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The Logistic Function Problem Set Math 140

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This Logistic Function Problem Set will give you practice with a realistic logistic function. It is an INDIVIDUAL assignment. Cite all resources used. You may ask your Math 140 instructor any question, but refrain from asking anyone else. If you violate the INDIVIDUAL nature of the assignment, document the unauthorized help you receive by name and by the nature of the help received. There will be an academic penalty depending on the extent of the unauthorized help, but you will not be guilty of plagiarism.

Undocumented unauthorized help will be treated as plagiarism.

Purpose:

• To extend your skill in using an extremely important generalized exponential function called the logistic function

• To develop, algebraically, the two forms (exponential and logarithmic) of the logistic

function .

• To recognize valid applications of the logistic function in scenarios of constrained

growth.

You will want the use of Microsoft Mathematics (think of it as a cool graphing calculator) or

other graphing tool (TI or Casio calculator or web-based applet, or Graphmatica), and the

Equation Editor built into MS Word 2007 or 2010, or MathType. MS Excel (or equivalent) can

also be used for graphing or other calculations.

The primary submission must be composed in Microsoft Word or any equivalent.

The rules are as follows:

Submission via email is due on Day Seven of Module/Week 6 (2359 Eastern Time Zone). It

must be in one of the following formats:

*.docx/*.xlsx

*.doc/*.xls (Google Docs is okay, but keep Private—do not share them or borrow others)

*.pdf (if you convert your work into Adobe Acrobat format)

If you use Open Source office suites, be sure to convert your documents into one of the

acceptable formats listed.

The asterisk is where you enter “LastName FirstName Logistic Problem Set” (no special

characters. Things like #, ? and * screw up your submission)

This assignment is due by the end of Day Two in Module 6, and is weighted 7.5% of the final

grade.

The Logistic Function Problem Set Math 140

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Math background.

Pure exponential growth is not a real-world phenomenon (there are a finite number of atoms in

the observable universe, and the exponential function is a continuous function), but it can be a

fine model for short periods of time. How fast does an exponential function grow? We’ll

discover shortly.

Even graphing an exponential function is far more difficult than textbooks lead you to believe.

Using a blackboard/whiteboard with horizontal units 1 cm apart, graphing the function y = 2x ,

the point (100 cm, 2100 cm) is not feasible to plot on any normal scale.

Task 1: Determine how far 2100 cm is, to the nearest light year.

IF one uses a logarithmic scale on the y – axis, though, then the graph would be a line (semi-log

paper)!

Nonetheless, if the growth is slow enough—on the order of a couple percent per year, for

example, an exponential model is simple and very good at tracking populations over a twenty

year interval, but not much longer. Such functions as y = (1.025)x grow slowly enough to be

realistic growth models for x values to 30 or so. Calculate 1.02530 to see why this is so…..

Logistic growth described

No living system exhibits exponential growth over any extended period of time, simply due to

the exhaustion of available resources to feed the “beast”. It is this resource limitation (among

other limiting factors) that causes an eventual “leveling” of the population (or size) of the

system.

With this in mind, think about resource-constrained exponential growth, also known as

logistic growth. Google or Bing a guy named Verhulst (he lived in the 1800’s) for background.

It will be similar to the notes of this project.

Task 2: In a couple of paragraphs, describe and react to the issues Verhulst wrote about.

Identify key famous people engaged in the debate of his day.

The Logistic Function Problem Set Math 140

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In order to model resource-constrained growth, you need a graph that looks more like the

following instead of the pure exponential growth curve:

The upper limit of this graph was set at y = 1 to represent 100% of a system’s capacity. As an

organism’s population or other resource-constrained variable increases, it exhibits almost-pure

exponential growth early until it gets to about 50% of the environment’s capacity, such as

“amount of easily arable farmland brought to production”, after which the pressures of limited

supply slow the growth in easy-to-use farmland. Or in the case of an species, until it runs out of

space or food supply.

The textbook provides you one of the many possible logistic growth models in Section 4.5,

without explanation or interpretation. Here is its exponential form:

( ) 1 bt

c f t

ae? =

(1)

In problems where 100% is the total capacity of a system (and for simplicity, one that won’t

change over time), we let c be the number 1. If the capacity increases, decreases, or oscillates

over time (such as the seasons), then we’d replace c with whatever capacity function applies.

When we start measuring the system, we assign t the value of zero (also known as the initial

condition of the system).

y = 1

The Logistic Function Problem Set Math 140

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Note that, no matter what t is, since c is nonzero, then so is f(t). Further, if you set c = 1, then f(t)

is the proportion of capacity the system achieves at time t.

Task 3:

a. In Equation 1, describe what f(0) means, and calculate it for Equation 1.

b. Now let t get enormous. Over a very long period of time, what value does f(t) approach in

Equation 1?

The logarithmic form of the logistic growth function

In Chapter 4, you’ve learned that the logarithm can be used to “free” a variable from the

exponent, if you will, for algebraic advantage. There are many good reasons to solve Equation 1

for t or b. Unfortunately, it takes a bit of algebra to get the equation into a form that would

permit a simple rewrite into its logarithmic form. So, let’s see what we can do algebraically to

get bt e? by itself:

Task 4: Have at it! Algebraically manipulate Equation 1 until bt e? is by itself. That is, solve for

bt e? . But first, replace f(t) by y to simplify the details.

Done right, you should get bt c y e

ay

? ? =

Now, using the definition of logarithm in Section 4.2, you can rewrite this equation to solve for

either b or t. For Equation (2), we’ll solve for t:

ln

1 ln

1 () ln (2) ( )

c y bt

ay

c y t

b ay

c ft t

b af t

? ? ? ? = ? ? ? ?

? ? ? ? = ? ? ? ? ?

? ? ? ? = ? ? ? ? ?

Now, you solve for b instead of t. Call it Equation 3.

Equations 2 and 3 are equivalent logarithmic forms of the logistic equation (1).

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One of the Big Ideas: If we know the population, or mass, or whatever quantity we are

measuring at t = 0, we can use Equation (1) to quickly determine the value of a.

Once we know a, Equation 3 can be used to solve for b. And then finally, once we know a and

b, we can find the time t it takes for a system to grow to whatever proportion of capacity we wish

to predict using Equation 2.

The Application—Understanding the Spread of Influenza

Influenza (Google it) is a highly mutative virus that generally starts in livestock populations;

eventually a strain develops that spreads to humans among the farmers of chickens and pigs in

3rd world countries not able to maintain strict standards of hygiene. New strains develop in the

animal population fairly continuously, and the strongest of these varieties kill or greatly weaken

their hosts. Symptoms manifest themselves more slowly than the disease spreads, hence its

effectiveness. The World Health Organization has a regular strategy to “trap” new strains as

they mutate before cross-species variants develop, and the CDC acts as coordinating agent in the

US to gather information for the WHO. They also are the principal developers of flu vaccines

(though in recent years, it had been outsourced to other nations to reduce costs. A move back to

insourcing our own vaccine production began in 2012.)

Since the vast majority of people survive the flu, it isn’t as critical to identify a new flu virus

right away. In fact, a prospective new strain needs to show the ability to spread fairly rapidly

and be rather potent before the government will spend large sums of money to build a vaccine

for any new strain. H5N1 (called avian flu), however, showed itself to be rather deadly in Asia

in the summer and fall of 2008, but it also proved to be tough to spread in the human population.

It was H5N1’s deadliness that led the Centers for Disease Control to set up a crash vaccine

creation program for it in early 2009. By 2010, it was incorporated into the mainstream Fall

season vaccine, along with two other strains showing early effectiveness in the 2009 season.

Vaccinations: The chief effect of a vaccine is to reduce the susceptible population size. If

50,000,000 people receive a vaccine and if it is timely and effective, then 50,000,000 people can

be considered to be safe from the disease (or even to have “caught” it already). Using the

logistic function as a model, this 16% reduction in the vulnerable population should also reduce

the number infected by 16% at any given time.

Vaccinations also make a disease like the flu harder to spread, because clusters of vulnerable

people in close quarters are fewer. This would reduce the infectivity of a virus, so that its spread

becomes even slower, until the flu season passes and the virus goes largely dormant. Out-ofseason

flu epidemics are pretty rare. (None of the Tasks involve the effects of vaccination)

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For the rest of the Problem Set, in Equation 1 we’ll set c = 1. It will be your job to find the

values for a and b under three separate Scenarios, explain what they represent for each scenario

in Equations 1, 2, and 3, and then find the time it takes for a given set of proportions of the

vulnerable population to be infected by it.

For U.S. population studies, where at least 250,000,000 people are vulnerable to a particular flu

virus, if we set the capacity of the flu virus to infect vulnerable people in the US at c = 1, the

value of the initial proportion f(0) is considerably smaller than 1, (as few as 300 people with flulike

symptoms may have their blood tested by the time a new virus is identified) and so we’d set

f(0) = 300/250,000,000, a REALLY small decimal.

Okay, back to the project tasks.

You will evaluate the effects of changing f(0) and changing b on the graphs of your logistic

equations. The value of b will be given as a proportion/day so that if b = .01, then 1% of the

vulnerable population is newly infected per day. Your value of f(0) must be computed from the

initial number of cases as a proportion of the 250,000,000 vulnerable. f(0) will be a very small

decimal number in most of the tasks. We’ll call Day Zero (that is, you set t = 0 on Day Zero) the

day when scientists have identified a new flu virus strain.

For all scenarios (except the one in Task 7), find a, and then express the logistic equation

using Equation 1 as the template using the values for a and b, with c = 1.

Plot a graph for each scenario, ideally using MS Math 4.0 (free), or Graphmatica 2.0 (shareware)

or even Excel if you already have the skills for plotting graphs in it. A Graphing calculator

really won’t be good enough for the plots—they’re pretty primitive.

Scenario 1: Let’s assume that b = .0075 and that on Day Zero, there are an estimated 10,000

infected people out of a vulnerable population of 250,000,000.

Scenario 2. Another strain of the flu is more virulent, with double the value of b as in Scenario

1. Let’s also assume 10,000 people have been infected by Day Zero (same vulnerable

population.

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Scenario 3. Strain 3 is in its second year in the US, so it is estimated that 500,000 people have

had that variant. Let us also assume that it was slightly less virulent than the second strain, so

that k = .008.

Note: You may use Equations (1), (2) or (3) or some combination of them to complete the

following tasks. You may even graph Equation (1) and use a Trace facility to approximate the

exact solutions.

Task 5: For each of the Scenarios above, determine when half the population (f(t) = 0.5) has

been infected, and estimate when 80% of the vulnerable population has “caught” that strain.

Remember, your units for t are in “days”, though that number may be large.

Task 6: Using the three Scenarios, compare the impact of the different b values, and compare

the impact of the different f(0) values. Which factor seems more important—the size of the

initial population, or the virulence?

In this last task, your job is to find b, and you are given a different set of assumptions.

Task 7: Assume you start with an initial population of 100,000 infectees. Calculate what

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