## Difficult Differential Equation System

The following is a model for HIV.
Infected cells = T
Concentration of viral particles = C

$\dot{T} = 0.06V - \frac{T}{2} , \dot{V} = 100T - cV$

where $c > 0$ is the rate constant for viral clearance.

1) Show that there are 2 possible types of critical points at the origin and one dividing case, and state the values of c which correspond to each case. In each case clearly state the stability of characteristics of the critical point at the origin.

2) If c = 5.5 what will the phase portrait of the system look like. (teacher said to look at the phase portrait non-uniformly, say take (-0.01, 0.01) * (-1,-1)).

3) Under treatment, the model changes to

$\dot{T} = 0.06v - \frac{T}{3}, \dot{V} = -cV$

What are the possible types of critical point at the origin in this model?

4) Introducing a new variable, $V_N$ which is the non-infected particles. The new system is

$\dot{T} = 0.06v - \frac{T}{3}, \dot{V} = -cV , \dot{V_N} = 100T - cV_N$

Prove that the eigenvalues of the linearized matrix are all negative in this case and find out what they are.
Why are the solutions to this no simply exponentials?

I know this is a massive question, so I am going to show my working in a separate post shortly.