Suppose we have a 5x5 matrix, how would we check the whether the eigenvalues are negative so that we can conclude they are locally stable?
As requested in the comment by Woody3:
here is the matrix:
m1 = {{-\[Beta]1 i2 - \[Beta]2 j - \[Beta]3 a - \[Mu],
0, -\[Beta]1 s, -\[Beta]2 s, -\[Beta]3 s}, {p \[Beta]1 i2 +
q \[Beta]2 j + r \[Beta]3 a, -b1, p \[Beta]1 s,
q \[Beta]2 s + \[Xi]1,
r \[Beta]3 s}, {(1 - p) \[Beta]1 i2 + (1 - q) \[Beta]2 j + (1 -
r) \[Beta]3 a, \[Epsilon], (1 - p) \[Beta]1 s -
b2, (1 - q) \[Beta]2 s + \[Xi]2, (1 - r) \[Beta]3 s}, {0, 0,
p1, -b3, 0}, {0, 0, 0, p2, -b4}} /. {s -> ss, i1 -> i1s,
i2 -> i2s, j -> js, a -> as};
where:
ss = FullSimplify[
Together[
Expand[(-b1 b2 b3 b4 + b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2)/(-b1 b3 b4 \[Beta]1 + b1 b3 b4 p \[Beta]1 -
b1 b4 p1 \[Beta]2 + b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] - p1 p2 r \[Beta]3 \[Epsilon])]]];
i1s = FullSimplify[
Together[
Expand[(-b2 b3^2 b4 p \[Beta]1 \[Nu] -
b2 b3 b4 p1 q \[Beta]2 \[Nu] - b2 b3 p1 p2 r \[Beta]3 \[Nu] -
b3 b4 p1 \[Beta]1 \[Nu] \[Xi]1 +
b3 b4 p p1 \[Beta]1 \[Nu] \[Xi]1 -
b4 p1^2 \[Beta]2 \[Nu] \[Xi]1 +
b4 p1^2 q \[Beta]2 \[Nu] \[Xi]1 -
p1^2 p2 \[Beta]3 \[Nu] \[Xi]1 +
p1^2 p2 r \[Beta]3 \[Nu] \[Xi]1 +
b3 b4 p p1 \[Beta]1 \[Nu] \[Xi]2 +
b4 p1^2 q \[Beta]2 \[Nu] \[Xi]2 +
p1^2 p2 r \[Beta]3 \[Nu] \[Xi]2 + (
b2 b3^2 b4 p \[Beta]1 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
b2 b3 b4 p1 q \[Beta]2 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
b2 b3 p1 p2 r \[Beta]3 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
b3 b4 p1 \[Beta]1 \[Mu] \[Xi]1 (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) - (
b3 b4 p p1 \[Beta]1 \[Mu] \[Xi]1 (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
b4 p1^2 \[Beta]2 \[Mu] \[Xi]1 (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) - (
b4 p1^2 q \[Beta]2 \[Mu] \[Xi]1 (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
p1^2 p2 \[Beta]3 \[Mu] \[Xi]1 (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) - (
p1^2 p2 r \[Beta]3 \[Mu] \[Xi]1 (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) - (
b3 b4 p p1 \[Beta]1 \[Mu] \[Xi]2 (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) - (
b4 p1^2 q \[Beta]2 \[Mu] \[Xi]2 (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) - (
p1^2 p2 r \[Beta]3 \[Mu] \[Xi]2 (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]))/(-b1 b2 b3^2 b4 \[Beta]1 -
b1 b2 b3 b4 p1 \[Beta]2 - b1 b2 b3 p1 p2 \[Beta]3 +
b3 b4 p1 \[Beta]1 \[Epsilon] \[Xi]1 +
b4 p1^2 \[Beta]2 \[Epsilon] \[Xi]1 +
p1^2 p2 \[Beta]3 \[Epsilon] \[Xi]1 +
b1 b3 b4 p1 \[Beta]1 \[Xi]2 + b1 b4 p1^2 \[Beta]2 \[Xi]2 +
b1 p1^2 p2 \[Beta]3 \[Xi]2)]]];
i2s = FullSimplify[
Together[
Expand[(-b1 b3^2 b4 \[Beta]1 \[Nu] + b1 b3^2 b4 p \[Beta]1 \[Nu] -
b1 b3 b4 p1 \[Beta]2 \[Nu] + b1 b3 b4 p1 q \[Beta]2 \[Nu] -
b1 b3 p1 p2 \[Beta]3 \[Nu] + b1 b3 p1 p2 r \[Beta]3 \[Nu] -
b3^2 b4 p \[Beta]1 \[Epsilon] \[Nu] -
b3 b4 p1 q \[Beta]2 \[Epsilon] \[Nu] -
b3 p1 p2 r \[Beta]3 \[Epsilon] \[Nu] + (
b1 b3^2 b4 \[Beta]1 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) - (
b1 b3^2 b4 p \[Beta]1 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
b1 b3 b4 p1 \[Beta]2 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) - (
b1 b3 b4 p1 q \[Beta]2 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
b1 b3 p1 p2 \[Beta]3 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) - (
b1 b3 p1 p2 r \[Beta]3 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
b3^2 b4 p \[Beta]1 \[Epsilon] \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
b3 b4 p1 q \[Beta]2 \[Epsilon] \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
b3 p1 p2 r \[Beta]3 \[Epsilon] \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]))/(-b1 b2 b3^2 b4 \[Beta]1 -
b1 b2 b3 b4 p1 \[Beta]2 - b1 b2 b3 p1 p2 \[Beta]3 +
b3 b4 p1 \[Beta]1 \[Epsilon] \[Xi]1 +
b4 p1^2 \[Beta]2 \[Epsilon] \[Xi]1 +
p1^2 p2 \[Beta]3 \[Epsilon] \[Xi]1 +
b1 b3 b4 p1 \[Beta]1 \[Xi]2 + b1 b4 p1^2 \[Beta]2 \[Xi]2 +
b1 p1^2 p2 \[Beta]3 \[Xi]2)]]];
js = FullSimplify[
Together[
Expand[(-b1 b3 b4 p1 \[Beta]1 \[Nu] +
b1 b3 b4 p p1 \[Beta]1 \[Nu] - b1 b4 p1^2 \[Beta]2 \[Nu] +
b1 b4 p1^2 q \[Beta]2 \[Nu] - b1 p1^2 p2 \[Beta]3 \[Nu] +
b1 p1^2 p2 r \[Beta]3 \[Nu] -
b3 b4 p p1 \[Beta]1 \[Epsilon] \[Nu] -
b4 p1^2 q \[Beta]2 \[Epsilon] \[Nu] -
p1^2 p2 r \[Beta]3 \[Epsilon] \[Nu] + (
b1 b3 b4 p1 \[Beta]1 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) - (
b1 b3 b4 p p1 \[Beta]1 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
b1 b4 p1^2 \[Beta]2 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) - (
b1 b4 p1^2 q \[Beta]2 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
b1 p1^2 p2 \[Beta]3 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) - (
b1 p1^2 p2 r \[Beta]3 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
b3 b4 p p1 \[Beta]1 \[Epsilon] \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
b4 p1^2 q \[Beta]2 \[Epsilon] \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
p1^2 p2 r \[Beta]3 \[Epsilon] \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]))/(-b1 b2 b3^2 b4 \[Beta]1 -
b1 b2 b3 b4 p1 \[Beta]2 - b1 b2 b3 p1 p2 \[Beta]3 +
b3 b4 p1 \[Beta]1 \[Epsilon] \[Xi]1 +
b4 p1^2 \[Beta]2 \[Epsilon] \[Xi]1 +
p1^2 p2 \[Beta]3 \[Epsilon] \[Xi]1 +
b1 b3 b4 p1 \[Beta]1 \[Xi]2 + b1 b4 p1^2 \[Beta]2 \[Xi]2 +
b1 p1^2 p2 \[Beta]3 \[Xi]2)]]];
as = FullSimplify[
Together[
Expand[(-b1 b3 b4 p1 p2 \[Beta]1 \[Nu] +
b1 b3 b4 p p1 p2 \[Beta]1 \[Nu] -
b1 b4 p1^2 p2 \[Beta]2 \[Nu] +
b1 b4 p1^2 p2 q \[Beta]2 \[Nu] - b1 p1^2 p2^2 \[Beta]3 \[Nu] +
b1 p1^2 p2^2 r \[Beta]3 \[Nu] -
b3 b4 p p1 p2 \[Beta]1 \[Epsilon] \[Nu] -
b4 p1^2 p2 q \[Beta]2 \[Epsilon] \[Nu] -
p1^2 p2^2 r \[Beta]3 \[Epsilon] \[Nu] + (
b1 b3 b4 p1 p2 \[Beta]1 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) - (
b1 b3 b4 p p1 p2 \[Beta]1 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
b1 b4 p1^2 p2 \[Beta]2 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) - (
b1 b4 p1^2 p2 q \[Beta]2 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
b1 p1^2 p2^2 \[Beta]3 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) - (
b1 p1^2 p2^2 r \[Beta]3 \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
b3 b4 p p1 p2 \[Beta]1 \[Epsilon] \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
b4 p1^2 p2 q \[Beta]2 \[Epsilon] \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]) + (
p1^2 p2^2 r \[Beta]3 \[Epsilon] \[Mu] (-b1 b2 b3 b4 +
b4 p1 \[Epsilon] \[Xi]1 +
b1 b4 p1 \[Xi]2))/(-b1 b3 b4 \[Beta]1 +
b1 b3 b4 p \[Beta]1 - b1 b4 p1 \[Beta]2 +
b1 b4 p1 q \[Beta]2 - b1 p1 p2 \[Beta]3 +
b1 p1 p2 r \[Beta]3 - b3 b4 p \[Beta]1 \[Epsilon] -
b4 p1 q \[Beta]2 \[Epsilon] -
p1 p2 r \[Beta]3 \[Epsilon]))/(-b1 b2 b3^2 b4^2 \[Beta]1 -
b1 b2 b3 b4^2 p1 \[Beta]2 - b1 b2 b3 b4 p1 p2 \[Beta]3 +
b3 b4^2 p1 \[Beta]1 \[Epsilon] \[Xi]1 +
b4^2 p1^2 \[Beta]2 \[Epsilon] \[Xi]1 +
b4 p1^2 p2 \[Beta]3 \[Epsilon] \[Xi]1 +
b1 b3 b4^2 p1 \[Beta]1 \[Xi]2 + b1 b4^2 p1^2 \[Beta]2 \[Xi]2 +
b1 b4 p1^2 p2 \[Beta]3 \[Xi]2)]]];
and:
r0 = (\[Nu]/\[Mu]) (1/ss);