Investigation 20 Doubling Time Exponential Growth Answer Key.zip
Destini Armstrong
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.
The Figure shows the same data as the table. The cumulative number of cases is curved (left hand plot), but its logarithm follows a straight line. (We can use the fitted line to estimate the doubling time: the slope of the fitted line is 0.251 log cases per day, and log(2)/0.251 = 2.76 days, similar to the 3 days we estimated by looking at the table.)
investigation 20 doubling time exponential growth answer key.zip
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Exponential growth matters because it is easy to underestimate. In the legend of the wheat and the chess board, a petitioner asks a king for a grain of wheat on the first square of a chess board; two grains of wheat on the second square; and so on, doubling the amount of wheat on each square until all 64 squares are full. The king accepts what he takes to be a modest request, only to discover that he has promised to deliver more wheat than the whole world can produce.
In real life problems, exponential growth has an upper limit. Our hypothetical red kites will breed, but they will also age and die, and the hypothetical large, bountiful place in which they live will look less large, and less bountiful, as the red kite population grows. In the case of disease, when the cases become a sizeable fraction of the total population, so that the susceptible population is significantly smaller, growth will be slower than exponential (for more detail, see this video lecture by Dr Robin Thompson).
Richard Stevens is an Associate Professor in Medical Statistics at the Nuffield Department of Primary Care Health Sciences and Course Director of the MSc in Evidence Based Health Care (Medical Statistics).
Carl Heneghan is Professor of Evidence-Based Medicine, Director of the Centre for Evidence-Based Medicine and Director of Studies for the Evidence-Based Health Care Programme. (Full bio and disclosure statement here)
Richard Hobbs is a GP and Nuffield Professor of Primary Care Health Sciences, Director, NIHR English School for Primary Care Research and Director, NIHR Applied Research Collaboration (NIHR ARC) Oxford
Jason Oke is a Senior Statistician at the Nuffield Department of Primary Care Health Sciences and Module Coordinator for Statistical Computing with R and Stata (EBHC Med Stats), and Introduction to Statistics for Health Care Research (EBHC), as part of the Evidence-Based Health Care Programme.
Disclaimer: the article has not been peer-reviewed; it should not replace individual clinical judgement, and the sources cited should be checked. The views expressed in this commentary represent the views of the authors and not necessarily those of the host institution, the NHS, the NIHR, or the Department of Health and Social Care. The views are not a substitute for professional medical advice.
The rate of natural increase of a population depends on birth and death rates, which are strongly influenced by the population age structure. Births occur primarily to people in the younger-adult age groups. If there are comparatively more young adults than older adults where mortality is highest, then even at replacement fertility levels (when each woman has about an average of two children) there will be more births than deaths.
In the United States, birth rates are higher than death rates at present, partly due to the relatively young age structure of the U.S. population. Immigrants, who are younger on average than the U.S.-born population, play a significant role in keeping the United States younger than most other developed countries. For example, among U.S. Hispanics, 40 percent of whom are foreign-born, there are approximately 10 births for every death.
The composition of a population as determined by the number or proportion of males and females in each age category. The age-sex structure of a population is the cumulative result of past trends in fertility, mortality, and migration. Information on age-sex composition is essential for the description and analysis of many other types of demographic data.
The number of children women are having today. The average number of children that would be born alive to a women during her childbearing years if she conformed to the age-specific fertility rates of a given year.
Aside from the total size, the most important demographic characteristic of a population is its age and sex structure, or the proportion of people at each age, by sex. The age-sex structure determines potential for future growth of specific age groups, as well as the total population. For these reasons, the age structure has significant government policy implications. A population of young people needs a sufficient number of schools and, later, enough jobs to accommodate them. Countries with a large proportion of older people must develop retirement systems and medical facilities to serve them. Therefore, as a population ages, needs change from childcare and schools to jobs, housing, and medical care.
Short-term fluctuations in birth and death rates that produce unusual bites or bulges in population pyramids, such as the baby boom, often can be traced to such historical events as wars, epidemics, economic booms, or depressions. The decline in the birth rate during the Great Depression caused a small bite in the U.S. pyramid for the group born between 1930 and 1934. World Wars I and II caused a deficit of older men in Germany. The impact of these events emphasizes the interrelationships among population change and economic, social, political, and health factors.
Experts are attempting to find quantitative ways to consider both consumption patterns and population size when determining the link between people and the environment. Environmentalists have been using an equation known as I=PAT, which attempts to factor both causes into determining environmental impacts.
The causes of tropical deforestation lay both in population growth in less developed countries and consumption levels in more developed countries. However, for some other environmental problems such as ozone depletion, most of the damage is due to the use of refrigerators and air conditioning systems in industrialized countries, not to population growth.
The adverse environmental impact of consumption patterns in more developed countries is likely to increase as less developed countries further industrialize and adopt consumption patterns similar to those of their more financially wealthy neighbors. Already, elites in the less developed countries mimic the prolific consumption of rich Americans or Europeans. Consumption has surged in China and India since the 1980s and, with the fall of the USSR, Eastern Europeans have increased their appetites for consumer goods. The most rapid growth in energy consumption now occurs in less developed countries because of rising affluence, consumption, and population.
The link between population growth and the environment is found somewhere between the view that population growth is solely responsible for all environmental ills and the view that more people means the development of new technologies to overcome any environmental problems. Most environmentalists agree that population growth is only one of several interacting factors that place pressure on the environment. High levels of consumption and industrialization, inequality in wealth and land distribution, inappropriate government policies, poverty, and inefficient technologies all contribute to environmental decline. In fact, population may not be a root cause in environmental decline, but rather just one factor among many that exacerbate or multiply the negative effects of other social, economic, and political factors.
The gains in food production have been a result of increased yields in fertile lands and new cultivation of marginal lands through industrial agriculture. However, improper use of machinery, chemicals, and extensive irrigation, has resulted in the degradation of land and water resources. Land is made vulnerable to wind and water erosion. Misguided irrigation practices can mean an increase in soil salinity and a greater demand on irreplaceable groundwater. Chemical runoff from fertilizers and pesticides also damage water resources.
Nonindustrial farming or traditional agriculture that continues to intensify in less developed countries often involves the cultivation of fragile soils that are difficult to farm, such as drylands, highlands, and forests. When farmland expands toward fragile lands in order to keep pace with the needs of a growing population in a region, it can lead to deforestation, erosion, and desertification.
Carbon dioxide emissions have grown dramatically in the past century because of human activity, chiefly the use of fossil fuels such as oil and coal, as well as changes in land use such as cutting down forests. These emissions are a key contributor to climate change that is expected to produce rising temperatures, lead to more extreme weather patterns, facilitate the spread of infectious diseases, and put more stress on the environment. The United States is the largest contributor of total carbon dioxide emissions, and has one of the highest per capita rates. The U.S. per capita emission rate has risen from 19.2 metric tons per person to 19.9 metric tons between 1990 and 2002, according to the World Resources Institute. Per capita use also has gone up in China, rising from 2.2 to 2.9 metric tons between 1990 and 2002. China is expected to surpass the United States in total carbon dioxide emissions by 2009.
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