Math 4600
Homework 6
Please, note that even though this homework is due after the midterm, the
problems here can and should be used for midterm practice. So, you are strongly
encouraged to do them before the midterm.
1. Here is the HIV dynamics model that we considered in class:
dT
T
) − dT T − kV T
= s + pT (1 −
dt
Tmax
dT ∗
= kV T − δT ∗
dt
dV
= N δT ∗ − cV
dt
The RT-inhibitor drug blocks the reverse transcription, i.e. in the presence of
this drug the virus is not able to infect T-cells.
a) Which parameter is affected by the RT-inhibitor? Explain
Imagine that the inhibitor is perfect, i.e. it sets the appropriate parameter
to zero. In the resulting system T ∗ and V equations are uncoupled from the T
equation. Use this system to answer questions b) and c).
b) In the (T ∗ , V ) system find the steady states and their stability analytically
(by computing the eigenvalues). Then, draw a phase plane with nullclines and
representative direction field arrows. Finally, draw qualitative time courses for
T ∗ (t) and V (t) starting with some high initial values. (Assume c = N kT0 if you
need to).
c) In the T equation with s = 0 find the steady states and stability analytically
and using the phase line. Discuss what will be the long-term condition of T-cell
population according to this equation.
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