Separable mixing: the general formulation and a particular example focusing on mask efficiency

TL;DR

This paper develops a general epidemic modeling framework with heterogeneity, simplifies it under separable mixing, and demonstrates that individual protection is most effective for population health, confirming previous network model results.

Contribution

It introduces a unified formulation of heterogeneous epidemic models and shows the robustness of the conclusion that individual protection benefits the entire population.

Findings

Separable mixing simplifies the epidemic model to a scalar renewal equation.

Protecting oneself is the most effective strategy for population protection.

The conclusion about individual protection's importance holds beyond network models.

Abstract

The aim of this short note is twofold. We formulate the general Kermack-McKendrick epidemic model incorporating static heterogeneity and show how it simplifies to a scalar Renewal Equation (RE) when separable mixing is assumed. A key feature is that all information about the heterogeneity is encoded in one nonlinear real valued function of a real variable. Inspired by work of R. Pastor-Satorras and C. Castellano, we next investigate mask efficiency and demonstrate that it is straightforward to rederive from the RE their main conclusion, that the best way to protect the population as a whole is to protect yourself. Thus we establish that this conclusion is robust, in the sense that it also holds outside the world of network models.