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Unformatted text preview: Chapter 9 Kermack and McKendrick and the Epic of Epidemiological Dynamics 9.1 Prelude The pillars upon which modern epidemiological theory for directlytransmitted infectious microparasitic disease (primarily viral and bacterial) is built are the deterministic model of Kermack and McKendrick (1927; also see Heth cote, 2000), the stochastic chain binomial models of Reed, Frost and Soper (Reed and Frost unpublished lecture notes; Soper 1929) later enriched through the application of GaltonWatson branching process theory (Daley and Gani, 1999). The field has only come of age over the past three decades, marked first by Baileys (1975) seminal text The Mathematical Theory of Infectious Diseases and its Application, later by Anderson and Mays (1991) synthetic tome Infectious Diseases of Humans: Dynamics and Control, and more re cently with three research survey volumes by the broader community, arising from the Isaac Newton Institutes 1993 focus on infectious disease modeling (Mollison, 1995; Grenfell and Dobson, 1995; Isham and Medley 1996). Also, texts by Daley and Gani (1999), Diekmann and Heesterbeek (2000), and Thieme (2003) provide wellcrafted introductory and advanced mathemati cal presentations, while several edited volumes provide an overview of current interests and directions (e.g. Dieckmann et al., 2002; Hudson et al., 2002; this volume). An important current area of research is the development of theory and methods to model epidemics in heterogeneous systems. Hetero geneity due to spatial and other population structures, such as age, social groups, or genetic variation, can arise in many different ways, and modeling studies dealing with various kinds of heterogeneity are being published at an increasing rate. Heterogeneity is grist for the evolutionary mill. In this and 2009 Getz 141 May 20, 2009 142 9.2. BASIC SUSCEPTIBLEINFECTED MODELS the next chapter, however, we focus only on the development of more co hesive theoretical and methodological approaches to heterogeneity, because further maturation of these fields are still needed to approach some of the most challenging problems in the coevolution of hostpathogen systems. Theme Enabling Mythology 1. Structurally lumped, spatially homogeneous population structure that is is closedthat is, migration processes are absent. 2. The population is embedded in a constant environment to the extent that even the resources perpetually maintain the same levels or fluxes irrespective of the density of the population. Vocabulary Explain use of S I V D and the fact that R is not used for recoverd or removed, but reserved for R , R * etc. 9.2 Basic SusceptibleInfected Models 9.2.1 Background Underlying all dynamical systems models of epidemiological processes is the susceptible ( S ) and infected ( I ) approachsocalled SI frameworkof Ker mack and McKendrick (1927) that was foreshadowed by the work of Enko (1889). Within this framework, a population infected by a microparasite(1889)....
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 Fall '10
 WayneM.Getz

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