I need to show that the steady state of this non-dimensional model is stable using minimal algebra however I am not sure how to approach this without long lines of working.
The model is:
$$\frac{dS}{dt}=e-R_0SI-eS,\qquad \frac{dI}{dt}=R_0SI-I+\sigma R_0I(1-S-I)$$
and the endemic steady state is:
$$I=\frac{e}{2\sigma R_0}\left[\sqrt{\left(\frac{1-R_0\sigma}{e}+\sigma \right)^2+4(R_0-1)\frac{\sigma}{e}} -\left(\frac{1-R_0\sigma}{e}+\sigma \right) \right]$$
I understand that this can be shown by working through long lines of working however the question states explicitly it should not have lots of algebra. Any help would be appreciated.