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Variable Susceptibility, Reinfection, and Immunity

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Age-Structured Population Dynamics in Demography and Epidemiology

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Abstract

It is widely recognized that dynamic changes in host susceptibility resulting from the evolution of infectious agents and changes in the host immunity distribution play important roles in the spread of infectious diseases. Even for common childhood diseases such as measles, the natural decay of host immunity occurs if the environmental virus disappears and the booster effect is lost. In this chapter, we first consider the Pease model for type A influenza epidemics, which was an early attempt to take into account the decay of host immunity due to antigenic changes in the virus population. A remarkable feature of this model is that for realistic parameter values, the antigenic drift of a dominant virus is a possible mechanism for recurrent outbreaks. Next, we formulate the Kermack–McKendrick reinfection model using the standard age-dependent population dynamics equations and examine its basic properties. The potential importance of the Kermack–McKendrick reinfection model is that it can take into account variable susceptibility and reinfection, and will thus be a useful starting point in considering the epidemiological life history of individuals. The Pease influenza model can be seen as a special case of this reinfection model and has a reinfection threshold. Moreover, subcritical endemic steady states may be created by a backward bifurcation. We show some realistic examples of the reproductivity enhancement that can create the backward bifurcation. Finally, we introduce Aron’s malaria model, which can be interpreted as a model for acquired immunity boosted by exposure to infection. It was originally developed to understand the functional relation between the force of infection and the reversion rate and the age prevalence curve of malaria. However, our purpose here is not to give a realistic account of the malaria epidemic, but to show that the age-dependent model is a useful tool for describing the boosting mechanism. Although our analysis is far from complete, we can show that the functional relationship in the endemic steady state is naturally induced from the basic structured population dynamics, and the age-dependent model is essential in formulating epidemiological indices.

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Notes

  1. 1.

    Even in these classical childhood diseases, however, the vaccinated population could give rise to more complex epidemiologic consequences. For example, see [33] and Sect. 8.3.

  2. 2.

    The fully immunized system is the disease-free steady state of the system where all newborns are immunized, so it is composed of the recovered population \(r^*(\tau )=\mu N e^{-\mu \tau }\), its effective size of susceptibility is \(N\theta ^*\) and the infection process is described by a renewal equation \(i_2(t,0)=\theta ^*N \int _{0}^{\infty }\beta _2(\tau )i_2(t,\tau )d\tau \).

  3. 3.

    In the malaria model by Aguas et al. (Sect. 8.3.1), reinfected individuals is defined as the (secondary) infected class. In Aron’s model, reinfection of recovered individuals is considered as the boosting of the immune status and formulated by the boundary feedback.

  4. 4.

    personal communication (2016).

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    Hisashi Inaba

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Inaba, H. (2017). Variable Susceptibility, Reinfection, and Immunity. In: Age-Structured Population Dynamics in Demography and Epidemiology. Springer, Singapore. https://doi.org/10.1007/978-981-10-0188-8_8

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  • DOI: https://doi.org/10.1007/978-981-10-0188-8_8

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