J. Math. Biol. (1992) 30:473-491 Journal of Mathematical 61olo9y © Springer-Vedag 1992 Small amplitude, long period outbreaks in seasonally driven epidemics Ira B. Schwartz US Naval Research Laboratory, Code 97003, Washington, DC 20375, USA Received March 3, 1991; received in revised form May 28, 1991 Abstract. It is now documented that childhood diseases such as measles, mumps, and chickenpox exhibit a wide range of recurrent behavior (periodic as well as chaotic) in large population centers in the first world. Mathematical models used in the past (such as the SEIR model with seasonal forcing) have been able to predict the onset of both periodic and chaotic sustained epi- demics using parameters of childhood diseases. Although these models possess stable solutions which appear to have the correct frequency content, the corre- sponding outbreaks require extremely large populations to support the epi- demic. This paper shows that by relaxing the assumption of uniformity in the supply of susceptibles, simple models predict stable long period oscillatory epidemics having small amplitude. Both coupled and single population models are considered. Key words: Chaos - Nonlinear coupled oscillators - Epidemiology - Dynamics 1 Introduction It is now well documented that childhood diseases which incur permanent immunity oscillate periodically, as well as exhibit chaotic behavior (Hethcote 1983; London and Yorke 1973; Yorke and London 1973; Schaeffer 1985; Yorke et al. 1979). Examples of such diseases are chickenpox, measles, and mumps. Another example of a disease which exhibits strong oscillations but does not fit exactly into the framework of the other childhood diseases is rubella, which exhibits outbreaks with periods as long as seven years. One common feature each of the abovementioned diseases has is that they have a maximum peak in their respective power spectrum of 1 year, which is reflected in the fact that peak-to-peak outbreaks have local maxima separated by one year. This annual behavior in childhood disease has been linked to the opening and closing of schools in the various cities (London and Yorke 1973; Fine and Clarkson 1980). Given that the childhood diseases all have an annual seasonal component, spectral analysis reveals that their outbreaks can exhibit a widely varying range of longer interepidemic periods. In measles, periods on the order of 2-3 years have been observed. Mumps exhibits periods from 3-4 years, while rubella has