Investigation 20 Doubling Time Exponential Growth Answer Key.zip
Sheron Norsworthy
The scenario and the three questions are presented to subjects in one of four frames. Each subject is randomly assigned to one frame. Frames vary along two dimensions, r vs. d and C vs. T. Dimension r vs. d concerns the way in which the growth of the disease is communicated: either in terms of the daily growth rate (r), or in the equivalent doubling time in days (d). Dimension C vs. T changes the perspective on the benefit of the mitigation measures: either in terms of the cases avoided within 30 days (C) or in terms of the time gained until 1 million cases are reached (T). This results in four frames, C-r, C-d, T-r, and T-d. Table 1 gives an overview of the four frames, and the complete questions are given under Procedure.
Fig 1 illustrates the underlying system. The parameters of the questions are set such that for the high exponential growth question the correct answer in frames C-r/C-d is about equal to the number of cases given in frames T-r/T-d, and, conversely, the correct answer in frames T-r/T-d is about equal to the amount of time given in frames C-r/C-d. The mitigation measures either reduce the number of cases in the country (C-r/C-d) or buy time for the country (T-r/T-d).
investigation 20 doubling time exponential growth answer key.zip
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The correct answers to the mitigation, high exponential growth and low exponential growth questions for frames T-r and T-d are given by MT, HT and LT, respectively. For frames C-r and C-d the answers are given by MC, HC and LC, respectively. The high exponential growth function is fH (26% per day/doubling time of 3 days), the low exponential growth function is fL (9% per day/doubling time of 8 days). Not drawn to scale.
A subject exhibits exponential growth bias if they underestimate exponential growth. In frames C-r/C-d, this means underestimating the number of cases that result after a given time. In frames T-r/T-d, this means overestimating the amount of time until a given number of cases is reached. In line with this, we define mitigation bias as underestimating the benefit of decelerating the exponential spread of the disease. In frames C-r/C-d, this means underestimating the number of cases avoided due to the mitigation measures and in frames T-r/T-d, it means underestimating the number of days gained due to the measures.
As one can see from the table, it turns out that the frame that least effectively communicates exponential growth is the same in all three questions: frame C-r. This frame communicates exponential growth in terms of the growth rate and asks for the number of cases. The share of biased subjects in this frame is statistically significantly higher than in all other frames, though only marginally so for frames T-r (p = 0.056) and C-d (p = 0.07). Regarding which frame most effectively communicates exponential growth, it turns out that one frame is most effective in all three questions: frame T-d, which communicates exponential growth in terms of doubling times and asks for an amount of time until a threshold of cases is reached. The share of biased subjects in this frame is statistically significantly lower than in all other frames, except for frame T-r (p = 0.14).
A: answers to the mitigation question plotted against the difference in answers to the exponential growth questions for frame C-r (n = 54). B: same plot for frame C-d (n = 50). Solid lines indicate the correct answer to the mitigation question, respectively, the difference between the correct answers to the exponential growth questions (about 1 million cases avoided). For observations on the dashed line, mitigation bias can be fully explained by exponential growth bias. Multiple identical answers are displayed by larger circles. Only subjects to whom the mitigation question was displayed prior to the exponential growth questions are included. Data points with non-positive values are excluded. One outlier in C-d is not shown.
A: Answers to the mitigation question plotted against the difference in answers to the exponential growth questions for frame C-r (n = 49). B: Same plot for frame C-d (n = 40). The solid line indicates the correct answer (about 1 million cases avoided). For observations on the dashed line, mitigation bias can be fully explained by exponential growth bias (28% in frame C-r, 23% in frame C-d). Multiple identical answers are displayed by larger crosses. Only subjects to whom the two exponential growth questions were displayed prior to the mitigation question are included. Data points with non-positive values are excluded.
Impact of varying period of observation and length ofobservation on doubling time during the course of anepidemic. The doubling time Td was estimated during the course of anepidemic, simulated by the SIR model, varying the period of observationand the length of observation. Doubling times were estimated in sixdifferent time periods, including those at the start of the exponentialphase, near the epidemic peak, and after the peak of the epidemic. Thepeak incidence was observed on Day 42.
Recovery of intrinsic growth ratefrom doubling time. The intrinsic growth rate rwas calculated using two types of methods (equations (3), (13) in themain text, represented by unfilled and filled bars) using doubling time.The growth rate was assumed to be 0.01 per day (horizontal dashedline).
World population growth accelerated after World War II, when the population of less developed countries began to increase dramatically. After millions of years of extremely slow growth, the human population indeed grew explosively, doubling again and again; a billion people were added between 1960 and 1975; another billion were added between 1975 and 1987. Throughout the 20th century each additional billion has been achieved in a shorter period of time. Human population entered the 20th century with 1.6 billion people and left the century with 6.1 billion.
A story said to have originated in Persia offers a classic example of exponential growth. It tells of a clever courtier who presented a beautiful chess set to his king and in return asked only that the king give him one grain of rice for the first square, two grains, or double the amount, for the second square, four grains (or double again) for the third, and so forth. The king, not being mathematically inclined, agreed and ordered the rice to be brought from storage. The eighth square required 128 grains, the 12th took more than one pound. Long before reaching the 64th square, every grain of rice in the kingdom had been used. Even today, the total world rice production would not be enough to meet the amount required for the final square of the chessboard. The secret to understanding the arithmetic is that the rate of growth (doubling for each square) applies to an ever-expanding amount of rice, so the number of grains added with each doubling goes up, even though the rate of growth is constant.
The number of years required for the population of an area to double its present size, given the current rate of population growth. Population doubling time is useful to demonstrate the long-term effect of a growth rate, but should not be used to project population size. Many more-developed countries have very low growth rates. But these countries are not expected to ever double again. Most, in fact, likely have population declines in their future. Many less-developed countries have high growth rates that are associated with short doubling times, but are expected to grow more slowly as birth rates are expected to continue to decline.
A simple way to look out for exponential growth is to try to spot a doubling time. A concerned newspaper reader in the Spring of 2020 might notice the apparent doubling between the 23rd and 26th of February, for example, and then keep watching the news to see if cases continue to double approximately every three days.
Sometimes population growth may be exponential. During exponential growth, a population experiences an accelerated increase in size. This is due to a constant increase in growth rate over time.
The population growth rate by country can be calculated using the exponential growth rate formula. For example, here is an example based on the country of Niger, which has experienced exponential growth. The time period of this example is from 1990 to 2020:
Exponential growth cannot continue forever. Exponential models, while they may be useful in the short term, tend to fall apart the longer they continue. Consider an aspiring writer who writes a single line on day one and plans to double the number of lines she writes each day for a month. By the end of the month, she must write over 17 billion lines or one-half-billion pages. It is impractical, if not impossible, for anyone to write that much in such a short period of time. Eventually an exponential model must begin to approach some limiting value and then the growth is forced to slow. For this reason, it is often better to use a model with an upper bound instead of an exponential growth model although the exponential growth model is still useful over a short term before approaching the limiting value.
Based on estimates from the first half of August, the monkeypox outbreak in the United States continued to grow exponentially, although there are recent signs the rate of growth is slowing. We estimated the outbreak doubling time is approximately 25 days, which has slowed from earlier estimates of doubling times of around 8 days throughout most of July (Figure 8). Our growth rate estimates are uncertain, and recent trend changes should be interpreted cautiously, given reporting delays and missing data. We have attempted to adjust for reporting delays, as shown in Figure 9 below, however adjustment is particularly sensitive to temporal changes in reporting delays. There are recent signs that case growth may have slowed further, but changes in data reporting processes have made it difficult to update this analysis with the most recent data.
Among these scenarios, we assess the monkeypox outbreak in the United States will most likely continue to grow very slowly over the next two to four weeks, likely with a declining growth rate. We have low confidence in this assessment. Given the estimated doubling time above, in this scenario cumulative cases could double to approximately 35,000 over the next month.
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