Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Dynamics of a reaction-diffusion SIRI model with relapse and free boundary

1 School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006, China
2 Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada

Special Issues: Spatial dynamics for epidemic models with dispersal of organisms and heterogenity of environment

This paper is concerned with the free boundary problem for a reaction-diffusion SIRI model with relapse and bilinear incidence rate. After studying the (global) existence and uniqueness of solutions, we provide some sufficient conditions on the disease spreading-vanishing dichotomies for both cases with and without relapse.
  Article Metrics

Keywords relapse; reaction-diffusion equation; free boundary; spreading-vanishing dichotomy

Citation: Qian Ding, Yunfeng Liu, Yuming Chen, Zhiming Guo. Dynamics of a reaction-diffusion SIRI model with relapse and free boundary. Mathematical Biosciences and Engineering, 2020, 17(2): 1659-1676. doi: 10.3934/mbe.2020087


  • 1. D. Tudor, A deterministic model for herpes infections in human and animal populations, SIAM Rev., 32 (1990), 136-139.
  • 2. H. Moreira, Y. Wang, Global stability in a SIRI model, SIAM Rev., 39 (1997), 497-502.
  • 3. C. Vargas-De-León, On the global stability of infectious diseases models with relapse, Abstraction Application, 9 (2013), 50-61.
  • 4. P. van den Driessche, X. Zou, Modeling relapse in infectious diseases, Math. Biosci., 207 (2007), 89-103.
  • 5. S. Liu, S. Wang, L. Wang, Global dynamics of delay epidemic models with nonlinear incidence rate and relapse, Nonlinear Anal. Real World Appl., 12 (2011), 119-127.
  • 6. P. Georgescu, A Lyapunov functional for an SIRI model with nonlinear incidence of infection and relapse, Appl. Math. Comput., 219 (2013), 8496-8507.
  • 7. Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.
  • 8. L. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971.
  • 9. J. Crank, Free and Moving Boundary Problem, Clarendon Press, Oxford, 1984.
  • 10. X. Chen, A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800.
  • 11. L. Caffarelli, S. Salsa, A Geometric Approach to Free Boundary Problems, Grad. Stud. Math. 68, American Mathematical Society, Providence, RI, 2005.
  • 12. K. Kim, Z. Lin, Q. Zhang, An SIR epidemic model with free boundary, Nonlinear Anal. Real World Appl., 14 (2013), 1992-2001.
  • 13. Y. Du, Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.
  • 14. O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural'0ceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.
  • 15. J. F. Cao, W. T. Li, J. Wang, F. Y. Yang, A free boundary problem of a diffusive SIRS model with nonlinear incidence, Z. Angew. Math. Phys., 68 (2017), 39.
  • 16. R. S. Cantrell, C. Cosner, Spatial ecology via reaction-diffusion equations. Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003.
  • 17. J. P. LaSalle, S. Lefschetz, Stability by Liapunov's Direct Method with Applications, Academic Press, New York, 1961.


Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved