Advancing Mathematical Epidemiology and Chemical Reaction Network Theory via Synergies Between Them

TL;DR

This paper explores the intersection of chemical reaction network theory and mathematical epidemiology, proposing new definitions, methods, and educational tools to enhance understanding and analysis in both fields.

Contribution

It introduces a formal definition of ME models using CRN concepts, promotes the next generation matrix approach for stability analysis, and provides a Mathematica package for learning both disciplines.

Findings

Proposed a new definition of ME models in the CRN framework

Highlighted the usefulness of the next generation matrix approach for boundary stability

Developed a Mathematica package 'BootCamp' for educational purposes

Abstract

Our paper reviews some key concepts in chemical reaction network theory and mathematical epidemiology, and examines their intersection, with three goals. The first is to make the case that mathematical epidemiology (ME), and also related sciences like population dynamics, virology, ecology, etc., could benefit by adopting the universal language of essentially non-negative kinetic systems as developed by chemical reaction network (CRN) researchers. In this direction, our investigation of the relations between CRN and ME lead us to propose for the first time a definition of ME models, stated in Open Problem 1. Our second goal is to inform researchers outside ME of the convenient next generation matrix (NGM) approach for studying the stability of boundary points, which do not seem suficiently well known. Last but not least, we want to help students and researchers who know nothing about either ME or CRN to learn them quickly, by offering them a Mathematica package "BootCamp", located at https://github.com/adhalanay/epidemiology_crns, including illustrating notebooks (and certain sections below will contain associated suggested notebooks; however, readers with experience may safely skip the bootcamp). We hope that the files indicated in the titles of various sections will be helpful, though of course improvement is always possible, and we ask the help of the readers for that.