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