Math 4600, Homework
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-healhy 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 (S∞ ). Show the
histogram for S∞ values. Notice its shape.
c) (computing) Repeat b) 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?
2. The model for random genetic drift in a population of bacteria that I
described in class has the following ingredients: The states of the markov chain
are marked by the number of Q’-type plasmids in the cell. There are N + 1
states (the number of Q’ plasmids can vary from 0 to N). In the n-th generation
a fraction of cells in each state is p[n] = (pn0 , . . . , pnN ). In the next generation
p[n+1] = p[n] P
where P is the transition matrix given by
2i 2N −2i
Pi,j =
j
N −j
2N
N
,
where n0 = 1 and nk = 0 for k > n, and i and j vary from 0 to N .
Consider the case when each cell has 4 plasmids. In the original cells we introduce half of the plasmids of type Q’ and half of the plasmid of type Q”, so
p02 = 1. Iterate to find what will be the distribution of plasmids Q’ and Q”
in the population after 50 generations. This phenomenon was first observed
experimentally and it was hypothesized that the plasmids interact so that they
cannot be picked up into the same cell. Your result shows that the separation
of plasmid types can happen simply as a result of random plasmid transfer.
3. Consider the Awake-Sleep Markov chain model that we discussed in class.
Consider the following transition probabilities: PAA = 0.9, PAS = 0.1, PSA =
0.4, PSS = 0.6.
a) Find analytically, by reducing the problem to 1d discrete dynamical system
and doing cobwebbing, the fraction of time spent awake and sleeping
b) (computing) Starting with p0A = 1, and p0S = 0, find the fraction of awake
and sleeping students at several subsequent time steps. Repeat iteration until
the distribution stabilizes. What are the resulting awake and asleep fractions?
Does it match the theoretical steady state?
c) (computing, extra credit) Do a Monte-Carlo (random) simulation of one
student. Start in awake state. Then ask matlab to toss the appropriate coin
and decide which state the student will be in at the next time step. Record the
sequence of states for 20 time steps. What proportion of time was the student
awake? Compare it with the steady state you estimated from a).
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