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An SIS epidemic model with diffusion

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Abstract

The aim of this paper is to study the dynamics of an SIS epidemic model with diffusion. We first study the well-posedness of the model. And then, by using linearization method and constructing suitable Lyapunov function, we establish the local and global stability of the disease-free equilibrium and the endemic equilibrium, respectively. Furthermore, in view of Schauder fixed point theorem, we show that the model admits traveling wave solutions connecting the disease-free equilibrium and the endemic equilibrium when R 0 > 1 and c > c*. And also, by virtue of the two-sided Laplace transform, we prove that the model has no traveling wave solution connecting the two equilibria when R 0 > 1 and c ∈ [0, c*).

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References

  1. S Ai, R Albashaireh. Traveling waves in spatial SIRS model, J Dynam Differential Equations, 2014, 26: 143–164.

    Article  MathSciNet  MATH  Google Scholar 

  2. Z Bai, S Zhang. Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay, Commun Nonlinear Sci Numer Simul, 2015, 22: 1370–1381.

    Article  MathSciNet  MATH  Google Scholar 

  3. F Braner, C Castillo-Chávez. Mathematical Models in Population Biology and Epidemiology, Springer, New York, 2001.

    MATH  Google Scholar 

  4. F Brauer, P van den Driessche. Models for transmission of disease with immigration of infectives, Math Biosci, 2001, 171: 143–154.

    Article  MathSciNet  MATH  Google Scholar 

  5. A Ducrot, M Langlais, P Magal. Qualitative analysis and travelling wave solutions for the SI model with vertical transmission, Commun Pure Appl Anal, 2012, 11: 97–113.

    Article  MathSciNet  MATH  Google Scholar 

  6. A Ducrot, P Magal. Travelling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 2011, 23: 2891–2911.

    Article  MathSciNet  MATH  Google Scholar 

  7. J K Hale, H Kocak. Dynamics and Bifurcations, Springer, New York, 1991.

    Book  MATH  Google Scholar 

  8. D Henry. Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981.

    Book  MATH  Google Scholar 

  9. H W Hethcote. The mathematics of infectious diseases, SIAM Rev, 2000, 42: 599–653.

    Article  MathSciNet  MATH  Google Scholar 

  10. W Hirsch, H Hamisch, J P Gabriel. Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Commun Pure Appl Math, 1985, 38: 733–753.

    Article  MathSciNet  MATH  Google Scholar 

  11. W T Li, G Lin, C Ma, FY. Yang. Travelling wave solutions of a nonlocal delayed SIR model with outbreak threshold, Discrete Contin Dyn Syst Ser B, 2014, 19: 467–484.

    Article  MathSciNet  MATH  Google Scholar 

  12. Y Li, WT Li, G Lin. Traveling waves of a delayed diffusive SIR epidemic model, Commun Pure Appl Anal, 2015, 14: 1001–1022.

    Article  MathSciNet  MATH  Google Scholar 

  13. R H Martin, HL Smith. Abstract functional differential equations and reaction-diffusion systems, Trans Amer Math Soc, 1990, 321: 1–44.

    MathSciNet  MATH  Google Scholar 

  14. J D Murray. Mathematical Biology, I and II, third edn, Springer, New York, 2002.

    Google Scholar 

  15. S Ruan, J Wu. Modeling spatial spread of communicable diseases involving animal hosts, In: Spatial Ecology, Chapman & Hall/CRC, Boca Raton, FL, 2009, 293–316.

    Google Scholar 

  16. C Vargas-De-León. On the global stablity of SIS, SIR and SIRS epidemic models with standard incidence, Chaos Solitons Fractals, 2011, 44: 1106–1110.

    Article  MATH  Google Scholar 

  17. H Wang, X S Wang. Travelling waves phenomena in a Kermack-McKendrick SIR model, J Dynam Differential Equations, 2016, 28: 143–166.

    Article  MathSciNet  MATH  Google Scholar 

  18. XS Wang, H Wang, J Wu. Travelling waves of diffusive predator-prey systems: disease outbreak propagation, Discrete Contin Dyn Syst, 2012, 32: 3303–3324.

    Article  MathSciNet  MATH  Google Scholar 

  19. Z C Wang, J Wu. Travelling waves of a diffusive Kermack-McKendrick epidemic model with nonlocal delayed transmission, Proc R Soc A, 2010, 466: 237–261.

    Article  MathSciNet  MATH  Google Scholar 

  20. P Weng, X Q Zhao. Spreading speed and traveling waves for a multi-type SIS epidemics model, J Differential Equations, 2006, 229: 270–296.

    Article  MathSciNet  MATH  Google Scholar 

  21. D V Widder. The Laplace transform, Princeton University Press, 1941.

    MATH  Google Scholar 

  22. J Wu. Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996.

    Book  MATH  Google Scholar 

  23. Z T Xu. Traveling waves in a Kermack-Mckendrick epidemic model with diffusion and latent period, Nonlinear Anal, 2014, 111: 66–81.

    Article  MathSciNet  MATH  Google Scholar 

  24. E Zeilder. Nonlinear Functional Analysis and its Applications, I: Fixed-point Theorems, Springer, New York, 1986.

    Google Scholar 

  25. T Zhang, W Wang. Existence of traveling wave solutions for influenza model with treatment, J Math Anal Appl, 2014, 419: 469–495.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors express their sincere gratitude to the editors and anonymous referees for the careful reading of the original manuscript and useful comments which have led to a significant improvement to our original manuscript.

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Zhi-ting Xu & Dan-xia Chen

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  1. Zhi-ting Xu
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  2. Dan-xia Chen
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Correspondence to Zhi-ting Xu.

Additional information

Partially supported by the NSF of Guangdong Province (2016A030313426) and the HLUCF of South China Normal University (2016YN30).

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Xu, Zt., Chen, Dx. An SIS epidemic model with diffusion. Appl. Math. J. Chin. Univ. 32, 127–146 (2017). https://doi.org/10.1007/s11766-017-3460-1

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  • DOI: https://doi.org/10.1007/s11766-017-3460-1

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