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The Theory of Basic Reproduction Ratios

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Dynamical Systems in Population Biology

Part of the book series: CMS Books in Mathematics ((CMSBM))

Abstract

The basic reproduction number (ratio) R 0 is one of the most important concepts in population biology, see, e.g., [16, 94, 148, 149, 78] and the references therein. In epidemiology, R 0 is the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual during the infectious period, and R 0 is also a commonly used measure of the effort needed to control an infectious disease. Diekmann, Heesterbeek and Metz [95] introduced the next generation matrices (NGM) approach to R 0 for models of infectious diseases in heterogeneous populations; van den Driessche and Watmough [376] developed the theory of R 0 for autonomous ordinary differential equations (ODE) models with compartmental structure; and Diekmann, Heesterbeek and Roberts [96] provided a recipe for the construction of the NGM for compartmental epidemic models. These works have found numerous applications in the study of various models of infectious diseases. For population models in a periodic environment, Bacaër and Guernaoui [24] proposed a general definition of R 0, that is, R 0 is the spectral radius of an integral operator on the space of continuous periodic functions. Wang and Zhao [388] characterized R 0 for periodic compartmental ODE models and proved that it is a threshold parameter for the local stability of the disease-free periodic solution. Further, Thieme [370] presented the theory of spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. Bacaër and Ait Dads [22, 23] also found a more biological interpretation of R 0 for periodic models and showed that it is the asymptotic ratio of total infections in two successive generations of the infection tree. Recently, Inaba [189] introduced the concept of a generation evolution operator to give a new definition of R 0 for structured populations in heterogeneous environments, which unifies two definitions in [95, 24] and has intuitively clear biological meaning.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, Canada

    **ao-Qiang Zhao

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Zhao, XQ. (2017). The Theory of Basic Reproduction Ratios. In: Dynamical Systems in Population Biology. CMS Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-56433-3_11

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