# Equilibrium of simple SIS model

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:

$$$$I ( bS - a - c ) = 0$$$$

so $$\hat{I} = 0$$ or $$\hat{S} = \frac{a+c}{b}$$ must be accord with $$\dot{I} = 0$$. The first means that:

$$$$\dot{S} = a(0) - b S (0) = 0$$$$

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.
• Okay, thanks. I see that at for $\hat{I}=0$ and $\hat{S}$ equal to any value, stability depends on the ratio $\frac{a+c}{b \hat{S}} > 1$, which means infections will increase before dying out. But how do I find the equilibrium value of $\hat{S}$? My simulations show that when that ratio is greater than 1, $S$ settles down to an equilibrium, but I am struggling to compute what that is. Thanks. – user_15 Nov 3 '20 at 11:10
• Thanks for your edited response. However, I'm still not understanding how we can derive the equilibrium value of $\hat{S}$ from your answer. Is it possible? Or can we only derive the conditions for when disease will grow in the population? – user_15 Nov 3 '20 at 13:42
• What you now have is an implicit equation. You can solve it numerically or via the Lambert-W function, which is only slightly better than giving the numerical method a name. But the value of $\hat S$ is uniquely given by that equation. – Lutz Lehmann Nov 3 '20 at 13:51