I have a simple SIS model with a non-constant population size due to disease-related mortality:
\begin{aligned} \frac{dS}{dt} &= a I - b S I\\\\ \frac{dI}{dt} &= b S I - a I - c I \end{aligned}
where $S$ and $I$ are the susceptible and infected classes, $a$ is the rate of transition back to the susceptible class after being infected, $b$ is the per-capita transmission rate, and $c$ is the disease-related mortality for infecteds.
Finding the equilibria of this model is stumping me for some reason. First, I factor the second equation to get:
\begin{equation} I ( bS - a - c ) = 0 \end{equation}
so $\hat{I} = 0$ or $\hat{S} = \frac{a+c}{b}$ must be accord with $\dot{I} = 0$. The first means that:
\begin{equation} \dot{S} = a(0) - b S (0) = 0 \end{equation}
and so $\hat{S}$ must be its initial condition, $S_0$.
However, there is another equilibrium where the disease spreads and dies out, meaning $\hat{S} < S_0$, which I assumed is equal to $\hat{S} = \frac{a+c}{b}$.
However, using simultation in $R$, this equilibrium isn't correct - $\frac{a+c}{b}$ is still off by a fair amount.
Can anyone see where I'm going wrong? Thanks!
For $I=0$ you can have any value of $S$ to get an equilibrium position, the equilibrium equations do not impose further conditions on $S$. If you add the stability at $I=0$, then you have to add the condition $bS−a−c<0$. Outside that region the values for $I$ will increase.
If you divide both equations, you find that $$ \frac{dS}{dI}=\frac{a-bS}{bS-a-c} $$ is a separable equation, so that $$ I+S-\frac{c}{b}\ln|bS-a| $$ is a conserved quantity. That is, for the limit with $\hat I=0$ you get the equation $$ \hat S-\frac{c}{b}\ln|b\hat S-a|=I_0+S_0-\frac{c}{b}\ln|bS_0-a| \\~\\ e^{b\hat S-a}(b\hat S-a)=e^{b(I_0+S_0)-a}(bS_0-a) \\~\\ \hat S=\frac1b\left(a+W_0(e^{b(I_0+S_0)-a}(bS_0-a))\right) $$ for the limit value $\hat S$. The sign of $bS-a$ has to stay constant, $W_0$ is the Lambert-W function.
This only works for the simplest of models, more complex models will usually not give rise to conserved quantities/constants of motion.