Traveling waves in the Kermack–McKendrick epidemic model with latent period

  • Junfeng HeEmail author
  • Je-Chiang Tsai


We study traveling waves for a diffusive susceptible–infected–recovery model, due to Kermack and McKendrick, of an epidemic with standard incidence and latent period included. In contrast to the classical case where the mass action incidence is employed, the total population is varied in the present model. It turns out that the governing equation for the recovery species cannot be decoupled from the other two equations for the susceptible and the infected species, and hence that the present model cannot be reduced to a two-component system as the classical one does. The existence of traveling waves of the model in this study can be completely characterized by the basic reproduction number of the system of ordinary differential equations associated with the present model. The model admits a continuum of traveling waves parameterized by wave speed c when waves do exist. Our approach is based on the fixed point theory and a delicately designed pair of super-/sub-solutions. This set of super-/sub-solutions also allows us to completely answer two unsolved questions in the existing literatures where the latent period is zero: (i) the existence of the minimal-speed wave which is believed to play a key role in the evolution of epidemic diseases and (ii) the existence of traveling waves does not depend on the relative ratio of the diffusivity of the infected species to the one of the recovery species.


Traveling wave Kermack–McKendrick model Minimal wave speed 

Mathematics Subject Classification

92D30 35K57 34B40 35B40 



  1. 1.
    Bai, Z., Wu, S.: Traveling waves in a delayed SIR epidemic model with nonlinear incidence. Appl. Math. Comput. 263, 221–232 (2015)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bernfeld, S.R., Lakshmikantham, V.: An Introduction to Nonlinear Boundary Value Problems. Mathematics in Science and Engineering, vol. 10. Academic Press, New York (1974)zbMATHGoogle Scholar
  3. 3.
    Berestycki, H., Hamel, F., Kiselev, A., Ryzhik, L.: Quenching and propagation in KPP reaction–diffusion equations with a heat loss. Arch. Ration. Mech. Anal. 178, 57–80 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brauer, F.: The Kermack–McKendrick epidemic model revisited. Math. Biosci. 198, 119–131 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brauer, F., van den Driessche, P., Wu, J.: Mathematical Epidemiology. Lecture Notes in Mathematics, vol. 1945. Springer, New York (2008)zbMATHGoogle Scholar
  6. 6.
    Castillo-Chavez, C., Cooke, K., Huang, W., Levin, S.A.: The role of long incubation periods in the dynamics of HIV/AIDS. Part 1: single populations models. J. Math. Biol. 27, 373–398 (1989)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Carr, J., Chmaj, A.: Uniqueness of travelling waves for nonlocal monostable equations. Proc. Am. Math. Soc. 132, 2433–2439 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dietz, K.: Overall patterns in the transmission cycle of infectious disease agents. In: Anderson R.M., May R.M. (eds.) Population Biology of Infectious Diseases. Life Sciences Research Report, vol. 25, p. 87. Springer, Berlin (1982)Google Scholar
  9. 9.
    Fu, S.C.: Traveling waves for a diffusive SIR model with delay. J. Math. Anal. Appl. 435, 20–37 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fu, S.C.: The existence of traveling wave fronts for a reaction–diffusion system modelling the acidic nitrate–ferroin reaction. Quart. Appl. Math. 72, 649–664 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42, 599–653 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hosono, Y., Ilyas, B.: Existence of traveling waves with any positive speed for a diffusive epidemic model. Nonlinear World 1, 277–290 (1994)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hosono, Y., Ilyas, B.: Traveling waves for a simple diffusive epidemic model. Math. Models Methods Appl. Sci. 5, 935–966 (1995)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Huang, G., Takeuchi, Y.: Global analysis on delay epidemiological dynamic models with nonlinear incidence. J. Math. Biol. 63, 125–139 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Källén, A.: Thresholds and travelling waves in an epidemic model for rabies. Nonlinear Anal. TMA 8, 851–856 (1984)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Keeling, M.J., Rohani, P.: Modeling Infectious Diseases in Humans and Animals. Princeton University Press, Princeton (2008)zbMATHGoogle Scholar
  17. 17.
    Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A 115, 700–721 (1927)CrossRefGoogle Scholar
  18. 18.
    Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics: II. Proc. R. Soc. Lond. B 138, 55–83 (1932)CrossRefGoogle Scholar
  19. 19.
    Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics: III. Proc. R. Soc. Lond. B 141, 94–112 (1933)CrossRefGoogle Scholar
  20. 20.
    Lewis, M.A., Li, B., Weinberger, H.F.: Spreading speed and linear determinacy for two-species competition models. J. Math. Biol. 45, 219–233 (2002)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Li, B., Weinberger, H.F., Lewis, M.A.: Spreading speeds as slowest wave speeds for cooperative systems. Math. Biosci. 196, 82–98 (2005)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Li, Y., Li, W.T., Lin, G.: Traveling waves in a delayed diffusive SIR epidemic model. Commun. Pure Appl. Anal. 14, 1001–1022 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Martcheva, M.: An Introduction to Mathematical Epidemiology. Texts in Applied Mathematics, vol. 61. Springer, New York (2015)zbMATHGoogle Scholar
  24. 24.
    Mena-Lorca, J., Hethcote, H.W.: Dynamic models of infectious diseases as regulators of population sizes. J. Math. Biol. 30, 693–716 (1992)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Murray, J.D.: Mathematical Biology. II: Spatial Models and Biomedical Applications. Springer, New York (2004)CrossRefGoogle Scholar
  26. 26.
    Nagumo, M.: Über die Differentialgleichung \(y^{\prime \prime } = f(x, y, y^{\prime })\). Proc. Phys. Math. Soc. Jpn. 19, 861–866 (1937)zbMATHGoogle Scholar
  27. 27.
    Pauwelussen, J.P.: Nerve impulse propagation in a branching nerve system: a simple model. Physica D 4, 67–88 (1981/82)Google Scholar
  28. 28.
    Ross, R.: An application of the theory of probabilities to the study of a priori pathometry: I. Proc. R. Soc. Lond. A 92, 204–230 (1916)CrossRefGoogle Scholar
  29. 29.
    Wang, H., Wang, X.: Traveling wave phenomena in a Kermack–McKendrick SIR model. J. Dyn. Differ. Equ. 28, 143–166 (2016)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Wang, X.-S., Wang, H., Wu, J.: Traveling waves of diffusive predator–prey systems: disease outbreak propagation. Discrete Contin. Dyn. Syst. A 32, 3303–3324 (2012)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Xu, Z.: Traveling waves in a Kermack–McKendrick epidemic model with diffusion and latent period. Nonlinear Anal. 111, 66–81 (2014)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Wang, Z., Wu, J.: Travelling waves of a diffusive Kermack–McKendrick epidemic model with non-local delayed transmission. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 466, 237–261 (2010)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1941)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsSouthern University of Science and TechnologyShenzhenPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsWuhan UniversityWuhanPeople’s Republic of China
  3. 3.Department of MathematicsNational Tsing Hua UniversityHsinchuTaiwan
  4. 4.National Center for Theoretical SciencesNational Taiwan UniversityTaipeiTaiwan

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