Variable Susceptibility With An Open Population: A Transport Equation Approach

TL;DR

This paper explores the use of transport equations to model variable individual responses in open populations within SIR epidemic frameworks, enabling finite-dimensional analysis and revealing qualitative similarities to traditional models.

Contribution

It extends transport equation methods to open populations in epidemic modeling, providing conditions for equivalence with nondistributed models and analyzing disease sterilization effects.

Findings

Transport equations convert infinite-dimensional systems into finite-dimensional ones.

Models with inherited parameters are qualitatively similar to nondistributed models.

Sterilization effects are fully analyzed within the transport equation framework.

Abstract

Variable individual response to epidemics may be found within many contexts in the study of infectious diseases (e.g., age structure or contact networks). There are situations where the variability, in terms of epidemiological parameter, cannot be neatly packaged along with other demographics of the population like spatial location or life stage. Transport equations are a novel method for handling this variability via a distributed parameter; where particular parameter values are possessed by various proportions of the population. Several authors (e.g., Kareva, Novozhilov, and Katriel) have studied such systems in a closed population setting (no births/immigrations or deaths/emigrations), but have cited restrictions to employing such methods when entry and removal of individuals is added to the population. This paper details, in the context of a simple susceptible-infectious-recovered (SIR) epidemic, how the method works in the closed population setting and gives conditions for initial, transient, and asymptotic results to be equivalent with the nondistributed case. Additionally, I show how the method may be applied to various forms of open SIR systems. Transport equations are used to transform an infinite dimensional system for the open population case into a finite dimensional system which is, at the very least, able to be numerically studied, a model with direct inheritance of the distributed parameter is shown to be qualitatively identical to the nondistributed case, and finally a model where disease results in sterilization is fully analyzed.