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GLOBAL ATTRACTIVITY OF A DELAYED SEIR EPIDEMIC MODEL OF COVID-19 WITH DIFFUSION

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Abstract

Some mathematical epidemic equation of SEIR with diffusion, which appears as a model of COVID-19 for the spread of disease-causing, is treated. The asymptotic properties of the diffusive equation are studied by applying the strong maximum principle, strong fading memory property and some Lyapunov function.

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Basically, our work proceeds within a theoretical and mathematical approach, and we confirm that especially all data in the example quoted during this study are included in this published article in the references below.

References

  1. R.M. Anderson and R.M. May, Population biology of infectious diseases, Part 1, Nature 280 (1979), 361-367.

    Article  Google Scholar 

  2. R.M. Anderson, H. Heesterbeek, D. Klinkenberg and T.D. Hollingsworth, How will country-based mitigation measures influence the course of the COVID-19 epidemic? The Lancet 395 (2020), 931-934.

    Article  Google Scholar 

  3. C. Bommer and S. Vollmer, Average detection rate of SARS-CoV-2 infections has improved since our last estimates but is still as low as nine percent on March 30th. URL: https://www. uni-goettingen.de/en/606540..html (04/09/2020).

  4. F. Brauer and C. Castillo-Chavez, Mathematical Model in Population Biology and Epidemiology, Vol. 2, Springer New York (2012), 3-47.

    MATH  Google Scholar 

  5. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Dordrecht, 1992.

    Book  MATH  Google Scholar 

  6. M. Granovetter, Threshold Models of Collective Behavior, The Sociological Quarterly, 1983.

  7. Y. Hamaya, On the asymptotic behavior of a diffusive epidemic model (AIDS), Nonlinear Analysis 36 (1999), 685-696.

    Article  MathSciNet  MATH  Google Scholar 

  8. Y. Hamaya and T. Arai, Permanence of an SIR epidemic model with diffusion, Nonlinear Stadies 17 (2010), 69-79.

    MathSciNet  MATH  Google Scholar 

  9. Y. Hamaya and K. Saito, Global asymptotic stability of a delayed SIR epidemic model with diffusion, Libertas Mathematica (new series), 36 (2016), 53-72.

    MathSciNet  MATH  Google Scholar 

  10. Y. Hamaya and K. Saito, On the global attractivity of a SEIR epidemic model with diffusion, submitted.

  11. J. Hellewell, S. Abbott, A. Gimma, N. I. Bosse, C. I. Jarvis, T. W. Russell, J. D. Munday, A. J. Kucharski and W. J. Edmunds, Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts, Lancet Glob. Health, 8 (2020), e488-e496.

    Google Scholar 

  12. H. Inaba, Mathematical Models for Demography and Epidemics, Expanded Edition, University of Tokyo Press, 2020.

  13. Johns Hopkins Corona virus Resource Center, 2020, https://coronavirus.jhu.edu/map.html.

  14. W. O. Kermack and A. G. Mckendrick, A contribution to the mathematical theory of epidemics, P. Roy. Soc. A, Math. Phys. Eng. Sci., 115 (1927), 700-721.

  15. N.M. Linton, T. Kobayashi, Y. Yang, K. Hayashi, A.R. Akhmetzhanov, S. Jung, B. Yuan, R. Kinoshita and H. Nishiura, Incubation period and other epidemiological characteristics of 2019 novel coronavirus infections with right truncation: a statistical analysis of publicly available case data, Jour. Clini. Medi. 9, (2020), 538.

    Article  Google Scholar 

  16. Y. Liu, A.A. Gayle, A. Wilder-Smith and J. Rocklov, The reproductive number of COVID-19 is higher compared to SARS coronavirus, Jour. Travel Medi. 27 (2020) taaa021.

  17. S. Murakami and Y. Hamaya, Global attractivity in an integrodifferential equation with diffusion, Differential Equations and Dynamical Systems, 3 (1995), 35-42.

    MathSciNet  MATH  Google Scholar 

  18. N.S. Padhye, Reconstructed diagnostic sensitivity and specificity of the RT-PCR test for COVID-19, medRxiv. (2020) https://doi.org/10.1101/2020.04.24.20078949.

  19. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Math. Sci. Springer-Verlag, New York, 1983.

    Book  MATH  Google Scholar 

  20. M.H. Protter and H.F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag New York Inc., 1984.

    Book  MATH  Google Scholar 

  21. R. Redlinger, On Volterra’s population equation with diffusion, SIAM J. Math. Anal. 16 (1985), 135-142.

    Article  MathSciNet  MATH  Google Scholar 

  22. C. del Rio and P. N. Malani, COVID-19 - New insights on a rapidly changing epidemic, JAMA, (2020), https://doi.org/10.1001/jama.2020.3072.

    Article  Google Scholar 

  23. K. Saito, On the global stability of an SIR epidemic discrete model, submitted.

  24. K. Saito, T. Kohno and Y. Hamaya, Global stability of a delayed SIR epidemic model with diffusion, International Journal of Differential Equations and Applications, 16 (2017), 123-145.

    Google Scholar 

  25. K. Saito, T. Kohno and Y. Hamaya, Asymptotic behavior of delayed SIR epidemic models of COVID-19 with diffusion, Advances in Pure Mathematics, 13 (2023), 221-225.

    Google Scholar 

  26. Statistics Bureau of Japan, Population estimates monthly report, URL:http://www. stat.go.jp/english/data/jinsui/tsuki/index.html (04/09/2020).

  27. Y. Takeuchi and W. Ma, Stability analysis on a delayed SIR epidemic model with density dependent birth process, Dynamical and Continuous Discrete Impul. Systems, 5 (1999), 171-184.

    MathSciNet  MATH  Google Scholar 

  28. World Health Organization, Coronavirus disease 2019 (CCOVID-19) situation reports, URL:http://www. who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports/ (15/09/2020).

  29. C. Yang and J. A. Wang, Mathematical model for the novel corona virus epidemic in Wuhan, China, Math. Biosci. Eng., 17 (2020), 2708-2724.

    Article  MathSciNet  MATH  Google Scholar 

  30. Y. Zhang, Xi. Yu, H. G. Sun, G. R. Tick, W. Wei and B. Jin, COVID-19 infection and recovery in various countries: Modeling the dynamics and evaluating the non-pharmaceutical mitigation scenarios, submitted on 31 Mar 2020 to Cornell University.

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Acknowledgements

The authors wish to thank referees gratitude for their many useful comments concerning the content of this paper.

Funding

We are partially supported by JSPS KAKENHI Grant Number 21K03318 in Japan.

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Correspondence to Yoshihiro Hamaya.

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Hamaya, Y., Saito, K. GLOBAL ATTRACTIVITY OF A DELAYED SEIR EPIDEMIC MODEL OF COVID-19 WITH DIFFUSION. J Math Sci 271, 378–399 (2023). https://doi.org/10.1007/s10958-023-06527-6

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