Abstract
Age-structured epidemic models have been used to describe either the age of individuals or the age of infection of certain diseases and to determine how these characteristics affect the outcomes and consequences of epidemiological processes. Most results on age-structured epidemic models focus on the existence, uniqueness, and convergence to disease equilibria of solutions. In this paper we investigate the existence of travelling wave solutions in a deterministic age-structured model describing the circulation of a disease within a population of multigroups. Individuals of each group are able to move with a random walk which is modelled by the classical Fickian diffusion and are classified into two subclasses, susceptible and infective. A susceptible individual in a given group can be crisscross infected by direct contact with infective individuals of possibly any group. This process of transmission can depend upon the age of the disease of infected individuals. The goal of this paper is to provide sufficient conditions that ensure the existence of travelling wave solutions for the age-structured epidemic model. The case of two population groups is numerically investigated which applies to the crisscross transmission of feline immunodeficiency virus (FIV) and some sexual transmission diseases.
Similar content being viewed by others
References
Al-Omari J. and Gourley S.A. (2002). Monotone travelling fronts in an age-structured reaction–diffusion model of a single species. J. Math. Biol. 45: 294–312
Anderson R.M. (1991). Discussion: the Kermack–McKendrick epidemic threshold theorem. Bull. Math. Biol. 53: 3–32
Bartlett M.S. (1956). Deterministic and stochastic models for recurrent epidemics. Proc. 3rd Berkeley Symp. Math. Stat. Prob. 4: 81–109
Berestycki H., Hamel F., Kiselev A. and Ryzhik L. (2005). Quenching and propagation in KPP reaction–diffusion equations with a heat loss. Arch. Rational Mech. Anal. 178: 57–80
Cruickshank I., Gurney W.S.C. and Veitch A.R. (1999). The characteristics of epidemics and invasions with thresholds. Theoret. Pop. Biol. 56: 279–292
Diekmann O. and Heesterbeek J.A.P. (2000). Mathematical Epidemiology of Infective Diseases: Model Building, Analysis and Interpretation. Wiley, New York
Ducrot A. (2007). Travelling wave solutions for a scalar age-structured equation. Dis. Con. Dynam. Syst. B7: 251–273
Ducrot, A., Magal, P.: Travelling wave solutions for an infection-age structured model with diffusion. Proc. R. Soc. Edinburgh Sect. A (accepted)
Fitzgibbon W.E., Langlais M., Parrott M.E. and Webb G.F. (1995). A diffusive system with age dependency modeling FIV. Nonlin. Anal. TMA 25: 975–989
Genieys S., Volpert V. and Auger P. (2006). Pattern and waves for a model in population dynamics with nonlocal consumption of resources. Math. Model. Nat. Phnem. 1: 65–82
Gurtin M.E. and MacCamy R.C. (1974). Nonlinear age-dependent population dynamics. Arch. Rational Mech. Anal. 54: 28l–300
Hosono Y. and Ilyas B. (1994). Travelling waves for a simple diffusive epidemic model. Math. Models Methods Appl. Sci. 5: 935–966
Iannelli M. (1994). Mathematical Theory of Age-Structured Population Dynamics. Giadini Editori e Stampatori, Pisa
Inaba H. (2001). Kermack and McKendrick revisited: the variable susceptibility model for infectious diseases. Jpn. J. Indust. Appl. Math. 18: 273–292
Kermack W.O. and McKendrick A.G. (1927). A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. 115: 700–721
Magal P. and Ruan S. (2007). On integrated semigroups and age-structured models in Lp space. Differ. Integral Equ. 20: 197–239
Murray J.D. (2002). Mathematical Biology II: Spatial Models and Biomedical Applications. Springer, Berlin
Rass, L., Radcliffe, J.: Spatial Deterministic Epidemics, Math Surveys Monogr. 102. American Mathematical Society, Providence, 2003
Ruan, S.: Spatial-temporal dynamics in nonlocal epidemiological models. In: Takeuchi, Y., Sato, K., Iwasa, Y. Mathematics for Life Science and Medicine, pp. 97–122. Springer, New York, 2007
So J.W.-H., Wu J. and Zou X. (2001). A reactiondiffusion model for a single species with age structure. I. Travelling wavefronts on unbounded domains. Proc. R. Soc. Lond. A457: 1841–1853
Thieme H.R. (2003). Mathematics in Population Biology. Princeton University Press, Princeton
Webb G.F. (1980). An age-dependent epidemic model with spatial diffusion. Arch. Rational Mech. Anal. 75: 91–102
Webb G.F. (1985). Theory of Nonlinear Age-Dependent Population Dynamics. Marcel Dekker, New York
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Rabinowitz
Rights and permissions
About this article
Cite this article
Ducrot, A., Magal, P. & Ruan, S. Travelling Wave Solutions in Multigroup Age-Structured Epidemic Models. Arch Rational Mech Anal 195, 311–331 (2010). https://doi.org/10.1007/s00205-008-0203-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-008-0203-8