In the spring of 2003, SARS (Severe Acute Respiratory Syndrome) spread rapidly in

If t is in years since 1990, one model for the population of the world, P, in billions, is P = 40 1 + 11e0.08t . (a) What does this model predict for the maximum sustainable population of the world? (b) Graph P against t. (c) According to this model, when will the earths population reach 20 billion? 39.9 billion?

Investigate the effect of the parameter C on the logistic curve P = 10 1 +Cet . Substitute several values for C and explain, with a graph and with words, the effect of C on the graph.

The following table shows the total sales, in thousands, since a new game was brought to market. (a) Plot this data and mark on your plot the point of diminishing returns. (b) Predict total possible sales of this game, using the point of diminishing returns. Month 0 2 4 6 8 10 12 14 Sales 0 2.3 5.5 9.6 18.2 31.8 42.0 50.8

Write a paragraph explaining why sales of a new product often follow a logistic curve. Explain the benefit to the company of watching for the point of diminishing returns.

(a) Draw a logistic curve. Label the carrying capacity L and the point of diminishing returns t0. (b) Draw the derivative of the logistic curve. Mark the point t0 on the horizontal axis. (c) A company keeps track of the rate of sales (for example, sales per week) rather than total sales. Explain how the company can tell on a graph of rate of sales when the point of diminishing returns is reached.

The following table gives the percentage, P, of households with cable television between 1977 and 2003.15 Year 1977 1978 1979 1980 1981 1982 1983 P 16.6 17.9 19.4 22.6 28.3 35.0 40.5 Year 1984 1985 1986 1987 1988 1989 1990 P 43.7 46.2 48.1 50.5 53.8 57.1 59.0 Year 1991 1992 1993 1994 1995 1996 1997 P 60.6 61.5 62.5 63.4 65.7 66.7 67.3 Year 1998 1999 2000 2001 2002 2003 P 67.4 68.0 67.8 69.2 68.9 68.0 (a) Explain why a logistic model is reasonable for this data. (b) Estimate the point of diminishing returns.What limiting value L does this point predict? Does this limiting value appear to be accurate, given the percentages for 2002 and 2003? (c) If t is in years since 1977, the best fitting logistic function for this data turns out to be P = 68.8 1 + 3.486e0.237t , What limiting value does this function predict? (d) Explain in terms of percentages of households what the limiting value is telling you. Do you think your answer to part (c) is an accurate prediction?

The Tojolobal Mayan Indian community in Southern Mexico has available a fixed amount of land.16 The proportion, P, of land in use for farming t years after 1935 is modeled with the logistic function P = 1 1 +3e0.0275t . (a) What proportion of the land was in use for farming in 1935? (b) What is the long-run prediction of this model? (c) When was half the land in use for farming? (d) When is the proportion of land used for farming increasing most rapidly?

In the spring of 2003, SARS (Severe Acute Respiratory Syndrome) spread rapidly in several Asian countries and Canada. Table 4.9 gives the total number, P, of SARS cases reported in Hong Kong17 by day t, where t = 0 is March 17, 2003. (a) Find the average rate of change of P for each interval in Table 4.9. (b) In early April 2003, there was fear that the disease would spread at an ever-increasing rate for a long time. What is the earliest date by which epidemiologists had evidence to indicate that the rate of new cases had begun to slow? (c) Explain why an exponential model for P is not appropriate. (d) It turns out that a logistic model fits the data well. Estimate the value of t at the inflection point. What limiting value of P does this point predict? (e) The best-fitting logistic function for this data turns out to be P = 1760 1 + 17.53e0.1408t . What limiting value of P does this function predict? Table 4.9 Total number of SARS cases in Hong Kong by day t (where t = 0 is March 17, 2003) t P t P t P t P 0 95 26 1108 54 1674 75 1739 5 222 33 1358 61 1710 81 1750 12 470 40 1527 68 1724 87 1755 19 800 47 1621

Substitute t = 0, 10, 20, . . . , 70 into the exponential function used in this section to model the US population 17901860. Compare the predicted values of the population with the actual values.

On page 214, a logistic function was used to model the US population. Use this function to predict the US population in each of the census years from 17901940. Compare the predicted and actual values.

A curve representing the total number of people, P, infected with a virus often has the shape of a logistic curve of the form P = L 1 +Cekt , with time t in weeks. Suppose that 10 people originally have the virus and that in the early stages the number of people infected is increasing approximately exponentially, with a continuous growth rate of 1.78. It is estimated that, in the long run, approximately 5000 people will become infected. (a) What should we use for the parameters k and L? (b) Use the fact that when t = 0, we have P = 10, to find C. (c) Now that you have estimated L, k, and C, what is the logistic function you are using to model the data? Graph this function. (d) Estimate the length of time until the rate at which people are becoming infected starts to decrease. What is the value of P at this point?

Find the point where the following curve is steepest: y = 50 1 + 6e2t for t 0.

A dose-response curve is given by R = f(x), where R is percent of maximum response and x is the dose of the drug in mg. The curve has the shape shown in Figure 4.77. The inflection point is at (15, 50) and f(15) = 11. (a) Explain what f(15) tells you in terms of dose and response for this drug. (b) Is f(10) greater than or less than 11? Is f(20) greater than or less than 11? Explain.

If R is percent of maximum response and x is dose in mg, the dose-response curve for a drug is given by R = 100 1 + 100e0.1x . (a) Graph this function. (b) What dose corresponds to a response of 50% of the maximum? This is the inflection point, at which the response is increasing the fastest. (c) For this drug, the minimum desired response is 20% and the maximum safe response is 70%. What range of doses is both safe and effective for this drug?

Dose-response curves for three different products are given in Figure 4.79. (a) For the desired response, which drug requires the largest dose? The smallest dose? (b) Which drug has the largest maximum response? The smallest? (c) Which drug is the safest to administer? Explain.

Explain why it is safer to use a drug for which the derivative of the dose-response curve is smaller.

In Figure 4.80, what range of doses appears to be both safe and effective for 99% of all patients? 10 20 30 25 50 75 100 Percent effective Percent lethal dose (mg) percent of patients showing given response Figure 4.80

In Figure 4.81, discuss the possible outcomes and what percent of patients fall in each outcome when 50 mg of the drug is administered. 20 40 60 80 20 40 60 80 100 Percent effective Percent lethal dose (mg) percent of patients showing given response Figure 4.81

A population, P, growing logistically is given by P = L 1 + Cekt . (a) Show that L P P = Cekt. (b) Explain why part (a) shows that the ratio of the additional population the environment can support to the existing population decays exponentially.

Cell membranes contain ion channels. The fraction, f, of channels that are open is a function of the membrane potential V (the voltage inside the cell minus voltage outside), in millivolts (mV), given by f(V ) = 1 1 + e(V +25)/2 . (a) Find the values of L, k, and C in the logistic formula for f: f(V ) = L 1 + CekV . (b) At what voltages V are 10% , 50% and 90% of the channels open?