Section1.8Projects for First-Order Differential Equations
Project1.8.1.Project—The Spruce Budworm.
Choristoneura fumiferana or the eastern spruce budworm is a species of moth native to the eastern United States and Canada. The caterpillars feed on the needles of spruce and fir trees. Populations can experience significant oscillations. The spruce budworm population remains at a relatively low, constant level most of the time. However, outbreaks have been recurring approximately every three decades, and studies suggest the spruce budworm has been breaking out in eastern North America for thousands of years.
Outbreaks of the spruce budworm have been responsible for some major deforestations in Canada and the United States. The eastern spruce budworm is considered one of the most destructive forest pests in North America.
You are acting as a consultant to the state department of forestry. Your task is to explain how outbreaks occur and how often the state can expect outbreaks.
The equation
\begin{equation}
x' = r \left( 1 - \frac{x}{K} \right) x - c \frac{x^2}{a + x^2}\tag{1.8.1}
\end{equation}
has been used to describe the dynamics of spruce budworm populations, where the variable \(x\) denotes the population or density of the insect [16]. One explanation that has been given for the occurrence of outbreaks is based on the multiple bifurcations that occur with this differential equation.
(a)
Explain how (1.8.1) can be used to model the spruce budworm population.
(b)
If \(a = 0.01\text{,}\)\(c = 1\text{,}\) and \(K = 1\text{,}\) we have a family of differential equations parameterized by \(r\text{,}\)
and plot the result in the \(xr\)-plane for \(0 \leq r \lt 1\text{.}\)
(c)
To find the bifurcation diagram for the spruce budworm equation, reflect the graph obtained in part Task 1.8.1.b about the line \(r= x\) line.
(d)
Estimate the two bifurcation values from your graph. Explain what happens to the population as \(r\) increases. That is, when does an outbreak occur? What happens after an outbreak?
(e)Your Final Report.
Your final report should contain a one-page executive summary. The executive summary should summarize your work in such a way that the reader can rapidly become acquainted with the material. It should contain a brief description of the problem, important background information, a discussion of pertinent assumptions, a short description of your methodology, concise analysis, and your main conclusions. Assume the reader is familiar with the basics of calculus and differential equations, so there is no need to walk through every step of your solution process or include equations. However, you should still describe the processes and mathematical techniques you used to reach your conclusions and explain why you used them. Refer the reader to the appendices as needed.
Appendices should be neatly formatted and present information in a logical manner. DO NOT simply print out Sage code. Consolidate your results and provide a short explanation of what it is the reader is seeing while also highlighting key pieces of information in the appendix.
Appendix A—Answers and analysis
Additional Appendices—Include additional appendices as necessary.
Project1.8.2.Project—The Spread of an Oil Spill.
Oil spills such as the Deepwater Horizon in the Gulf of Mexico or the Amoco Cadiz off the coast of Brittany can have disastrous consequences for society—economically, environmentally, and socially. Even smaller spills such as the Exxon Valdez can have have a huge impact on the surrounding environment due to the remoteness of the site or the difficulty of mounting an emergency environmental response. Cleanup and recovery from an oil spill is difficult and depends upon many factors, including the type of oil spilled, the temperature of the water, and the types of shorelines and beaches involved. Spills may take weeks, months or even years to clean up. For this reason the timeliness of an emergency response is critical. 1
Suppose an oil spill occurs off the coast of Texas. From time to time, but irregularly, a helicopter is dispatched by the Coast Guard to photograph the oil slick. On each trip, the helicopter arrives over the slick, the pilot takes a photo, waits 10 minutes, takes a second photo, and heads home. On each of seven trips the size (in area) of the slick is measured from both photographs. The data is given in Table 1.8.1
Table1.8.1.Data on spread of an oil spill
Initial Observation
10 Minutes Later
1.047
1.139
2.005
2.087
3.348
3.413
5.719
5.765
7.273
7.304
8.410
8.426
9.117
9.127
(a)
Build a model for the size of the oil slick at time \(t\text{.}\)
(b)
Predict the future size of the oil slick, say at \(t=10\) min, \(t=20\) min, \(t=120\) min.
(c)
Plot your model of the size of the oil slick as a function of time.
(d)
Find the time at which the oil slick is 8 square miles.
(e)
Determine the time of each of the observations for the first, third, fifth, and seventh initial observations.
(f)Your Final Report.
You have been retained as a consultant to the United States Coast Guard (USCG) to analyze this data and submit a report of your findings. Your final report should contain a one-page executive summary. The executive summary should summarize your work in such a way that the reader can rapidly become acquainted with the material. It should contain a brief description of the problem, important background information, a discussion of pertinent assumptions, a short description of your methodology, concise analysis, and your main conclusions. Do not assume that the reader knows anything about calculus or differential equations, but does have experts that can verify your model and calculations, which should appear in an appendix.
Appendices should be neatly formatted and present information in a logical manner. DO NOT simply print out Sage code or a series of equations. Consolidate your results and provide a short explanation of what it is the reader is seeing while also highlighting key pieces of information in the appendix.
Appendix A—Answers and analysis
Additional Appendices—Include additional appendices as necessary.
Project1.8.3.Project—Malaria Control.
Your company has been contracted to build a hospital in sub-Saharan Africa. The company is aware of the malaria threat in the region and has asked you to analyze malaria preventive measures for the employees that will be sent to build the hospital. 2
Malaria is a serious and sometimes fatal disease. In 2018, it was estimated that 228 million people worldwide contracted malaria. The estimated number of deaths was approximately 405,000 people. Children under the age of 5 years are the most vulnerable group, and they accounted for 67% (272,000) of all malaria deaths worldwide. Ninety-three percent of malaria cases are in Africa with six countries accounting or more than half of all malaria cases worldwide: Nigeria (25%), the Democratic Republic of the Congo (12%), Uganda (5%), and Côte d’Ivoire, Mozambique and Niger (4% each). 3
People who get malaria are typically very sick with high fevers, shaking chills, and flu-like illness. Although malaria can be a deadly disease, illness and death from malaria can usually be prevented. The disease is caused by a parasite that infects red blood cells. The parasite is transmitted from person to person through the bite of mosquitoes. Avoiding mosquito bites is the only sure method to prevent malaria infection. Malaria can be cured with prescription drugs. The type of drugs and length of treatment depend on the type of malaria, where the person was infected, their age, whether they are pregnant, and how sick they are at the start of treatment. 4
Your first task is to analyze the anti-malarial drug dosing regimen for those building the hospital. The primary concerns are how soon to start treatment before everyone arrives and the potential risks if employees miss one or two of their scheduled doses. These questions require an understanding of pharmacokinetics.
Pharmacokinetics is the study of what the body does to a drug. Specifically, it refers to the movement of the drug into, through, and out of the body. Understanding the pharmacokinetics of a drug allows medical professionals to develop appropriate drug dosing regimens for their patients. A complete understanding of the pharmacokinetics of a drug requires information on the processes of absorption, distribution, metabolization, and excretion of the drug from the body. In order to simplify this problem, we will assume that the anti-malarial medication is immediately absorbed into the blood stream and that everyone begins treatment with zero anti-malarial medication in their body. After the drug is absorbed into the blood, the rate of elimination of the drug from the body is proportional to the amount of drug currently in the blood. In other words, the more of the drug in the blood stream, the faster it is removed from the body. We will also assume that the rate of excretion of the drug is the same for everyone.
Atovaquone/proguanil, sold under the trade names Malarone among others, is a combination of two antimalarial medication atovaquone and proguanil. It is used to treat and prevent malaria, including chloroquine-resistant malaria. A standard adult pill of Malarone contains 250mg of atovaquone and 100mg of proguanil. A typical dose is one pill taken at the same time each day. An individual must maintain at least 300mg of atovaquone and 30mg of proguanil in their blood at all times in order to prevent contracting malaria. A typical adult eliminates atovaquone at a rate that corresponds to a half-life (time to eliminate one half of the original dose) of 48 hours. The half-life of proguanil is 12 hours. 5
At \(t = 0\text{,}\) a typical adult ingests one pill of Malarone. Write an initial value problem (IVP) that models the mass (mg) of the drug proguanil in the adult’s blood for \(t \geq 0\text{.}\) Solve this IVP analytically. After a person takes the first pill of Malarone, if they take no further medication, for how many hours can they expect to have enough proguanil in his blood to prevent contracting malaria?
(ii)
Approximate the solution to the IVP you developed in Task 1.8.3.a.i over the next seven days (168 hours) by implementing Euler’s method in Sage with two step sizes: \(h = 6\) hours and \(h = 1\) hour (see utmost-sage-cell.org/diff-equations). Compute the error for both step sizes by comparing your estimate to the analytic solution.
(iii)
Approximate the solution to the IVP over the next seven days by implementing the RK4 or classic Runge–Kutta method in Sage with step size: \(h = 1\) hour. Compute the error for both step sizes by comparing your estimate to the analytic solution (see utmost-sage-cell.org/diff-equations).
(iv)
Create a table summarizing the results of all numerical methods (Task 1.8.3.a.ii and Task 1.8.3.a.iii). This table should include columns for time, the actual mass of proguanil in each subject’s blood (from Task 1.8.3.a.i), and the approximate values of the solution for all methods and step sizes at \(t = 0, 24, 48, \ldots, 168\) only. Your table should have a total of 5 columns. Include an additional entry at the bottom of each of the last 3 columns providing the error for that method and step size. Discuss your results. Your analysis should include a comparison of error for different step sizes and for different methods.
(v)
Write an IVP that models the mass (mg) of the drug atovaquone in a typical adult’s blood for \(t \geq 0\) if a person ingests only one pill of Malarone at \(t = 0\text{.}\) Approximate the solution to this IVP using the RK4 method in Sage with step size: \(h = 1\) hour.
(vi)
Using your solutions to parts Task 1.8.3.a.iii and Task 1.8.3.a.v, if an adult ingests one pill of Malarone at the same time each day, how long before your construction crew departure to Africa should your crew begin taking their pills? In other words, how long until everyone has enough proguanil AND atovaquone in their blood to prevent contracting malaria? Illustrate your result using a graph or table.
(vii)
Assume a typical person reaches a “steady state” level of proguanil and atovaquone in their body after their 8th dose of Malarone. If after reaching “steady state” a person misses one dose, are they at risk of contracting malaria? How about after missing two consecutive doses? Explain.
(b)Mosquito Population Control.
Unfortunately anti-malarial drugs are not 100% effective at preventing the spread of malaria. The only sure method to prevent contracting malaria is to avoid mosquito bites. In light of this, you must look at methods to control the mosquito population in the construction area near the living quarters. Your objective is to reduce the mosquito population in both the short (less than 6 months) and long (greater than 6 months) term while also minimizing the environmental impact and potential health problems associated with the mosquito control measures that are implemented.
There are two typical approaches to mosquito control. The first approach is to directly kill mosquitoes using an insecticide spray. The effectiveness of this method depends on the frequency of spraying and the concentration of the insecticide. However, frequent spraying of insecticide has potential negative health effects on both humans and nearby wildlife. This method is also temporary as the mosquito population will recover once routine spraying has stopped. The second approach to mosquito population control is to eliminate the resources that support the mosquito population. This is done through the destruction of mosquito breeding grounds and involves everything from filling in small puddles to the draining of swamps and marshes. Keeping in consideration manpower and budget constraints, you must find the combination of these two approaches that best meets these objectives over the next six months.
Studies on mosquito control indicate that the rate of change of the mosquito population, \(P\text{,}\) (measured in millions of mosquitoes) in the area can be accurately modeled using the following Initial Value Problem:
where \(E\) is a constant that represents the effectiveness of insecticide spraying efforts, M is the local carrying capacity and is affected by efforts to reduce mosquito breeding grounds, \(k\) is the population growth constant, \(P_0\) is the initial population, and time \(t\) is measured in months. Current estimates suggest that there are approximately 10 million mosquitoes living near the proposed site of the hospital. Data indicates that \(M = 11\) million and \(k = 1.2\) for this mosquito population.
Use the information provided and the RK4 method to evaluate the following three mosquito population scenarios. Recommend a scenario and support your recommendation with mathematics. Be sure to include other considerations (not just what your mathematical model tells you) in your analysis. Tables, graphs, and other visual representations that summarize and support your analysis and recommendation are encouraged and should be included in an appendix.
(i)Scenario A.
In this scenario, you would devote most of your available resources for mosquito control to the use of insecticide. You would spray highly concentrated and expensive insecticide in the employee living areas daily. Due to the emphasis on the use of insecticide would be limited to destroying small mosquito breeding grounds only on the base camp. You estimate a value for \(E\) of \(E_A\) (see Table 1.8.2) and a new carrying capacity of \(M_A\) million mosquitoes that would be in effect immediately.
Table1.8.2.Values for \(M\) and \(E\)
Scenario A
Scenario B
Scenario C
\(M_A\)
\(E_A\)
\(M_B\)
\(E_B\)
\(M_C\)
\(E_C\)
Set 1
10.5
0.55
6.0
0.10
9.0
0.40
Set 2
10.0
0.60
5.5
0.10
8.5
0.45
Set 3
10.5
0.50
6.5
0.10
9.0
0.40
(ii)Scenario B.
In this scenario, you would devote most of your resources towards destroying mosquito breeding grounds. This would involve the permanent draining of a nearby marsh that is believed to be the primary breeding ground for mosquitoes at the construction site. Draining the marsh would take time to complete, delaying any impact on the mosquito population by two months, but would lower the carrying capacity to \(M_B\) million mosquitoes. You would still spray insecticide but less frequently and with less potency than in Scenario A. It is estimated that spraying would immediately result in a value for \(E\) of \(E_B.\)
(iii)Scenario C.
This scenario seeks to balance the insecticide and breeding ground destruction approaches. You would use the same, more potent, insecticide as Scenario A but spray less frequently. This would free up manpower to destroy small mosquito breeding grounds. You estimate a new value for \(E\) of \(E_C\) and a new carrying capacity of \(M_C\) million mosquitoes that would be in effect immediately.
(c)Your Final Report.
Your final report should contain a one-page executive summary. The executive summary should summarize your work for Task 1.8.3.a and Task 1.8.3.b in such a way that the reader can rapidly become acquainted with the material. It should contain a brief description of the problem, important background information, a discussion of pertinent assumptions, a short description of your methodology, concise analysis, and your main conclusions. Assume the reader is familiar with the basics of calculus and differential equations, so there is no need to walk through every step of your solution process or include equations. However, you should still describe the processes and mathematical techniques you used to reach your conclusions and explain why you used them. Refer the reader to the appendices as needed.
Appendices should be neatly formatted and present information in a logical manner. DO NOT simply print out Sage code. Consolidate your results and provide a short explanation of what it is the reader is seeing while also highlighting key pieces of information in the appendix.
Appendix B—Provide a table that summarizes the results of your analysis for all three scenarios in Task 1.8.3.b. Also include a copy of all work you used to analyze your recommended scenario.
Additional Appendices—Include additional appendices as necessary.
Project1.8.4.Project—Oligarchies.
An oligarch is an individual holding a non-negligible fraction of the country’s total wealth. 6
This project is adopted from Boghosian, Bruce and Borgers, Christoph (2024) “ODE Models of Wealth Concentration and Taxation,” CODEE Journal: Vol. 17, Article 10. Available at: scholarship.claremont.edu/codee/vol17/iss1/10
The concentration of wealth in the hands of a small number of individuals is considerable in many countries around the world, including the United States. In 2017, Forbes found that just three individuals (Jeff Bezos, Warren Buffett and Bill Gates) held more wealth than the bottom half of the population. 7
Kirsch, Noah. “The 3 Richest Americans Hold More Wealth Than Bottom 50% Of The Country, Study Finds” Forbes.
Individuals who hold a significant fraction of society’s total wealth have considerable political power.
Assume that \(w \in (0, 1)\) is the fraction society’s wealth. We assume that in a short time \(dt\text{,}\) the oligarchs, by virtue of the power that their wealth gives them, acquire a small fraction \(\lambda \, dt\) of the wealth not yet in their hands:
As \(w\) increases, acquiring wealth becomes easier for the oligarchs. They can hire more lobbyists and better lawyers. They can afford riskier but more profitable investments. Therefore, it is reasonable to assume that \(\lambda\) is proportional to \(w\text{,}\) so
\begin{equation*}
\lambda = gw
\end{equation*}
for some constant \(g \gt 0\text{.}\) We thereby arrive at the logistic growth equation,
Suppose that the government taxes the oligarchs by taxing away a small fraction of their wealth, say \(rw\text{.}\) This is called a flat tax. Our equation now becomes
We now have a one-parameter family of equations. Discuss the stability of equilibrium solutions. When does a bifurcation occur? What are the implications for a flat tax on the existence of oligarchies?
(ii)Flat Income Tax.
To model the taxation of income (or capital gains), assume the government takes away a fixed fraction \(\theta \in (0, 1)\) of the wealth gained by the oligarchs in time \(dt\text{,}\)
What are the equilibrium solutions. Discuss their stability. What effect does a flat income tax have on an oligarchy?
(iii)Progressive Wealth Tax.
A progressive wealth tax would mean that \(r\) itself grows with \(w\text{.}\) Assume that \(r = r_0 P(w)\text{,}\) where \(r_0\) is a fixed positive constant, and \(P\) is a continuous, monotonically increasing function of \(w \in [0, 1]\) with \(P(0) = 1\text{.}\) Our equation now becomes
What are the equilibrium solutions for this equations. Examine the stability when \(r_0 \lt g\) and \(r_0 \gt g\text{.}\) What happens to oligarchies in both cases?
(iv)Progressive Income Tax.
In a flat income tax the government takes away a fixed fraction \(\theta \in (0, 1)\) of the wealth gained by the oligarchs in time \(dt\text{,}\)
In a progressive income tax, the government will take away a larger fraction as income increases. In other words, suppose that government collects taxes as a function \(Q\) of \(x = gw(1-w)\text{.}\) Suppose that \(Q\) is a monotoniically increasing function of \(x \geq 0\text{,}\) where \(Q(0) \geq 0\text{.}\) The greater an oligarch’s gain per unit time, the greater percentage will be collected as tax. Assuming that the government will never tax at a rate of 100%, we have \(Q(x) \lt 1\) for all \(x\text{.}\) The equation for a progressive income tax is now
What will happen to oligarchs under a progressive income tax?
(b)Your Final Report.
Your final report should contain a one-page executive summary. The executive summary should summarize your work for work in a way that the reader can rapidly become acquainted with the material. It should contain a brief description of the problem, important background information, a discussion of pertinent assumptions, a short description of your methodology, concise analysis, and your main conclusions. Assume the reader is familiar with the basics of calculus and differential equations, so there is no need to walk through every step of your solution process or include equations. However, you should still describe the processes and mathematical techniques you used to reach your conclusions and explain why you used them. Refer the reader to the appendices as needed.
Appendices should be neatly formatted and present information in a logical manner. DO NOT simply print out Sage code or write isolated equations. Consolidate your results and provide a short explanation of what it is the reader is seeing while also highlighting key pieces of information in the appendix.
