Math 4600, Homework 6
1. Recall the HIV dynamics model that we considered in class: dT dt dT
∗ dt dV
= kV T − δT
∗
= N δT
∗
− cV dt
= s + pT (1 −
T
T max
) − d
T
T − kV T
Consider the model during the RT-inhibitor treatment. Assume that the inhibitor is perfect; it sets the appropriate parameter to zero. In the resulting system T ∗ and V equations are uncoupled from the T equation. In the ( T
∗
, V ) system find the steady states and their stability analytically (by computing the eigenvalues in terms of the parameters). 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.
2.
Computing A simple model of HIV dynamics is dV dt
= P − cV
Where P is virus production rate, V is virus concentration (per mL), and c is the clearance rate. Suppose a protease inhibitor drug is given, resulting in production becoming 0.
a.
The text file (“hw6 data”) on the website records how a patient’s virus concentration changes with time. Time (measured in days) is scaled so that treatment begins on day 0. Use this data to estimate c , assuming that once treatment starts, P = 0. This can be done by plotting the natural log of virus concentration vs time after time 0, and fitting a line through the resulting scatterplot. Make a plot of this scatterplot, and a line going through the post-treatment points.
b.
What is the production rate, P, at equilibrium given this data? At steady state, the production rate of the virus must equal its clearance rate ( cV
0
). This requires estimating V
0
, the equilibrial virus concentration, from the data.
c.
What is the total virus production per day (pre-treatment) if the person’s fluid volume is estimated to be 15 liters? Previously, it was thought that virus production during the latent period was small.
This calculation completely changed how people thought of HIV dynamics.
Sidenote: Since P does not instantly go to 0 when the inhibitor is given, this is actually a lower bound of c. A better estimate (found by data fitting) is c = 3 .
1 per day.
3.
Computing Recall that the effect of glucose in this original Beta Bursting model (proposed by Atwater,
1980) is to increase the parameter k c
, the removal of calcium from the system. On hw 5, you found 3 k c parameter values that produced 3 very different dynamics in the full model.
a.
Use these k c values (or different ones, if you wish) to produce 3 bifurcation diagrams of V
∗ vs.
c .
b.
Add in the c nullcline to each diagram with direction arrows on either side (these can be added by hand, or in R using the “arrows” function), as well as a representative solution trajectory (using R).
c.
Write a few sentences about why the dynamics are the way they are, in terms of the various bifurcations, and the placement of the c-nullcline.
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