Consider the following matrix:
$$ \left( \begin{array}{ccc} -1/\tau & x(Y_E-X_E) & -gX_I-xX_E & -gY_I-xX_E \\ x(\lambda/\sigma)(X_E-Y_E) & -2/\tau & (\lambda/\sigma)(g^2X_I-xY_E) & (\lambda/\sigma)(g^2Y_I-xY_E)\\ xX_E+gX_I & xY_E+gX_I & -1/\tau & g(X_I-Y_I)\\ (\lambda/\sigma)(xX_E-g^2Y_I) & (\lambda/\sigma)(xY_E-g^2Y_I) & g^2(\lambda/\sigma)(X_I-Y_I) & -2/\tau\end{array} \right) $$
Take $\tau=1$. It seems pretty obvious to me that two of the eigenvalues are -1 and -2.
According to this dude whose paper I'm reading, the other two eigenvalues are negative if
$$ xX_E - gX_I + (xX_E + g^2Y_I - xg(g + 1)[X_EY_I-X_IY_E])\frac{\lambda}{2\sigma}<1. $$
How did he get to that conclusion?
Update
Following the advice of a commenter below, I computed the characteristic polynomial of this matrix.
It is
$$ (t + \frac{2}{\tau})(t + \frac{1}{\tau})(t^2+\frac{3}{\tau} t+\frac{2}{\tau^2} + \ldots) + (\frac{2}{\tau})(t + \frac{2}{\tau})gxX_EX_I + (\frac{1}{\tau})(t + \frac{2}{\tau})(g^2X_I^2+x^2X_E^2)+(t+\frac{2}{\tau})(\ldots)+(\frac{2}{\tau})(t + \frac{1}{\tau})\frac{\lambda^2}{\sigma^2}g^2xY_EY_I+(\frac{1}{\tau})(t+\frac{1}{\tau})(-\frac{\lambda^2}{\sigma^2}x^2Y_E^2-\frac{\lambda^2}{\sigma^2}g^4Y_I^2)+(t+\frac{1}{\tau})(\ldots)+(\frac{2}{\tau}t-\frac{2}{\tau})(\frac{\lambda}{\sigma}g^3X_I^2), $$
where $\dots$ means a lot of terms (without $t$ or $\tau$).
When I divide out the known roots -- i.e., divide by $(t + \frac{2}{\tau})$ and $(t + \frac{1}{\tau})$ -- I get
$$ (t^2+\frac{3}{\tau} t+\frac{2}{\tau^2} + \ldots) + \frac{(\frac{2}{\tau})gxX_EX_I}{(t + \frac{1}{\tau})} + \frac{(\frac{1}{\tau})(g^2X_I^2+x^2X_E^2)}{(t + \frac{1}{\tau})}+\frac{(\ldots)}{(t+\frac{1}{\tau})}+\frac{(\frac{2}{\tau})\frac{\lambda^2}{\sigma^2}g^2xY_EY_I}{(t + \frac{2}{\tau})}+\frac{(\frac{1}{\tau})(-\frac{\lambda^2}{\sigma^2}x^2Y_E^2-\frac{\lambda^2}{\sigma^2}g^4Y_I^2)}{(t+\frac{2}{\tau})}+\frac{(\ldots)}{(t+\frac{2}{\tau})}+\frac{(\frac{2}{\tau}t-\frac{2}{\tau})(\frac{\lambda}{\sigma}g^3X_I^2)}{(t+\frac{1}{\tau})(t+\frac{2}{\tau})}. $$
Where do I go from here?