Some pathogens change their antigenic properties at an intermediate rate in the sense that antigenes remain constant during a single infectious episode while antigenic change is fast enough that a host during its lifespan may experience several reinfections with antigenically cross-reacting strains. Thus the pathogens utilize a distinct life history strategy where evolutionary changes in the pathogen allow them to recolonize hosts in several infection episodes. The most well documented example of intermediate antigenic change is Influenza A during a period of antigenic drift [1]. Extensions of the SIR-model can describe the epidemiology of such diseases as well as the natural selection on the pathogen strains [2-4]. In general models of the natural selection are rather complicated because it is necessary to keep track of all possible host immune histories. However if cross-reaction acts by reducing the infectivity of a second infection, the model may be simplified considerably by subdividing for each strain k, the uninfected hosts according to their degree of cross-immunity to k [5].

[1] C.M. Pease. An evolutionary epidemiological mechanism with applications to type A influenza. Theor. Popul. Biol. 31: 422-452, 1987.

[2] V. Andreasen et al. A model of Influenza A drift evolution. Z. Angew. Math. Mech., 76: Suppl. 2, 421-424, 1996.

[3] V. Andreasen et al. The dynamics of cocirculating Influenza strains conferring partial cross-immunity. J. Math. Biol., 35: 825-842, 1997.

[4] C. Castillo-Chavez et al. Epidemiological models with age structure, proportionate mixing, and cross-immunity. J. Math. Biol. 27: 233-258, 1989.

[5] S. Gupta et al. The maintenance of strain structure in populations of recombining infectious agents. Nature Med., 2: 437-442, 1996.

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1998-1999 Mathematics in Biology