Math 4600 Homework 9
1. A particular ion channel can be in one of 4 states and can transition
between states as follows. Closed channel (State 1) can move to state 2 with
prob 0.2 and state 3 with prob 0.3. From state 2 it can move to state 4 with
prob .6 or go back to closed with prob .3. From State 1 it can move to state 4
with probability .8 and it can close with probability 0.1. Once it in State 4 it is
open, and can move back states 2 and 3 with probabilities 0.2 each.
a) Draw a transition diagram for this Markov chain
b) What is the transition matrix
c) Start one such channel in the closed state and simulate what it will do for
the first 10 time steps.
d) Suppose that now we have a whole population of such channels. They all
start at the closed state. What will be the distribution of channels across the
states after a long time? What fractions of channels will be open after a long
time? What is the probability that a particular channel is closed at a particular
(large) time step?
e) Reduce transition probabilities leading to the open state. How does it affect
the long-term fraction of open channels?
2. Drug concentration in the body on day n is Mn . It satisfies Mn+1 =
(1 − σ)Mn + c. Find steady state and its stability (either by cobwebbing or by
finding the derivative of the right hand side), if σ is just above or just below 1.
What happens to the solution in both these cases if c = 1 and M0 = 0.5? (Use
cobwebbing to find the solution)
3. In the model for disease spread in a family, consider the following conditions: A family consists of 6 people, p = 0.2, S1 = 5, I1 = 1.
a) Find distribution of S2 . Illustrate it with a histogram. Also draw these probabilities as transition probabilities between relevant states. What is average
number of still-healthy people on the second day of the disease?
b) (computing) Simulate a course of disease in 100 such families. For each
family record the final number of people that remained heathy after the disease
runs out (S∞ ). Show the histogram for S∞ values. Notice its shape.
c) (computing) Repeat c) with p = 0.4 and p = 0.6. If you are a healh care
official dealing with a serious disease that has p = 0.6 and you have an expensive
drug that can reduce infectivity p to 0.4. Does it make sense to introduce it?
What about a drug that reduces p to 0.2?
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