1. Here is the HIV dynamics model that we considered in class: dT dt
= s + pT (1
−
T
T max
)
− d
T
T
− kV T dT ∗ dt dV dt
=
= kV T
N δT ∗
−
−
δT ∗ cV
Consider the model during the RT-inhibitor treatment. Assume 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 a) and b).
a) 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 kT
0 if you need to).
b) 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 at different parameter values. Explain why these are reasonable outcomes at these parameter ranges.
Next 2 problems focus on the HIV-model we last considered in class: when the protease inhibitor completely blocks the production of infectious virus particles. (As a side note: if you are wondering what happens when the drug is assumed to be only patially effective, for a look at some real data, and much more information, see the review article posted under “Lecture material” part of the class website.)
2. In an experiment the concentration of infectious viral particles V
I
( t ) was measured in one patient.
a) When plotted on a log-scale the graph was approximately linear with the slope of -3.1 day −
1 . Use these data to estimate value of c from the model solution we obtained in class.
b) What is the half-life of virus particles with this parameter value? (half time is the time in which he population reduces to half its initiial size)
3. Pre-treatment estimate of viral load in a patient was found to be 130
×
10 3 virus particles per ml of plasma.
a) What is the clearance rate of the virus per ml of plasma (change in virus concentration), using your estimate of c from above?
b) At steady state, the production rate of the virus must equal its clearance rate. What is the total viral production per day pre-treatment if the person’s
1

fluid volume is estimated to be 15 liters? This is a very high number of viruses produced daily. When it was first estimated, it was a surprise.
4.
Computing.
It has been estimated from data that δ = 0 .
5 day −
1 . Plot in Matlab total number of viruses V ( t ) as functions of time in regular and logscale (assume same values for c and V
0 as above. You will need to solve the equations analytically for the infectious and non-infectious virus populations.
You can assume also N kT
0
T = T
0
. Equation for T ∗ can be solved with T ∗
0
= kV
0
T
0
/δ ,
= c .). Notice that there is a “shoulder” in the beginning, indicative of a slower initial decay rate of V . Suggest a possible reason for this observation.
2