Building a better epidemic model

Transcription

1 Precalculus Honors The Logistic Growth Model November 29, 2005 Mr. DeSalvo In the previous section, we simulated a fictitious bird flu epidemic and determined that the bounded growth model does not work well to describe the spread of the disease. To create a better epidemic model, we need to create a more accurate description (using more detailed assumptions) of how the disease spreads. Building a better epidemic model Our new epidemic model is based on two assumptions. Assumption #1: The number of people newly infected each day with the bird flu virus is directly proportional to the number of encounters between infected and uninfected people that day. This follows from the unstated assumption that the primary means of infection is personal contact between infected and uninfected people. However, only a certain percentage of these contacts will actually result in infection. This percentage is the constant of proportionality. Assumption #2: The number of encounters between infected and uninfected people each day is directly proportional to the product of the number of infected people times the number of uninfected people. The two assumptions above can be turned into a recursive equation as follows: Let p n be the number of people infected at the start of day n, and suppose that the size of the entire population is L. Then p n+1 p n = kp n (L p n ) for some constant k. The Logistic Growth Model The above difference equation is the recursive part of the logistic growth model. In general, this model has the form shown below, where L is the maximum possible size or carrying capacity of the population. p 0 = [an initial value] p n+1 p n = kp n (L p n ) for some nonzero constant k

2 EXERCISES 1. Bird Flu Epidemic. In a community of 200 people, 5 are infected with a form of bird flu on the first day of the month, and 3 more become newly infected on the second day, making 8 infected people total. Assume the epidemic spreads logistically. (a) Write the values of p 0, p 1, and L. (b) Use your answers to part (a) to find the value of k. (c) State the complete recursive formula for the logistical sequence P. (d) Make a spreadsheet (or use your TI) to generate and graph the sequence from part (c), using enough terms to see the long-term behavior. (e) Your graph from (d) should have an S shape that we saw when we did the simulation in section 4.2. Such a graph is called a logistic curve or a sigmoid. How is this graph similar to that of the bounded growth epidemic model from section 4.1? How is it different? In particular, analyze how the rates at which the disease spreads for the two different models change over time. (f) Find the equilibrium value of the logistic model algebraically. (g) In part (f), you should have found two equilibrium values. Explain the significance of these answers. (h) If you have not already done so, make a spreadsheet of this sequence. Experiment with different values of p 0. Make sure you try a wide range of values, including values greater then the equilibrium value. How does changing p 0 affect the behavior of the model? (i) In part (h) you should have found that as you increase p 0, there comes a point where the overall shape of the graph changes so that it is no longer sigmoidal. Experiment on your spreadsheet to find this changeover point. What relationship does it have to the equilibrium value?

3 2. In problem 8 of section 4.1, we analyzed the bounded growth model of bacteria in a petri dish. We will now examine this situation using a logistic growth model. You place 100 bacteria in a petri dish that can support a maximum of 3500 bacteria. The next day, there are 130 bacteria in the dish. Suppose that the population grows according to the logistic growth model. Let p n be the number of bacteria in the dish after n days. (a) Write a recursive formula for sequence P. (b) Create a spreadsheet which generates and graphs this sequence, showing enough terms to make the long-term behavior evident. Save your work as logistic bacteria SS and print it out. (c) Confirm both graphically and algebraically that your model has an equilibrium point equal to the carrying capacity of the petri dish. (d) How long does it take for the population to reach equilibrium? Explain your reasoning. 3. The point of fastest increase in a logistic model. This problem refers to the problem situation in question 1 (bird flu). (a) Open your spreadsheet logistic epidemic SS from problem 1. Make sure the parameters are set to p 0 = 5, k = 1/325, and L = 200. (b) Create a new column that computes p n+1 p n. For example, if you use column C, then cell C2 will compute p 1 p 0, cell C3 will compute p 2 p 1, and so on. (c) According to the data in column C, when is the population growing the fastest? Estimate the population at this time. (d) Repeat part (c) for several different values of p 0, keeping k and L the same. (e) Generalize about the size of the population at the point where it is growing the fastest. In particular, what do you notice about the size of the population at this point compared to the equilibrium value?

4 (f) Describe the behavior of the graph of sequence P at the point where the population is growing the fastest. 4. Modeling the spread of technology. Diesel locomotives were first introduced in Initially, only a few railroad companies purchased diesel engines, since they were reluctant to convert to an expensive new technology which might turn out to be less profitable than the steam engines already in use. However, companies soon realized that diesels ran faster and were cheaper to operate than steam engines, so more and more companies began using them until eventually all companies had switched completely to diesel engines. (a) This problem situation describes a quantity (the number of companies using diesel engines), which starts out small and then grows at an increasing rate. Note that both the logistic and exponential models have this growth pattern. Why is the logistic growth a better model to use here than exponential growth? (b) In the early twentieth century there were 25 railroad companies in the U.S. In 1925, one railroad company had converted to diesel engines. By 1928, only one additional company had made the change. Create a recursive sequence which uses the logistic model to compute the number of companies using diesel engines since (c) Create a spreadsheet which generates and graphs your sequence from part (b). Save your work as locomotive SS and print it out. (d) According to your model, how long did it take for all 25 companies to convert to diesel? 5. Lysol attack. This problem refers to question 2 (bacteria in a petri dish). Suppose, without your knowing it, your lab partner secretly sprays your petri dish with Lysol at the end of each day, killing 25 bacteria with each spraying. For example, at the end of day 1, there were 130 bacteria, but then your partner sprays your dish killing 25, leaving 105 bacteria alive for the next day. (a) Modify the sequence you wrote in problem 2 to include the Lysol spraying. (b) Modify your spreadsheet from problem 2 to take into account the Lysol spraying. Save your work as Lysol bacteria SS and print it out.

5 (c) Describe the effects that your lab partner s Lysol spraying has on the bacteria population. In particular, compare its growth with and without Lysol. (d) Now suppose that your partner sprays the petri dish twice at the end of each day, killing 50 bacteria total at the end of each day. Modify your recursive sequence and spreadsheet to take into account this new spraying. Describe what happens to the population of bacteria over time. (e) What is the maximum number of bacteria that your partner can kill each day without causing your population of bacteria to eventually be wiped out? Justify your answer.