Hermine E. Smikle
Topic: The spread of diseases
Purpose: To use mathematical models to determine the spread of diseases.
Background Information:
The spread of infectious diseases randomly through a community can be modeled by N = ΒSI, where S is the number of susceptible people.
I is the number of infected people
N is the product of the rate of mixing and the concentration of susceptible and infected persons, and Β is the transmission coefficient (the probability of transmission). The probability of transmission will vary between micro organisms and their route of transmission.
Problems:
1.
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Assume that the relative spread of a disease is constant. Suppose that in the course of any given year the number of cases of the disease is reduced by 20%.
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a)
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If there are 10,000 cases today, how many years will it take to reduce the
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number to 100,000?
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b)
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How long will it take to eradicate the disease that is to reduce the number
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of cases to less than 1?
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2.
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The spread of the number of people infected by malaria in a certain Africa country is given by P(t) = 200 / (1 + e
5.3-t
)
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3.
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The spread of the swine flu is modeled by the equation P (t) = 100 / (1 + e
3-t
) where P (t) is the total number of students infected after t days after their trip to Mexico.
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a)
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Estimate the initial number of students infected with the flu.
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b)
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Haw fast is the flu spreading after 3 days?
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c)
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When will the flu spread at its maximum rate? What is the maximum rate?
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4.
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The measles virus is a pathogen that is spread from people to people. The table shows the percentage of students infected from an outbreak among children in a daycare center.
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a)
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Compute the average rates of change of the infection over the intervals [ 0, 12], [20, 32], [ 40, 52 ] and [30, 50].
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b)
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Draw the tangent line at t = 40 and estimate the slope. Explain what that means.
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c)
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Discuss the rate of infection over the following intervals [0,12], [20,32] and [40, 52].
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5. An epidemiologist finds that the percentage N(t) of susceptible children who were infected on day t during the first three weeks of a measles outbreak is given to a reasonable approximation by the function N(t) = (100t
2
) / (t
3
+ t
2
- 100t + 380)
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Graph the function
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a)
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Draw the secant line and use it to approximate the average rate of
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increase between i) [4, 6] ii [12, 14].
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b)
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Find the average rate of change on day 12.
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