A model fix gone wrong: investigation of an SIS model with saturating treatment

ABSTRACT

A textbook model of a contagious disease, the dynamics of which are represented by the SIS epidemic model with saturating treatment, is considered. I show that this model, as originally formulated, is not dimensionally consistent. The model can be fixed by including a dimensional constant a of value one (with units ${\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{d}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{s}}^{-1}$). At first sight, it appears that this is a cosmetic book-keeping exercise that will have no flow-on effects on the analysis of the model. In fact, we will discover that setting the value $a=1{\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{d}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{s}}^{-1}$ produces very unrealistic outcomes and that the parameter must have value $a\ll 1{\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{d}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{s}}^{-1}$. Necessary skills for students to work through this model are an understanding of steady-state solutions and linearized stability for a scalar differential equation and, most importantly, an understanding of units.

KEYWORDS:

• Modelling
• SIS
• steady-state solutions
• units

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 97M10

Acknowledgments

The author thanks the referees for their constructive comments which led to a greatly improved paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).