Math 4600
Homework 6
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
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 kT0 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 VI (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 130×103
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
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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 V0 as above. You will need to solve the
equations analytically for the infectious and non-infectious virus populations).
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.
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