Math 4600, Homework 9
1. 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 R-type plasmids in the
cell. There are N + 1 states (the number of R 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
Pi,j =
2i
j
2N −2i
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 R and half of the plasmid of type Q, so p02 = 1. Iterate to find what will be the distribution of plasmids
R 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. In the sickle cell anemia model, consider 2 different situations: a society with strong genetic counseling
(R = 0.9, T = 0.1, S = 0.1) and a society in tropics where carrying a sickle cell allele makes people less
likely to die from malaria (R = .2, T = 0.1, S = 0.4 and also see below).
a) Explain why this choice of parameters fits the word description of each society
b) What will happen to disease in each population in the long run (use cobwebbing by hand)?
c) How will the model change if individuals with bb genotype are sterile (cannot reproduce)?
The precise mechanism by which sickle cell trait imparts resistance to malaria is unknown. Malaria
parasites multiply within red blood cells. Red blood cells in trait-carrier individuals are mostly normal,
but can become sickled in low-oxigen condition. It seems that infestation of a cell may create low-oxigen
conditions and effectively destroy the cell.
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