://doi.org/10.1007/s12346-018-0289-9

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