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Epidemiological Models With Parametric Heterogeneity :Deterministic Theory for Closed Populations

Published online by Cambridge University Press:  06 June 2012

A.S. Novozhilov
A.S. Novozhilov*
Affiliation:
Applied Mathematics–1, Moscow State University of Railway Engineering Obraztsova 9, bldg. 9, Moscow 127994, Russia
*
⋆ Corresponding author. E-mail: anovozhilov@gmail.com
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Abstract

We present a unified mathematical approach to epidemiological models with parametricheterogeneity, i.e., to the models that describe individuals in the population as havingspecific parameter (trait) values that vary from one individuals to another. This is anatural framework to model, e.g., heterogeneity in susceptibility or infectivity ofindividuals. We review, along with the necessary theory, the results obtained using thediscussed approach. In particular, we formulate and analyze an SIR model with distributedsusceptibility and infectivity, showing that the epidemiological models for closedpopulations are well suited to the suggested framework. A number of known results from theliterature is derived, including the final epidemic size equation for an SIR model withdistributed susceptibility. It is proved that the bottom up approach of the theory ofheterogeneous populations with parametric heterogeneity allows to infer the populationlevel description, which was previously used without a firm mechanistic basis; inparticular, the power law transmission function is shown to be a consequence of theinitial gamma distributed susceptibility and infectivity. We discuss how the generaltheory can be applied to the modeling goals to include the heterogeneous contactpopulation structure and provide analysis of an SI model with heterogeneous contacts. Weconclude with a number of open questions and promising directions, where the theory ofheterogeneous populations can lead to important simplifications and generalizations.

Keywords

SIR modelheterogeneous populationsdistributed susceptibilityfinal epidemic sizeheterogeneous contact structurepower law transmission function
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 3: Epidemiology , 2012 , pp. 147 - 167
Copyright
© EDP Sciences, 2012

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