How to factor the following?

Question

Consider the Jacobian:

$$ J = \begin{bmatrix} -\frac{\beta I}{N} -\xi & -\frac{\beta S}{N} -\xi \\[1ex] \phantom{-}\frac{\beta I}{N} & \phantom{-}\frac{\beta S}{N} -\gamma \end{bmatrix}. $$

Finding the eigenvalues of this matrix at its endemic equilibrium we have:

$$ \lambda_1=(-be-e^2 - \sqrt[]{e} \sqrt[]{-4 b g^2 + 4 g^3 + b^2 e - 8 b g e + 8 g^2 e - 2 b e^2 + 4 g e^2 + e^3}/(2 (g + e))$$, $$\lambda_2=(-b e - e^2 + \sqrt[]{e} \sqrt[]{-4 b g^2 + 4 g^3 + b^2 e - 8 b g e + 8 g^2 e - 2 b e^2 + 4 g e^2 + e^3}/(2 (g + e)$$

Where $b=\beta, \;g=\gamma,\; e=\xi$.

Mathematica wouldn't simplify(Fullsimplify doesn't work!) it to the following which I did by hand:

\begin{align} \lambda_1 &= \frac{-\xi\left(\beta +\xi\right) + \sqrt{\xi^2 \left(\beta +\xi\right)^2 -4\xi \left(\gamma+\xi\right)^2\left(\beta -\gamma\right)} }{2\left(\gamma + \xi\right)}, \\[2ex] \lambda_2 &= \frac{-\xi\left(\beta +\xi\right) - \sqrt{\xi^2 \left(\beta +\xi\right)^2 -4\xi \left(\gamma+\xi\right)^2\left(\beta -\gamma\right)} }{2\left(\gamma + \xi\right)}. \end{align}

Now my question is, how would I do something similar to the following:

(i)$$ J = \begin{bmatrix} -\frac{\beta I}{N} -\xi -\nu & -\frac{\beta S}{N} -\xi \\[1ex] \phantom{-}\frac{\beta I}{N} & \phantom{-}\frac{\beta S}{N} -\gamma -\nu \end{bmatrix}. $$

Finding the eigenvalues at the endemic equilibrium we have:

$${(1/(2 (e + g + v)))(-b e - e^2 - b v - e v - \sqrt[]{e + v} \sqrt[]{b^2 e - 2 b e^2 + e^3 - 8 b e g + 4 e^2 g - 4 b g^2 + 8 e g^2 + 4 g^3 + b^2 v - 6 b e v + 5 e^2 v - 8 b g v + 16 e g v + 12 g^2 v - 4 b v^2 + 8 e v^2 + 12 g v^2 + 4 v^3}, (1/( 2 (e + g + v)))(-b e - e^2 - b v - e v + \sqrt[]{e + v} \sqrt[]{b^2 e - 2 b e^2 + e^3 - 8 b e g + 4 e^2 g - 4 b g^2 + 8 e g^2 + 4 g^3 + b^2 v - 6 b e v + 5 e^2 v - 8 b g v + 16 e g v + 12 g^2 v - 4 b v^2 + 8 e v^2 + 12 g v^2 + 4 v^3}}$$ where $b=\beta$, $e=\xi$, $g=\gamma$, and $v=\nu$.

(ii)$$ J = \begin{bmatrix} -\frac{\beta I}{N} -\nu & -\frac{\beta S}{N} +\gamma \\[1ex] -\sigma & -\gamma -\sigma-\nu \end{bmatrix}. $$

Finding the eigenvalues at the endemic equilibria we have:

$$(1/(2 m (g + m + v)))(-g^2 m - b m^2 - g m^2 - m^3 - 2 g m v - 2 m^2 v - m v^2 - m \sqrt[]{g^4 - 2 b g^2 m + 6 g^3 m + b^2 m^2 - 6 b g m^2 + 11 g^2 m^2 - 2 b m^3 + 6 g m^3 + m^4 + 8 g^3 v - 4 b g m v + 28 g^2 m v - 4 b m^2 v + 28 g m^2 v + 8 m^3 v + 18 g^2 v^2 - 2 b m v^2 + 38 g m v^2 + 18 m^2 v^2 + 16 g v^3 + 16 m v^3 + 5 v^4}, (1/(2 m (g + m + v)))(-g^2 m - b m^2 - g m^2 - m^3 - 2 g m v - 2 m^2 v - m v^2 + m \sqrt[]{(g^4 - 2 b g^2 m + 6 g^3 m + b^2 m^2 - 6 b g m^2 + 11 g^2 m^2 - 2 b m^3 + 6 g m^3 + m^4 + 8 g^3 v - 4 b g m v + 28 g^2 m v - 4 b m^2 v + 28 g m^2 v + 8 m^3 v + 18 g^2 v^2 - 2 b m v^2 + 38 g m v^2 + 18 m^2 v^2 + 16 g v^3 + 16 m v^3 + 5 v^4}$$

where $b=\beta$, $m=\sigma$, $g=\gamma$, and $v=\nu$.

How can I factor the above like in the first example?

ADDENDUM

(iii)$$ J = \begin{bmatrix} -\frac{\beta I}{N} -\nu & -\frac{\beta S}{N} \\[1ex] -\sigma & -\sigma-\nu \end{bmatrix}. $$

Once again, finding the eigenvalues at the endemic equilibria we have

$$ (-b m^2 - m^3 - 2 m^2 v - m v^2 - m \sqrt[]{b^2 m^2 - 2 b m^3 + m^4 - 4 b m^2 v + 8 m^3 v - 2 b m v^2 + 18 m^2 v^2 + 16 m v^3 + 5 v^4})/(2 m (m + v)), (-b m^2 - m^3 - 2 m^2 v - m v^2 + m \sqrt[]{b^2 m^2 - 2 b m^3 + m^4 - 4 b m^2 v + 8 m^3 v - 2 b m v^2 + 18 m^2 v^2 + 16 m v^3 + 5 v^4})/(2 m (m + v)) $$

Using Fullsimplify we have

$$-((b m + (m + v)^2 + \sqrt[]{ b^2 m^2 - 2 b m (m + v)^2 + (m + v)^3 (m + 5 v)})/(2 (m + v))), -(( b m + (m + v)^2 - \sqrt[]{ b^2 m^2 - 2 b m (m + v)^2 + (m + v)^3 (m + 5 v)})/(2 (m + v)))$$

Considering Blue's solution to my other problems, is this the simplest it will get?

ADDENDUM 2

(iv)$$ J = \begin{bmatrix} -\frac{\beta I}{N} -\nu & -\frac{\beta S}{N} \\[1ex] \frac{\beta I}{N} & \frac{\beta S}{N} - \gamma - \nu \end{bmatrix}. $$

Once again, finding the eigenvalues at the endemic equilibria we have

$$ (-b v - \sqrt[]v \sqrt[]{-4 b g^2 + 4 g^3 + b^2 v - 8 b g v + 12 g^2 v - 4 b v^2 + 12 g v^2 + 4 v^3})/(2 (g + v)), (-b v + \sqrt[]v \sqrt[]{-4 b g^2 + 4 g^3 + b^2 v - 8 b g v + 12 g^2 v - 4 b v^2 + 12 g v^2 + 4 v^3})/(2 (g + v)) $$

Using Fullsimplify we have

$$-((b v + \sqrt[]v \sqrt[]{b^2 v - 4 b (g + v)^2 + 4 (g + v)^3})/( 2 (g + v))), (-b v + \sqrt[]v \sqrt[]{b^2 v - 4 b (g + v)^2 + 4 (g + v)^3})/(2 (g + v))$$

Answer 1

For (i), I get to $$\frac{-h k \mp \sqrt{\;h^2 k^2 + 4k\ell^2(\ell-h)\;}}{2\ell} \tag1$$ where $h:=b+e$, $k:=e+v$, $\ell:=e+g+v$.

For (ii),

$$\frac{ - (p^2 + q) \mp \sqrt{ (p^2 - q)^2 + 4 v p^3 }}{2 p} \tag2$$ where $p:=g+m+v$ and $q:=m(b-g)$.

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Addendum. OP has provided a bit of backstory to (ii), so I thought I'd use it to illustrate how sneaking-up on a solution can sometimes be easier than trying to find patterns in a final result generated by brute-force application of a computer algebra system. (This a common strategy for me.)

Take OP's Jacobian $J$ for (ii) $$J = \begin{bmatrix}-\frac{bI}{N}-v & -\frac{bS}{N}+g \\ -m & -g-m-v\end{bmatrix} \tag{A.1}$$ and substitute $$S\to \frac{N(g+v)(m+v)}{bm} \qquad I\to\frac{Nm(g+v)(m+v)}{bm(g+m+v)}\left(\frac{bm}{(g+v)(m+v)}-1\right) \tag{A.2}$$ Now, generally, if the Eigenvalues command (with FullSimplify or whatever) gives a pretty result, take it and run. If the result is a mess, step back and take a semi-manual approach. Here, use CharacteristicPolynomial to get a polynomial of the form $Ax^2+Bx+C$. The coefficients may still be messes, but they're smaller messes. To isolate them, use CoefficientList (with, say, Factor). The results in this case are $$\begin{align} A &= g+m+v \\ B &= g^2 + b m + g m + m^2 + 2 g v + 2 m v + v^2 \\ C &= (g + m + v) (b m - g m - g v - m v - v^2) \end{align} \tag{A.3}$$ The forms of $A$ and $C$ suggest that $g+m+v$ wants to be a unified quantity; let's call it $p$. Note that we can find less-obvious instances of $p$ elsewhere in the coefficients: $$\begin{align} A &= p &&= p\\ B &= b m + g^2 + m^2 + v^2 + g m + 2 g v + 2 m v &&= bm + p^2 - gm = (b-g)m+p^2 \\ C &= p (b m - g m - g v - m v - v^2) &&= p((b-g)m-pv) \end{align} \tag{A.4}$$ These forms suggest that $(b-g)m$ also wants to be a unified quantity; call it $q$. So, we have: $$A = p\qquad B=p^2+q \qquad C = p(q-pv)$$ From here, applying the Quadratic Formula is almost effortless, and we get roots (ie, eigenvalues of $J$) of the form $$\frac{-B\pm\sqrt{B^2-4AC}}{2A}=\frac{-(p^2+q)\pm \sqrt{\Delta}}{2p} \tag{A.5}$$ where $$\begin{align} \Delta &:= (p^2+q)^2-4p^2(q-pv) \\ &\;= p^4+2p^2q+q^2\;-4p^2q+4p^3v \\ &\;= p^4-2p^2q+q^2\;+4p^3v \\ &\;= (p^2-q)^2+4p^3v \end{align} \tag{A.6}$$ Thus, $(A.5)$ gives us $(2)$ as before. $\square$

Finding a cleaner form of $\Delta$ was far easier in $pq$-form than when sifting through the $22$-term counterpart given in the original question. Seeing components in earlier stages of a problem can inform how they want to be organized. (I suspect that the groupings $g+m+v$ and $m(b-g)$ are meaningful even in the earliest stages of the problem, reflecting how parameters in the SEI(S) models influence each other.)