15.4: The SI Equations

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SIR is also a “frequency-dependent” model, at one end of a spectrum which has “density-dependent” models at the other end. Frequency dependence approximates situations in which infection propagules are limited, while density dependence approximates situations in which potential victims are limited.

But do not be concerned with the full SIR model for now. We shall simplify it here to reveal its basic properties. First, suppose there is no recovery— this is an incurable disease that, once contracted, stays with its victim forever. Many viral diseases approximate this situation—herpes and HIV, for example. In gray below are all the terms that will drop out if there is no recovery.

\[\frac{dS}{dt}\,=\,b(S+I\color{grey}{+R}\color{black}\,)\,-\beta\,I\frac{S}{S+I\color{grey}{+R}}\,-\delta\,S\]

\[\frac{dI}{dt}\,=\beta\,I\frac{S}{S+I\color{grey}{+R}}\,\color{grey}{-\gamma\,I}\color{black}{\,-\alpha\,I}\]

\[\color{grey}{\frac{dR}{dt}\,=\gamma\,I\,-\delta\,R}\]

Removing those terms gives an “SI” model.

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

\[\frac{dI}{dt}\,=\beta\,I\frac{S}{S+I}\,-\alpha\,I\]

But we won’t be concerned with this model just now.