Epidemiological models with parametric heterogeneity: Deterministic theory for closed populations

TL;DR

This paper introduces a unified mathematical framework for epidemiological models with individual parametric heterogeneity, enabling better understanding of disease spread in populations with varying susceptibility and infectivity.

Contribution

It formulates and analyzes an SIR model with distributed susceptibility and infectivity, providing a mechanistic basis for population-level descriptions and deriving known results like the final epidemic size.

Findings

Power law transmission function arises from gamma distributed susceptibility and infectivity.

The bottom-up approach infers population dynamics from individual heterogeneity.

The framework can incorporate heterogeneous contact structures.

Abstract

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.