# 15.3: The SIR Equations

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

The entire process depicted in Figure 15.2.1 is equivalent to the following set of equations.

$$\frac{dS}{dt}\,=\,b(S+I+R)\,-\beta\,I\frac{S}{S+I+R}\,-\delta\,S$$

$$\frac{dI}{dt}\,=\beta\,I\frac{S}{S+I+R}\,-\gamma\,I\,-\alpha\,I$$

$$\frac{dR}{dt}\,=\gamma\,I\,-\delta\,R$$

At the left in each equation is the net rate of change of each box, accounting for all arrows transferring individuals out of one box and into another. Again, $$S$$ is the density of susceptible individuals, $$I$$ the density of infected individuals, and $$R$$ the density of recovered individuals. Note that the terms are balanced— the term $$\gamma\,I$$, for example, representing individuals entering the recovered box in the last equation, is balanced by the complementary term $$-\gamma\,I$$, leaving the infected box in the middle equation.

The SIR model is another “macroscale model.” With recent changes in computation, “microscale models,” processing tens or hundreds of millions of individual hosts, are becoming more widely used. They can reliably take you beyond what purely mathematical formulations can do. More about them in later chapters.

This page titled 15.3: The SIR Equations is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Clarence Lehman, Shelby Loberg, & Adam Clark (University of Minnesota Libraries Publishing) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.