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EpiMaterial - Chapter 9 Kermack and McKendrick and the Epic...

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Chapter 9 Kermack and McKendrick and the Epic of Epidemiological Dynamics 9.1 Prelude The pillars upon which modern epidemiological theory for directly-transmitted 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(1927; also see Heth- cote, 2000), the stochastic chain binomial models of Reed, Frost and Soper (Reed and Frost unpublished lecture notes; Soper 1929) later enriched through the application of Galton-Watson branching process theory (Daley and Gani, 1999).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 well-crafted 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
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142 9.2. BASIC SUSCEPTIBLE-INFECTED 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 host-pathogen systems. Theme Enabling Mythology 1. Structurally lumped, spatially homogeneous population structure that is is closed—that 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 0 , R * etc. 9.2 Basic Susceptible-Infected Models 9.2.1 Background Underlying all dynamical systems models of epidemiological processes is the susceptible ( S ) and infected ( I ) approach—so-called S-I framework—of Ker- mack and McKendrick (1927) that was foreshadowed by the work of Enko (1889). Within this framework, a population infected by a microparasite (e.g. a bacterial or viral pathogen, or protist) is divided at its most ba- sic level into susceptible and infected (assumed to also be infective) groups,
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