Expert: Scott A Wilson - 5/7/2010

Question
Dutch elm disease is a fungal disease that kills elm trees. It is spread by certain species of bark beetle. The disease was first reported in the United States in 1928, with the beetles believed to have arrived in a shipment of logs from the Netherlands destined for use as veneer in the Ohio furniture industry. The disease spread slowly from New England westward and southward, almost completely destroying the famous elms in the “Elm City” of New Haven, reaching the Detroit area in 1950, the Chicago area by 1960, and Minneapolis by 1970. (Source: Wikipedia)
Suppose a region has 8000 healthy elm trees. In 1950, bark beetles harboring the fungus arrive in the region. In each subsequent year, the number of elm trees that contract the disease is 20% of the number of healthy trees at the end of the previous year. Once a tree is infected, it stays infected.
(a)    Find a difference equation describing the spread of Dutch elm disease through the region’s 8000 elm trees. Clearly define each variable.
(b)    Use your difference equation to find the total number of infected elm trees for each of the first six years after the arrival of the beetles.
(c)    Make a large, well-labeled graph showing the spread of Dutch elm disease, with total number of infected elm trees on the vertical axis and time in years on the horizontal axis. Number each axis appropriately. Include at least six years. Use graph paper or a computer program.
(d)    Solve the difference equation.
(e)    Use the solved difference equation to find the number of infected elm trees 20 years after the arrival of the beetles.
(f)    Use the solved difference equation to find in what calendar year (like 1983) the number of infected elm trees will be 7560. You must solve the solved difference equation; you may not “guess and check”. Get an equation for n and solve it.

Answer
(a) Since there are 8000 trees and 20% chance of being infected, the number of healthy trees would be 80% of the trees that were healthy the year before.  Thus, the equation is
f(x) = 8000(1-0.8^x) where x is the years that have passed and f(x) is the number of sick trees.
As can be seen, after only 3 years almost half of the trees are infected.

(b)
0 8000
1 6400
2 5120
3 4096
4 3276.8
5 2621.44
Now these should really be in integers, so year 4 is 3277, which makes 5 into 2621.6,
which rounds up to 2622.

(c) Make a graph, with the x-axis as years and the y-axis as the number of trees.

(d) Already solved it.

(e) I did the first column by rounding each time and the second by using numbers past the decimal.  As can be seen, there is little difference between them.

Year   Uninfected Trees
1   0   0
2   1600   1600
3   2880   2880
4   3904   3904
5   4723   4723.20
6   5378   5378.56
7   5902   5902.85
8   6322   6322.28
9   6658   6657.82
10   6926   6926.26
11   7141   7141.01
12   7313   7312.81
13   7450   7450.24
14   7560   7560.20
15   7648   7648.16
16   7718   7718.53
17   7774   7774.82
18   7819   7819.86
19   7855   7855.88
20   7884   7884.71

(f) As can be seen, 7560 is in the column above; the year is the number of the left.