Background: simple SIS model Let’s start by refreshing the basics of the SIS model. Susceptible individuals (\(S\)) become infected and move into the infected class (\(I\)), and then infected individuals who recover move straight back to the susceptible class (so there’s no period of immunity like in the SIR model). For simplicity we will ignore births and death, and also use a proportional model i.e. \(S\) + \(I\) = \(N\) = 1.

Introduction In class we covered the SIR model with births and deaths. As a quick refresher: susceptible individuals (\(S\)) become infected and move into the infected class (\(I\)), and then infected individuals who recover move into the recovored, or immune, class (\(R\)). Assuming the birth rate is equal to the death rate (\(\mu\)) gives:
\[\begin{align*} \frac{dS}{dt} &= \mu N -\beta S I - \mu S\\ \frac{dI}{dt} &= \beta S I - \gamma I - \mu I\\ \frac{dR}{dt} &= \gamma I - \mu R.

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