Basic Models in Epidemiology

Abstract

The longstanding debate as to whether or not chaotic dynamics occur in natural populations (Hastings et al. 1993) arises because various natural populations exhibit apparently “random” fluctuations in abundance. Furthermore, in several instances these fluctuations will look random not only to the naked eye but also through the “eyes” of some standard time series analysis (Ellner 1989). Several simple nonlinear mathematical ecological models that exhibit chaos for apparently reasonable parameter values have been studied over the last few years. Initially chaos was found in simple discrete time models for single species (May 1974, May 1976; May and Oster 1976). Chaotic dynamics have also been encountered in discrete models with age structure (Ebenman 1987; Guckenheimer et al. 1977; Levin 1981; Levin and Goodyear 1980; in discrete models with two species (Allen 1989a,Allen 1989b, Allen 1990a,Allen 1990b, Allen 1991; Beddington et al. 1975; Bellows and Hassell 1988; May 1974; May and Oster 1976; Neubert and Kot 1992); in simple discrete models of parasites (May 1985); in models for host-parasitoid-pathogen systems (Hochberg et al. 1990); in discrete demographic models with two sexes (Caswell and Weeks 1986); and in models that include frequency-dependent selection (Altenberg 1991; Cressman 1988; May and Anderson 1983a).