Assignment: Limited Logistic Growth Model Activity

Activity

Part 1- Characteristics of the Limited Logistic Growth Function

In the first part of the activity, you will analyze an equation and graph of a generic model of limited logistic growth, f (t) , with c = 1, a = 20, and b = 0.5, as shown below:

Answer the following questions:

1. Describe, in 2-3 sentences, the notable differences and similarities between the behaviors of a graph of pure exponential growth vs. that of limited logistic growth (shown above) as time increases.

2. Looking at the limited logistic growth graph above, determine the approximate value off (0). Show the calculation of the exact value off (0), and check the reasonableness of your answer with the value from the graph. What does the value off (0) physically represent for a limited logistic growth model?

3. Looking at the limited logistic growth graph above, as t increases, what value does f (t) approach? What would we call the horizontal line drawn along this value in "math terms"? What does the line physically represent for a limited logistic growth model?

4. Let us consider a different logistic growth function: g(t) = 2 / (1+10e^{-0.3t}); t ≥ 0 (c = 2, a =10, and b = 0.3). Without graphing the function, as t increases, what value does g (t) approach? Justify your answer.

Part 2 - Application: the Spread of Influenza

In the second part of the activity, you will evaluate the effects of changing the values of a and b in a limited logistic growth model f (t) for an influenza outbreak, by plotting graphs for different scenarios.

5. On your 1^{st} plot, compare the graphs of scenarios 1, 2, and 3. Briefly discuss, in 3-4 sentences, the differences or similarities in each of the following for the different values of b:

a. The percentage of the population infected on Day Zero (i.e., the value off (0)). Day Zero, or t = 0, represents the beginning of the outbreak, at which time a certain percentage of the population is already infected.

b. The time at which the limiting value of the function (i.e., the limit to the percentage of people infected) is approached.

c. The time at which 50% of the population became infected.

d. The general shape of the curve.

6. On your 2^{nd} plot, compare the graphs of scenarios 1, 4, and 5. Briefly discuss, in 3-4 sentences, the differences or similarities in each of the following for the different values of a:

a. The percentage of the population infected on Day Zero (i.e., the value off (0)).

b. The time at which the limiting value of the function (i.e., the limit to the percentage of people infected) is approached.

c. The time at which 50% of the population became infected.

d. The general shape of the curve.

7. In 3-4 sentences, summarize the physical meaning of the values of a and b within the logistic function for modeling the outbreak of influenza. In other words, what aspects of the trend of the outbreak are represented by the values of a and b?

8. This question has two parts. Graphical solutions will not be accepted. To solve them, you need to recall how to solve exponential equations.

f(t) = 100 / (1 + ae^{-0.75t}); t ≥ 0

a) So, c = 100 and b = .075. Also, on day zero (t = 0), .04% of the population is already infected. Using this information, first find the value of a.

b) Now determine when 50% of the population became infected.

9. Just like # 8, this question has two parts. Graphical solutions will not be accepted.

f(t) = 100 / (1 + ae^{-bt}); t ≥ 0

a) So, c = 100. Also, on day zero (t = 0), .02% of the population is already infected. Using this information, first find the value of a. (Hint: Notice, b - value is not given. A careful observation will show you that b - value is not needed to find the a - value)

b) Now calculate what value of b will result in 60% of the population being infected as of Day 300.

**Attachment:-** Assignment.pdf

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