Reducing an SIR model

Question

Past Paper Question:

Given a SIRS model:

\begin{align} \dfrac{dS}{dt} & = -\alpha IS+\gamma R \tag1 \\[8pt] \dfrac{dI}{dt} & = \alpha IS-\beta I\\[8pt] \dfrac{dR}{dt} & = \beta I-\gamma R, \end{align}

where $S$, $I$ and $R$ denote the number of susceptible, infected and recovered individuals respectively and $t$ is time.

Show that the total population N is constant, and hence reduce the system to two equations for $S$ and $I$.

My Attempt:

Since we are only looking over a small time interval, natural births and deaths are negligible hence we have a closed population so: $$S\left( t\right) +I\left( t\right) +R\left( t\right) =N \tag4 $$

My Question:

How do you use this to reduce the above to a system of two ODE's?

Answer 1

Hints:

1. Add the three equations together. What do you get?

2. If $S+I+R$ is constant, say it equals $N$, then one of the three variables $S$, $I$, $R$ is redundant. So eliminate $R$ from the first equation by substituting $R=N-S-I$. (Since $R$ is redundant, the third equation will follow from the other two, and can be ignored.)

Answer 2

Eliminate $R$ from (4) and (1) ( i.e., by plugging in (4) into (1) ):

$$ S'=-S( \alpha I +\gamma) + \gamma (N-I) $$

Only the rest two $S,I$ of first degree ODEs ( above equation of $S$ and the other $I$ as it is ) need be considered.