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Basic Ideas in Epidemic Modeling

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Age-Structured Population Dynamics in Demography and Epidemiology

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Abstract

In this chapter, we start by studying the early Kermack–McKendrick epidemic model, and introduce the basic ideas and ingredients of epidemic modeling. A crucial point is that we cannot precisely interpret the basic ideas and indices of infectious disease epidemiology without knowing the underlying nonlinear population dynamics. The early Kermack–McKendrick model is an infection-age-dependent outbreak model, and its extensions in the late 1970 s opened the door to the recent developments in mathematical epidemiology. The key idea of analyzing epidemic models is the basic reproduction number \(R_0\) and its well-known threshold principle: if \(R_0>1\), the final size of the epidemic is positive no matter how small the initial infected population, whereas if \(R_0<1\), the final size becomes zero as the initial number of infected individuals goes to zero. We demonstrate the threshold principle based on the original definition of the final size given by Kermack and McKendrick, although there are slightly different definition for the final size. We then extend the original model to account for the heterogeneity of individuals and derive the pandemic threshold theorem. Subsequently, we introduce the demography of the host population and prove the endemic threshold theorem: if \(R_0>1\), there exists at least one endemic steady state, whereas if \(R_0<1\), there is no endemic steady state. This principle, however, does not hold under certain conditions. We provide examples in which subcritical endemic steady states exist even when \(R_0<1\) because of the reinfection of recovered individuals.

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Notes

  1. 1.

    Reader are referred to [28] for the origin of epidemic models.

  2. 2.

    The time between the receipt of infection and when the infected individual becomes infective is called the latent period . The time between the receipt of infection and the appearance of symptoms is called the incubation period . The period during which infectious organisms are discharged is called the infectious period [6]. Readers are referred to [13] for more complicated compartment assumptions.

  3. 3.

    For more rigorous treatment to characteristic equation (5.71), readers are referred to Remark 3.1 or [50, Chap. IV].

  4. 4.

    Kuniya and Wang [72] studied the above model with spatial diffusion terms.

  5. 5.

    The Pease model studied in Chap. 8 can be seen as an age-dependent extension of this reinfection model on the subset \(\varOmega _0\).

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    Hisashi Inaba

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Inaba, H. (2017). Basic Ideas in Epidemic Modeling. In: Age-Structured Population Dynamics in Demography and Epidemiology. Springer, Singapore. https://doi.org/10.1007/978-981-10-0188-8_5

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