Question: Local Stability Analysis of Disease Free and Endemic Equilibrium using Routh-Hurwitz Criterion

I am currently facing an issue as regards stability analysis and I will be glad if anyone can offer me some suggestions.  My aim is to have a Theorem in my research stating that the Disease free equilibrium is locally asymptotically stable if R0 (Basic reproduction number) < 1 and unstable if R0 > 1. The same applies to the endemic equilibrium.  I am trying to carry out a stability analysis using Linearization. However, due to the nature of my model, not all my eigenvalues are negative. Below is my Jacobian Matrix:

ROW 1                           ROW 2                       ROW 3                         ROW 4

Column 1:=                                                                                                      

Column 2:=                 -psi                                     0                                   M

Column 3:=                                                                                                           

Column 4:=                 0                                0                                0                        +  M +  -

By using the Routhz Hurwitz criterion, I reduced the matrix block from 4 by 4 to 3 by 3 by using column 4 as co-factor, however, after this matrix reduction, I was still unable to get all terms relating to my basic reproduction number for me to compare and see if the conditions in Routh Criterion is satisfied using the condition R0 < 1 holds.

My Basic Reproduction Number is as follows:

Are there any other methods to  carry out local stability analysis? and Can local stability analysis be done using maple?

Thank you in advance