The Logistic Function Problem Set Math 140Page 1 of 7 pages0214This Logistic Function Problem Set…

The Logistic Function Problem Set Math 140Page 1 of 7 pages0214This Logistic Function Problem Set will give you practice with a realistic logistic function. It isan INDIVIDUAL assignment. Cite all resources used. You may ask your Math 140instructor any question, but refrain from asking anyone else.If you violate the INDIVIDUAL nature of the assignment, document the unauthorized help youreceive by name and by the nature of the help received. There will be an academic penaltydepending 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 functioncalled the logistic function• To develop, algebraically, the two forms (exponential and logarithmic) of the logisticfunction .• To recognize valid applications of the logistic function in scenarios of constrainedgrowth.You will want the use of Microsoft Mathematics (think of it as a cool graphing calculator) orother graphing tool (TI or Casio calculator or web-based applet, or Graphmatica), and theEquation Editor built into MS Word 2007 or 2010, or MathType. MS Excel (or equivalent) canalso 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). Itmust 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 theacceptable formats listed.The asterisk is where you enter “LastName FirstName Logistic Problem Set” (no specialcharacters. 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 finalgrade.The Logistic Function Problem Set Math 140Page 2 of 7 pages0214Math background.Pure exponential growth is not a real-world phenomenon (there are a finite number of atoms inthe observable universe, and the exponential function is a continuous function), but it can be afine model for short periods of time. How fast does an exponential function grow? Weâ€lldiscover 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-logpaper)!Nonetheless, if the growth is slow enough—on the order of a couple percent per year, forexample, an exponential model is simple and very good at tracking populations over a twentyyear interval, but not much longer. Such functions as y = (1.025)x grow slowly enough to berealistic growth models for x values to 30 or so. Calculate 1.02530 to see why this is so…..Logistic growth describedNo living system exhibits exponential growth over any extended period of time, simply due tothe exhaustion of available resources to feed the “beast”. It is this resource limitation (amongother limiting factors) that causes an eventual “leveling” of the population (or size) of thesystem.With this in mind, think about resource-constrained exponential growth, also known aslogistic growth. Google or Bing a guy named Verhulst (he lived in the 1800â€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 140Page 3 of 7 pages0214In order to model resource-constrained growth, you need a graph that looks more like thefollowing instead of the pure exponential growth curve:The upper limit of this graph was set at y = 1 to represent 100% of a systemâ€s capacity. As anorganismâ€s population or other resource-constrained variable increases, it exhibits almost-pureexponential growth early until it gets to about 50% of the environmentâ€s capacity, such as“amount of easily arable farmland brought to production”, after which the pressures of limitedsupply slow the growth in easy-to-use farmland. Or in the case of an species, until it runs out ofspace 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 btc f tae? = +(1)In problems where 100% is the total capacity of a system (and for simplicity, one that wonâ€tchange over time), we let c be the number 1. If the capacity increases, decreases, or oscillatesover time (such as the seasons), then weâ€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 initialcondition of the system).y = 1The Logistic Function Problem Set Math 140Page 4 of 7 pages0214Note 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 inEquation 1?The logarithmic form of the logistic growth functionIn Chapter 4, youâ€ve learned that the logarithm can be used to “free” a variable from theexponent, if you will, for algebraic advantage. There are many good reasons to solve Equation 1for t or b. Unfortunately, it takes a bit of algebra to get the equation into a form that wouldpermit a simple rewrite into its logarithmic form. So, letâ€s see what we can do algebraically toget bt e? by itself:Task 4: Have at it! Algebraically manipulate Equation 1 until bt e? is by itself. That is, solve forbt e? . But first, replace f(t) by y to simplify the details.Done right, you should get bt c y eay? ? =Now, using the definition of logarithm in Section 4.2, you can rewrite this equation to solve foreither b or t. For Equation (2), weâ€ll solve for t:ln1 ln1 () ln (2) ( )c y btayc y tb ayc ft tb 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).The Logistic Function Problem Set Math 140Page 5 of 7 pages0214One of the Big Ideas: If we know the population, or mass, or whatever quantity we aremeasuring 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 andb, we can find the time t it takes for a system to grow to whatever proportion of capacity we wishto predict using Equation 2.The Application—Understanding the Spread of InfluenzaInfluenza (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 in3rd world countries not able to maintain strict standards of hygiene. New strains develop in theanimal population fairly continuously, and the strongest of these varieties kill or greatly weakentheir hosts. Symptoms manifest themselves more slowly than the disease spreads, hence itseffectiveness. The World Health Organization has a regular strategy to “trap” new strains asthey mutate before cross-species variants develop, and the CDC acts as coordinating agent in theUS 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 toinsourcing our own vaccine production began in 2012.)Since the vast majority of people survive the flu, it isnâ€t as critical to identify a new flu virusright away. In fact, a prospective new strain needs to show the ability to spread fairly rapidlyand be rather potent before the government will spend large sums of money to build a vaccinefor any new strain. H5N1 (called avian flu), however, showed itself to be rather deadly in Asiain the summer and fall of 2008, but it also proved to be tough to spread in the human population.It was H5N1â€s deadliness that led the Centers for Disease Control to set up a crash vaccinecreation program for it in early 2009. By 2010, it was incorporated into the mainstream Fallseason 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. If50,000,000 people receive a vaccine and if it is timely and effective, then 50,000,000 people canbe considered to be safe from the disease (or even to have “caught” it already). Using thelogistic function as a model, this 16% reduction in the vulnerable population should also reducethe number infected by 16% at any given time.Vaccinations also make a disease like the flu harder to spread, because clusters of vulnerablepeople in close quarters are fewer. This would reduce the infectivity of a virus, so that its spreadbecomes even slower, until the flu season passes and the virus goes largely dormant. Out-ofseasonflu epidemics are pretty rare. (None of the Tasks involve the effects of vaccination)The Logistic Function Problem Set Math 140Page 6 of 7 pages0214For the rest of the Problem Set, in Equation 1 weâ€ll set c = 1. It will be your job to find thevalues for a and b under three separate Scenarios, explain what they represent for each scenarioin Equations 1, 2, and 3, and then find the time it takes for a given set of proportions of thevulnerable population to be infected by it.For U.S. population studies, where at least 250,000,000 people are vulnerable to a particular fluvirus, if we set the capacity of the flu virus to infect vulnerable people in the US at c = 1, thevalue of the initial proportion f(0) is considerably smaller than 1, (as few as 300 people with flulikesymptoms may have their blood tested by the time a new virus is identified) and so weâ€d setf(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 logisticequations. The value of b will be given as a proportion/day so that if b = .01, then 1% of thevulnerable population is newly infected per day. Your value of f(0) must be computed from theinitial number of cases as a proportion of the 250,000,000 vulnerable. f(0) will be a very smalldecimal number in most of the tasks. Weâ€ll call Day Zero (that is, you set t = 0 on Day Zero) theday 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 equationusing 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 calculatorreally wonâ€t be good enough for the plots—theyâ€re pretty primitive.Scenario 1: Letâ€s assume that b = .0075 and that on Day Zero, there are an estimated 10,000infected 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 Scenario1. Letâ€s also assume 10,000 people have been infected by Day Zero (same vulnerablepopulation.The Logistic Function Problem Set Math 140Page 7 of 7 pages0214Scenario 3. Strain 3 is in its second year in the US, so it is estimated that 500,000 people havehad that variant. Let us also assume that it was slightly less virulent than the second strain, sothat k = .008.Note: You may use Equations (1), (2) or (3) or some combination of them to complete thefollowing tasks. You may even graph Equation (1) and use a Trace facility to approximate theexact solutions.Task 5: For each of the Scenarios above, determine when half the population (f(t) = 0.5) hasbeen 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 comparethe impact of the different f(0) values. Which factor seems more important—the size of theinitial 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 whatvalue of b will result in 60% of the population being infected as of Day 300.

 

Do you need a similar assignment done for you from scratch? We have qualified writers to help you. We assure you an A+ quality paper that is free from plagiarism. Order now for an Amazing Discount!
Use Discount Code “Newclient” for a 15% Discount!

NB: We do not resell papers. Upon ordering, we do an original paper exclusively for you.

The post The Logistic Function Problem Set Math 140Page 1 of 7 pages0214This Logistic Function Problem Set… appeared first on Affordable Nursing Writers.