Abstract
In this work we study the asymptotic behaviour of the Kermack–McKendrick reaction-diffusion system in a periodic environment with non-diffusive susceptible population. This problem was proposed by Kallen et al. as a model for the spatial spread for epidemics, where it can be reasonable to assume that the susceptible population is motionless. For arbitrary dimensional space we prove that large classes of solutions of such a system have an asymptotic spreading speed in large time, and that the infected population has some pulse-like asymptotic shape. The analysis of the one-dimensional problem is more developed, as we are able to uncover a much more accurate description of the profile of solutions. Indeed, we will see that, for some initially compactly supported infected population, the profile of the solution converges to some pulsating travelling wave with minimal speed, that is to some entire solution moving at a constant positive speed and whose profile’s shape is periodic in time.
Similar content being viewed by others
References
Anderson RM, Jackson HC, May RM, Smith AM (1981) Population dynamics of fox rabies in Europe. Nature 289:765–771
Beaumont C, Burie J-B, Ducrot A, Zongo P (2012) Propagation of Salmonella within an industrial hens house. SIAM J Appl Math 72:1113–1148
Berestycki H, Hamel F, Kiselev A, Ryzhik L (2005) Quenching and propagation in KPP reaction-diffusion equations with a heat loss. Arch Ration Mech Anal 178:57–80
Berestycki H, Hamel F, Nadin G (2008) Asymptotic spreading in heterogeneous diffusive media. J Funct Anal 255:2146–2189
Berestycki H, Hamel F, Roques L (2005) Analysis of the periodically fragmented environment model: I—species persistence. J Math Biol 51:75–113
Berestycki H, Hamel F, Roques L (2005) Analysis of the periodically fragmented environment model: II-Biological invasions and pulsating travelling fronts. J Math Pures Appl 84:1101–1146
Berestycki H, Hamel F, Rossi L (2007) Liouville type results for semilinear elliptic equations in unbounded domains. Annali Mat Pura Appl 186:469–507
Bramson M (1983) Convergence of solutions of the Kolmogorov equation to travelling waves. Mem Am Math Soc 44(285): iv+190 pp
Britton NF (1986) Reaction-diffusion equations and their applications to biology. Academic Press, London
Britton NF (1991) An integral for a reaction-diffusion system. Appl Math Lett 4:43–47
Diekmann O (1977) Limiting behaviour in an epidemic model. Nonlinear Anal TMA 1:459–470
Diekmann O (1978) Thresholds and travelling waves for the geographical spread of infection. J Math Biol 6:109–130
Ducrot A, Magal P (2009) Travelling wave solutions for an infection-age structured model with diffusion. Proc Roy Soc Edinb Sect A Math 139:459–482
Ducrot A, Magal P, Ruan S (2010) Travelling wave solutions in multi-group age-structured epidemic models. Arch Ration Mech Anal 195:311–331
Giletti T (2010) KPP reaction-diffusion equations with a non-linear loss inside a cylinder. Nonlinearity 23:2307–2332
Giletti T (2013) Convergence to pulsating traveling waves with minimal speed in some KPP heterogeneous problems, preprint arXiv:1304.0832
Hamel F, Nolen J, Roquejoffre J-M, Ryzhik L (2012) The logarithmic delay of KPP fronts in a periodic medium, preprint arXiv:1211.6173
Hamel F, Ryzhik L (2010) Travelling waves for the thermodiffusive system with arbitrary Lewis numbers. Arch Ration Mech Anal 195:923–952
Hosono Y, Ilyas B (1995) Traveling waves for a simple diffusive epidemic model. Math Models Meth Appl Sci 5:935–966
Kallen A (1984) Thresholds and travelling waves in an epidemic model for rabies. Nonlinear Anal Theor Meth Appl 8:851–856
Kallen A, Arcuri P, Murray JD (1985) A simple model for the spatial spread of rabies. J Theor Biol 116:377–393
Kermack WO, McKendrick AG (1927) A contribution to the mathematic theory of epidemics. Proc Roy Soc Lond 115:700–721
Lau K-S (1985) On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov. J Differ Equ 59(1):44–70
Murray JD (2003) Mathematical biology, 3rd edn. Springer, Berlin
Murray JD, Stanley EA, Brown DL (1986) On the spatial spread of rabies among foxes. Pro. Roy Soc Lond B 229:111–150
Nadin G (2010) The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator. SIAM J Math Anal 41:2388–2406
Rass L, Radcliffe J (2003) Spatial deterministic epidemics. In: Mathematical surveys and monographs, vol 102. AMS, Providence
Thieme HR (1977) A model for the spatial spread of an epidemic. J Math Biol 4:337–351
Thieme HR (2003) Mathematics in population biology. Princeton University Press, Princeton
Uchiyama K (1978) The behavior of solutions of some nonlinear diffusion equations for large time. J Math Kyoto Univ 18(3):453–508
Weinberger H (2002) On spreading speeds and traveling waves for growth and migration models in a periodic habitat. J Math Biol 45:511–548
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ducrot, A., Giletti, T. Convergence to a pulsating travelling wave for an epidemic reaction-diffusion system with non-diffusive susceptible population. J. Math. Biol. 69, 533–552 (2014). https://doi.org/10.1007/s00285-013-0713-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-013-0713-3
Keywords
- Kermack–McKendrick reaction-diffusion system
- Periodic environment
- Pulsating travelling wave
- Asymptotic behaviour