steady state or equilibrium

Question

I need to find the equilibrium point of the following system of equation:

$ds/dt = u - asi - bsw - us$
$di/dt = asi + bsw - (u + g)i$
$dw/dt = e(i - w)$

I found that $$s = \frac{u}{ai + bw + u}$$ when I plug into $i$ that's where I got confused. I have the following
$aiu + bwu -i(u+g)(ai+bw+u) = 0$
and I don't know what to do. Can somebody help me to figure out the value of $i$ with what I actually tried?

Answer 1

You need the right sides of all three equations to be $0$.

Start with the last equation: assuming $e \ne 0$, you need $i=w$. Then the other equations say

$$ \eqalign{ u - (a + b) s w - u s &= 0\cr (a + b) s w - (u + g) w &= 0\cr}$$ The second of these factors nicely, and says either $w=0$ or $s = (u+g)/(a+b)$. If $w=0$ and $u \ne 0$, the first equation says $s=1$. If $s = (u+g)/(a+b)$, the first equation says ...