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

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.

Figures (6)

• Figure 1: The FHJ graph of the SIRS with demography \ref{['SIRS']} renders clear that the CRN is not weakly reversible. The orders of the eight reactions are (0,2,1,1,1,1,1,1).
• Figure 2: The "monomolecular SIRS" has one linkage class, is weakly reversible (WR), and has zero deficiency (ZD) 4 $-$ 3 $-$ 1 = 0.
• Figure 3: Chart flow/species graph of the SI$^2$R model, with two infected classes and extra deaths at rate $\delta$. The red edge corresponds to the entrance of susceptibles into the disease classes, the brown edges are the rate of the transition matrix V, and the cyan dashed lines correspond to the rate of loss of immunity. The remaining black lines correspond to the inputs and outputs of the birth and natural death rates, respectively, which are equal in this case.
• Figure 4: The Feinberg--Horn--Jackson graph of the "SAIR network" with $n_V=9$ vertices $(S,A,I,R,S+A,S+I, 2A, A+I,0)$ (where the 0 node represents the exterior), 12 edges, and 3 linkage classes. The deficiency is $n_V-rank(\Gamma)-n_C=9-4-3=2$, and weak reversibility does not occur, so neither of the conditions for having complex-balanced equilibria holds.
• Figure 5: Chart flow/species graph of the SLIAR model \ref{['SLIARG']}. The red edge corresponds to the entrance of susceptibles into the entrance disease class L, the cyan dashed lines correspond to the rates of recovery, and the brown and black edges are the rates of the transition matrix V towards the interior and exterior, respectively.

Theorems & Definitions (46)

• Remark 1
• Remark 2
• Remark 3
• Definition 1
• Example 1
• Lemma 1
• Definition 2
• Example 2
• Remark 4
• Remark 5