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Mathematical Epidemiology pp 229–293Cite as

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Distribution Theory, Stochastic Processes and Infectious Disease Modelling

Distribution Theory, Stochastic Processes and Infectious Disease Modelling

  • Ping Yan7 
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  • 18 Citations

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Part of the Lecture Notes in Mathematics book series (LNMBIOS,volume 1945)

The occurrence of a major outbreak, the shape of the epidemic curves, as well as the final sizes of outbreaks, are realizations of some stochastic events with some probability distributions. These distributions are manifested through some stochastic mechanisms. This chapter divides a typical outbreak in a closed population into two phases, the initial phase and beyond the initial phase. For the initial phase, this chapter addresses several aspects: the invasion (i.e. the risk of a large outbreak); quantities associated with a small outbreak; and characteristics of a large outbreak. In a large outbreak beyond the initial phase, the focus is on its final size. After a review of distribution theories and stochastic processes, this chapter separately addresses each of these issues by asking questions such as: Are the latent period and/or the infectious period distributions playing any role? What is the role of the contact process for this issue? Is the basic reproduction number R 0 sufficient to address this issue? How many stochastic mechanisms may manifest observations that may resemble a power-law distribution, and how much detail is really needed to address this specific issue? etc. This chapter uses distribution theory and stochastic processes to capture the agent–host–environment interface during an outbreak of an infectious disease. With different phases of an outbreak and special issues in mind, modellers need to choose which detailed aspects of the distributions and the stochastic mechanisms need to be included, and which detailed aspects need to be ignored. With these discussions, this chapter provides some syntheses for the concepts and models discussed in some proceeding chapters, as well as some food for thought for following chapters on case studies.

Keywords

  • Random Graph
  • Basic Reproduction Number
  • Infectious Period
  • Large Outbreak
  • Infectious Individual

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Authors and Affiliations

  1. Centre for Communicable Diseases and Infection Control, Infectious Diseases and Emergency Preparedness Branch, Public Health Agency of Canada, 100 Elangtine Drive, AL0602-B, Tunney's Pasture, Ottawa, ON, K1A 0K9, Canada

    Ping Yan

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  1. Ping Yan
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Editors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada

    Fred Brauer

  2. Department of Mathematics and Statistics, University of Victoria, 3060 STN CSC, Victoria, B.C. V8W 3R4, Canada

    Pauline van den Driessche

  3. Center for Disease Modeling Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada

    Jianhong Wu

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Yan, P. (2008). Distribution Theory, Stochastic Processes and Infectious Disease Modelling. In: Brauer, F., van den Driessche, P., Wu, J. (eds) Mathematical Epidemiology. Lecture Notes in Mathematics, vol 1945. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78911-6_10

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