Math 4600 Homework 9
1. (computing) (From “Modeling the dynamics of Life”) Use the normal
approximation to binomial or poisson distribution (as appropriate) to find the
follwing probabilities. Do not forget the continuity correction.
a) Thirty percent of cells in a small organism are not functioning. What is
the probability that an organ consisting of 250 cells is functioning if it requires
170 cells to work?
b) Ten percent of people are known to carry a certain gene. In a sample of
200, what is the probability that between 5 and 15 people (not including 5 or
15) carry the gene?
c) Molecule enters a cell at a rate of 0.045 per second. What is the probability
that between 150 and 200 (inclusive) enter during an hour?
2. Drug concentration in the body on day n is Mn . It satisfies
Mn+1 = (1 − σ)Mn + c.
a) Suggest the meaning of parameters σ and c.
b)Find steady state and its stability , 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
iteration and information you found about steady states and their stability)
3. A particular 2-subunit 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 3 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) (computing) Start one such channel in the closed state and simulate
what it will do for the first 10 time steps. At every time step you will need
to compute transition probabilities from your current state and to generate a
random number to choose a transition to make.
4. 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. What is average
number of still-healthy people on the second day of the disease?
b) We derived in class the probability distribution for Sn+1 . To complete the
model we also need to say what the number of infected and recovered individuals will be on that day. Find expressions for In+1 and Rn+1 . Explain how you
found them
c) (computing) Simulate a course of disease in 100 such families. For each
family record the final number of people that remained heathy (S∞ ). Show the
histogram for S∞ values. Notice its shape.
d) (computing) Repeat c) with p = 0.4 and p = 0.6. If you are a healh care
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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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