MATH 3900, Spring 2009
Homework 8 (Practice for midterm)
Due 03/13/09
First, good news: there will be fewer problems on the test than in this list. The
problems will be very similar to these. Also, there will be no homework to do
over the spring break!
1. Here is the HIV dynamics model that we considered in class:
dT = s + pT (1 T ) d T kV T
T
dt
Tmax
dT = kV T T dt
dV = NT 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 aected 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 syem to answer questions b) and c).
b) In the (T ; V ) system nd the steady states and their stability analytically
(by computing the eigenvalues). Then, draw a phase plane with nullclines and
representative direction eld arrows. Finally, draw qualitative time courses for
T (t) and V (t) starting with some high initial values. (Assume c = NkT0 if you
need to).
c) In the T equation with s = 0 nd the steady states and stability analytically
and using the phase line. Discuss what will be the long-term condition of T-cel
population according to this equation.
2. Here is a 1969 model of glucose-insulin interaction:
g_ = m1 g + m2 h
h_ = m3 g + m4 h
where g is the displacement of plasama glucose level from fasting conditions,
and h is the displacment of the insulin level.
a) For each of the constants m1 ; m2 ; m3 ; m4 decide whether it should be positive or negative, based on the type of interaction between glucose and insulin
(whether one substance should increase or decrease in response to the other)
b) Suppose that now we replace naturally produced insulin by injected one. It
has exactly the same eect, but is less ecient in its inuence on glucose. What
parameter would that change and in what way?
3. Below is the diagram of a thermostat (basically, a bifurcation diagram).
Horizontal axis is the slowly-changing room temperature, and the vertical axis
1
indicates how hard the heating device is working.
a) Start with room temperature held steady at 80. What will the heating device
do? Then allow the temperature to vary in the following way: the temperature
is increasing if the heating device is on and is decreasing if the heating device
is o. Mark on the diagram what will be happening.
b) Sketch the time course of the temperature dynamics and on the same graph
the time course of the heating device dynamics.
ON
OFF
60
65
70
75
80
4. Build a model for the spread of rumors in a town. People can be informed
about the rumor (I), uninformed (U) or disinterested (D - these are people
who know the rumor and don't care about it). If an informed person meets
an uninformed one, the uninformed turns into informed with rate and into
disinterested with rate . Uninformed can also become informed by reading the
local newspaper, this happens with rate . Informed people become disinterested
at rate .
Write equations for the dynamics of rumor spread. What will be the outcome for
a town with population of size P ? (hint: nd steady states and their stability)
5. Draw a schematic of the (periodic) arterial pulse. Describe what the
heart is doing during each phase of the pulse in terms of heart valves opening
and closing, and heart chambers volume.
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