Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bai, Z., Wu, S.: Traveling waves in a delayed SIR epidemic model with nonlinear incidence. Appl. Math. Comput. 263, 221–232 (2015)
Bernfeld, S.R., Lakshmikantham, V.: An Introduction to Nonlinear Boundary Value Problems. Mathematics in Science and Engineering, vol. 10. Academic Press, New York (1974)
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)
Brauer, F.: The Kermack–McKendrick epidemic model revisited. Math. Biosci. 198, 119–131 (2005)
Brauer, F., van den Driessche, P., Wu, J.: Mathematical Epidemiology. Lecture Notes in Mathematics, vol. 1945. Springer, New York (2008)
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)
Carr, J., Chmaj, A.: Uniqueness of travelling waves for nonlocal monostable equations. Proc. Am. Math. Soc. 132, 2433–2439 (2004)
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)
Fu, S.C.: Traveling waves for a diffusive SIR model with delay. J. Math. Anal. Appl. 435, 20–37 (2016)
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)
Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42, 599–653 (2000)
Hosono, Y., Ilyas, B.: Existence of traveling waves with any positive speed for a diffusive epidemic model. Nonlinear World 1, 277–290 (1994)
Hosono, Y., Ilyas, B.: Traveling waves for a simple diffusive epidemic model. Math. Models Methods Appl. Sci. 5, 935–966 (1995)
Huang, G., Takeuchi, Y.: Global analysis on delay epidemiological dynamic models with nonlinear incidence. J. Math. Biol. 63, 125–139 (2011)
Källén, A.: Thresholds and travelling waves in an epidemic model for rabies. Nonlinear Anal. TMA 8, 851–856 (1984)
Keeling, M.J., Rohani, P.: Modeling Infectious Diseases in Humans and Animals. Princeton University Press, Princeton (2008)
Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A 115, 700–721 (1927)
Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics: II. Proc. R. Soc. Lond. B 138, 55–83 (1932)
Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics: III. Proc. R. Soc. Lond. B 141, 94–112 (1933)
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)
Li, B., Weinberger, H.F., Lewis, M.A.: Spreading speeds as slowest wave speeds for cooperative systems. Math. Biosci. 196, 82–98 (2005)
Li, Y., Li, W.T., Lin, G.: Traveling waves in a delayed diffusive SIR epidemic model. Commun. Pure Appl. Anal. 14, 1001–1022 (2015)
Martcheva, M.: An Introduction to Mathematical Epidemiology. Texts in Applied Mathematics, vol. 61. Springer, New York (2015)
Mena-Lorca, J., Hethcote, H.W.: Dynamic models of infectious diseases as regulators of population sizes. J. Math. Biol. 30, 693–716 (1992)
Murray, J.D.: Mathematical Biology. II: Spatial Models and Biomedical Applications. Springer, New York (2004)
Nagumo, M.: Über die Differentialgleichung \(y^{\prime \prime } = f(x, y, y^{\prime })\). Proc. Phys. Math. Soc. Jpn. 19, 861–866 (1937)
Pauwelussen, J.P.: Nerve impulse propagation in a branching nerve system: a simple model. Physica D 4, 67–88 (1981/82)
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)
Wang, H., Wang, X.: Traveling wave phenomena in a Kermack–McKendrick SIR model. J. Dyn. Differ. Equ. 28, 143–166 (2016)
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)
Xu, Z.: Traveling waves in a Kermack–McKendrick epidemic model with diffusion and latent period. Nonlinear Anal. 111, 66–81 (2014)
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)
Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1941)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
He, J., Tsai, JC. Traveling waves in the Kermack–McKendrick epidemic model with latent period. Z. Angew. Math. Phys. 70, 27 (2019). https://doi.org/10.1007/s00033-018-1072-0
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00033-018-1072-0