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Threshold of a Stochastic Delayed SIR Epidemic Model with Saturation Incidence

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Mathematical Analysis and Applications in Modeling (ICMAAM 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 302))

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Abstract

A stochastic SIR epidemic model with time delay and saturation incidence is formulated in this paper. We show that the disease dynamics of the stochastic delayed SIR model can be governed by its related threshold \(R_0^S\), whose value completely determines the disease to go extinct and prevail for any size of the white noise. The results are improved and the method is simpler than the previously-known literature. And the related results recover the known results in the earlier literature as special cases. The presented results are illustrated by numerical simulations.

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References

  1. Ji, C., Jiang, D., Shi, N.: The behavior of an SIR epidemic model with stochastic perturbation. Stoch. Anal. Appl. 30, 755–773 (2012)

    Article  MathSciNet  Google Scholar 

  2. Gray, A., Greenhalgh, D., Hu, L., Mao, X., Pan, J.F.: A stochastic differential equation SIS epidemic model. SIAM J. Appl. Math. 71, 876–902 (2011)

    Article  MathSciNet  Google Scholar 

  3. Jiang, D., Yu, J., Ji, C., Shi, N.: Asymptotic behavior of global positive solution to a stochastic SIR model. Math. Comput. Model. 54, 221–232 (2011)

    Article  MathSciNet  Google Scholar 

  4. Artalejo, J., Economou, A., Lopez-Herrero, M.: On the number of recovered individuals in the SIS and SIR stochastic epidemic models. Math. Biosci. 228, 45–55 (2010)

    Article  MathSciNet  Google Scholar 

  5. Lu, Q.: Stability of SIRS system with random perturbations. Phys. A 388, 3677–3686 (2009)

    Article  MathSciNet  Google Scholar 

  6. Zhao, Y., Jiang, D., O’Regan, D.: The extinction and persistence of the stochastic SIS epidemic model with vaccination. Phys. A 392, 4916–4927 (2009)

    Article  MathSciNet  Google Scholar 

  7. Zhao, Y., Jiang, D.: The threshold of a stochastic SIS epidemic model with vaccination. Appl. Math. Comput. 243, 718–727 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Ji, C., Jiang, D.: Threshold behaviour of a stochastic SIR model. Appl. Math. Model. 38, 5067–5079 (2014)

    Article  MathSciNet  Google Scholar 

  9. Lin, Y., Jiang, D., Wang, S.: Stationary distribution of a stochastic SIS epidemic model with vaccination. Phys. A 394, 187–197 (2014)

    Article  MathSciNet  Google Scholar 

  10. Lahrouz, A., Settati, A.: Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation. Appl. Math. Comput. 233, 10–19 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Lu, Q., Chen, Q., Jiang, D.: The threshold of a stochastic delayed SIR epidemic model with temporary immunity. Phys. A 450, 115–125 (2016)

    Article  MathSciNet  Google Scholar 

  12. Yang, Q., Jiang, D., Shi, N.: The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence. J. Math. Anal. Appl. 388, 248–271 (2012)

    Article  MathSciNet  Google Scholar 

  13. Zhao, D.: Study on the threshold of a stochastic SIR epidemic model and its extensions. Commun. Nolinerar Sci. Numer. Simulat. 38, 172–177 (2016)

    Article  MathSciNet  Google Scholar 

  14. Mao, X.: Stochastic Differential Equations and Applications, 2nd edn. Horwood, Chichester, UK (2008)

    Book  Google Scholar 

  15. Lipster, R.: A strong law of large numbers for local martingale. Stochastics 3, 217–228 (1980)

    Article  MathSciNet  Google Scholar 

  16. Zhao, Y., Jiang, D.: The threshold of a stochastic SIRS epidemic model with saturated incidence. Appl. Math. Lett. 34, 90–93 (2014)

    Article  MathSciNet  Google Scholar 

  17. Kyrychko, Y., Blyuss, K.: Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate. Nonlinear Anal. Real World Appl. 6, 495–507 (2005)

    Article  MathSciNet  Google Scholar 

  18. Naresh, R., Tripathi, A., Tchuenche, J., Sharma, D.: Stability analysis of a time delayed SIR epidemic model with nonlinear incidence rate. Comput. Math. Appl. 58, 348–359 (2009)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Shanghai University of Medicine and Health Sciences, Shanghai, 201318, China

    Yanli Zhou

Authors
  1. Yanli Zhou
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Correspondence to Yanli Zhou .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Jadavpur University, Kolkata, West Bengal, India

    Priti Kumar Roy

  2. College of Science, Beijing Technology and Business University, Beijing, China

    Xianbing Cao

  3. College of Mathematics and Information Science, Henan Normal University, Xinxiang, Xinjiang, China

    Xue-Zhi Li

  4. Department of Mathematics, Jadavpur University, Kolkata, West Bengal, India

    Pratulananda Das

  5. Harish-Chandra Research Institute, Allahabad, Uttar Pradesh, India

    Satya Deo

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Zhou, Y. (2020). Threshold of a Stochastic Delayed SIR Epidemic Model with Saturation Incidence. In: Roy, P., Cao, X., Li, XZ., Das, P., Deo, S. (eds) Mathematical Analysis and Applications in Modeling. ICMAAM 2018. Springer Proceedings in Mathematics & Statistics, vol 302. Springer, Singapore. https://doi.org/10.1007/978-981-15-0422-8_27

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