Abstract: The Front Matter contains the Editor-in-Chief's Foreword, a Dedicatory by Associate Editor Douglas Meade, a Preface by the Special Editors Bev West and Samer Habre, and the Table of Contents. PubDate: Thu, 25 Jan 2024 17:46:03 PST

Abstract: This is the full issue (front matter and all papers) of the Third CODEE Special Issue, with the theme, "Engaging the World: Differential Equations can Influence Public Policies." PubDate: Thu, 25 Jan 2024 17:46:00 PST

Authors:Maila Hallare et al. Abstract: Many examples of 2x2 nonlinear systems in a first-course in ODE or a mathematical modeling class come from physics or biology. We present an example that comes from the business or management sciences, namely, the Bass diffusion model. We believe that students will appreciate this model because it does not require a lot of background material and it is used to analyze sales data and serve as a guide in pricing decisions for a single product. In this project, we create a 2x2 ODE system that is inspired by the Bass diffusion model; we call the resulting system the Bass Competition Model. We apply this model to investigate the competition between two smartphone brands, Apple and Samsung. We use existing data that capture social behavior of the users to further investigate the competition dynamics between Apple and Samsung. The goal is to present an example of a competition model that uses real data on a topic that is relatable to the 21st-century student. PubDate: Tue, 16 Jan 2024 11:41:31 PST

Authors:Christopher Evrard et al. Abstract: A Sand Tank Groundwater Model is a tabletop physical model constructed of plexiglass and filled with sand that is typically used to illustrate how groundwater water flows through an aquifer, how water wells work, and the effects of contaminants introduced into an aquifer. Mathematically groundwater flow through an aquifer can be modeled with the heat equation. We will show how a Sand Tank Groundwater Model can be used to simulate groundwater flow through an aquifer with a no flow boundary condition. PubDate: Wed, 03 Jan 2024 15:07:18 PST

Authors:Victor J. Donnay Abstract: How does mathematics connect with the search for solutions to the climate emergency' One simple connection, which can be explored in an introductory differential equations course, can be found by analyzing the energy generated by solar panels or wind turbines. The power generated by these devices is typically recorded at standard time intervals producing a data set which gives a discrete approximation to the power function $P(t)$. Using numerical techniques such as Euler’s method, one can determine the energy generated. Here we describe how we introduce the topic of solar power, apply Euler’s method to determine the energy generated, and provide a variety of lesson extensions which engage students in an exploration of policy issues related to climate change. I use these examples in my course on mathematical modeling and sustainability. There I not only want my students to understand how mathematics can be used to examine important real-world issues like climate change, but I also want to empower them to use their mathematical skills to help create solutions. To that end, the course has a community-based field component in which the students assist a community partner by analyzing a sustainability problem of relevance to the partner organization. PubDate: Wed, 03 Jan 2024 15:07:12 PST

Authors:Christoph Borgers et al. Abstract: This paper presents mathematics relevant to the question whether voting should be mandatory. Assuming a static distribution of voters’ political beliefs, we model how politicians might adjust their positions to raise their share of the vote. Various scenarios can be explored using our app at https: //centrism.streamlit.app/. Abstentions are found to have great impact on the dynamics of candidates, and in particular to introduce the possibility of discontinuous jumps in optimal candidate positions. This is an unusual application of ODEs. We hope that it might help engage some students who may find it harder to connect with the more customary applications from the natural sciences. PubDate: Wed, 03 Jan 2024 15:07:03 PST

Authors:Bruce Boghosian et al. Abstract: We refer to an individual holding a non-negligible fraction of the country’s total wealth as an oligarch. We explain how a model due to Boghosian et al. can be used to explore the effects of taxation on the emergence of oligarchs. The model suggests that oligarchs will emerge when wealth taxation is below a certain threshold, not when it is above the threshold. The underlying mechanism is a transcritical bifurcation. The model also suggests that taxation of income and capital gains alone cannot prevent the emergence of oligarchs. We suggest several opportunities for students to explore modifications of the model. PubDate: Wed, 03 Jan 2024 15:06:57 PST

Authors:Catherine Cavagnaro Abstract: Real-world applications can demonstrate how mathematical models describe and provide insight into familiar physical systems. In this paper, we apply techniques from a first-semester differential equations course that shed light on a problem from aviation. In particular, we construct several differential equations that model the distance that an aircraft requires to become airborne. A popular thumb rule that pilots have used for decades appears to emanate from one of these models. We will see that this rule does not follow from a representative model and suggest a better method of ensuring safety during takeoff. Aircraft safety is definitely a matter of public concern, although it is the FAA (Federal Aviation Administration) that makes the regulations. PubDate: Wed, 03 Jan 2024 15:06:52 PST

Authors:Scott A. Strong et al. Abstract: This article presents existing mathematical models associated with mountain pine beetle populations in lodgepole pine forests, whose reproductive cycle requires the destruction of colonized host trees, decreasing timber availability/quality, and providing fuel sources for wildfires. With the existence of a positive-feedback loop with environmental warming, the need for intervention and management is clear. However, the legislative responses to the focusing events from our 2000-2010 North American epidemics are characterized as under-leveraged. While the reasons for this are multifaceted, increasing the capacity of STEM-informed individuals to take part in quantitative modeling of the underlying ecosystem generates awareness and provides pathways connecting the ``how'' of public land management to the ``what'' and ``why'' of policy creation. To this end, we survey existing mathematical models for the mountain pine beetle, ranging from a simple planar model to a chemotactic model involving pheromone/kairomone signaling between beetles and host trees. The latter can be localized to a host tree characterized by a nonlinear system of ordinary differential equations whose solution is reasonable to approximate numerically. PubDate: Wed, 03 Jan 2024 15:06:42 PST

Authors:Li Zhang Abstract: We present an intriguing topic in an undergraduate mathematical modeling course where predator-prey models are taught to our students. We describe modeling activities and the use of technology that can be implemented in teaching this topic. Through modeling activities, students are expected to use the numerical and graphical methods to observe the qualitative long-term behavior of predator and prey populations. Although there are other choices of predators and prey, we find that using blue whales and krill as predator and prey, respectively, would be most beneficial in strengthening our students' awareness of protecting endangered species and its impact on climate change and global habitability. PubDate: Wed, 03 Jan 2024 15:06:37 PST

Authors:James Sandefur Abstract: In this paper, we develop differential equations that model the sustainable harvesting of species having different characteristics. Specifically, we assume the species satisfies one of two different types of density dependence. From these equations, we consider maximizing sustainable harvests. We then introduce a cost function for fishing and study how maximizing profit affects the harvesting strategy. We finally introduce the concept of open access which helps explain the collapse of many fish stocks.The equations studied involve relatively simple rational and exponential functions. We analyze the differential equations using phase-line analysis as well as graphing approximate solutions using Euler's method, which is really a discrete form of the differential equation. PubDate: Wed, 03 Jan 2024 15:06:32 PST

Authors:Michelle L. Ghrist Abstract: In this paper, I discuss two environmentally-focused writing assignments that I developed and implemented in recent integral calculus and differential equations courses. These models of carbon storage and PCB’s in a river provide interesting applications of one-compartment mixing problems. The assignments were intended to focus student attention on sustainability concerns while also developing other essential skills. I discuss these assignments and their effect on my students’ technical writing and environmental awareness. Detailed introductory instructions and mostly complete solutions to these assignments appear in the appendices, to include sample student work. PubDate: Wed, 03 Jan 2024 15:06:27 PST

Authors:Biyong Luo Abstract: In this article, I share my decade-long experience teaching an intensive five-week summer Differential Equation course covering complex topics and tips for creating an interactive and supportive learning environment to optimize student engagement. This article provides my detailed approach to planning and teaching an asynchronous course with rigor and flexibility for each student. An interactive teaching approach and variety of learning activities will augment students’ mathematical fluency and appreciation of the importance of differential equations in modeling a wide variety of real-world situations with special attention to ways differential equations can be relevant to creating public policy. PubDate: Wed, 03 Jan 2024 15:06:22 PST

Authors:D. Chloe Griffin et al. Abstract: Common mechanistic models include Susceptible-Infected-Removed (SIR) and Susceptible-Exposed-Infected-Removed (SEIR) models. These models in their basic forms have generally failed to capture the nature of the COVID-19 pandemic's multiple waves and do not take into account public policies such as social distancing, mask mandates, and the ``Stay-at-Home'' orders implemented in early 2020. While the Susceptible-Vaccinated-Infected-Recovered-Deceased (SVIRD) model only adds two more compartments to the SIR model, the inclusion of time-dependent parameters allows for the model to better capture the first two waves of the COVID-19 pandemic when surveillance testing was common practice for a large portion of the population. We find that the SVIRD model with time-dependent and piecewise parameters accurately fits the 2019-2020 experimental data from Spartanburg County, South Carolina. These additions give insight into the changing social response toward the COVID-19 pandemic within Spartanburg County. PubDate: Wed, 03 Jan 2024 15:06:15 PST

Authors:Carrin Goosen et al. Abstract: This is an account of a modelling scenario that uses the sir epidemic model. It was used in a third year applied mathematics subject. All students were enrolled in a mathematics degree of some type. Students are presented with the results of a test carried out on 100 individuals in a community containing 3000 people. From this they determined the number of infectious and recovered individuals in the population. Given the per capita recovery rate and making a suitable assumption about the number of infectious individuals at the start of the epidemic, they then estimate the infectious contact rate and from this the basic reproduction number. The mayor has asked the students to determine what will happen if no action is taken and to evaluate four policy options. They are asked to recommend the best course of action.This scenario provides students with a problem where parameter values must be inferred from the information provided (one cannot be determined). They use the sir model to provide public health recommendations, reinforcing their appreciation for the usefulness of mathematical modelling.Our paper gives details of student presentations, and errors on the final exam, along with feedback to and from the instructor and the two student coauthors. PubDate: Wed, 03 Jan 2024 15:06:05 PST

Authors:Christina Joy Edholm et al. Abstract: In May 2020, administrators of residential colleges struggled with the decision of whether or not to open their campuses in the Fall semester of 2020. To help guide this decision, we formulated an ODE model capturing the dynamics of the spread of COVID-19 on a residential campus. In order to provide as much information as possible for administrators, the model accounts for the different behaviors, susceptibility, and risks in the various sub-populations that make up the campus community. In particular, we start with a traditional SEIR model and add compartments representing relevant variables, such as quarantine compartments and a hospitalized compartment. We then duplicated the model for ten interacting sub-populations, resulting in a large system of differential equations. The model predicts possible outcomes based on hypothetical administrative policies such as masking, social distancing, and quarantining. As the pandemic developed, we updated the model to account for new policies, such as testing and vaccination and calibrated the model to data gathered from local sources. To complete the modeling process, we describe the parameter-fitting procedure, in which we used publicly available data from the county, as well as specific descriptions of our student body, faculty, and staff. The final stage of the work involved performing numerical simulations and designing an interactive application that allows non-mathematicians to experiment with a range of scenarios. We then extrapolate the findings of our model to a general audience, which along with our plots and app makes model conclusions accessible to all, democratizing the policy-making process. PubDate: Wed, 03 Jan 2024 13:00:55 PST

Authors:Yagub N. Aliyev Abstract: In this paper a nonlinear differential equation arising from an elementary geometry problem is discussed. This geometry problem was inspired by one of the proofs of the first remarkable limit discussed in a typical first semester undergraduate Calculus course. It is known that the involved differential equation can be reduced to Abel’s differential equation of the first kind. In this paper the problem was solved using an approximate geometric method which constructs a piecewise linear solution approximation for the curve. The compass tool of GeoGebra was extensively used for these constructions. At the end of the paper, some generalizations are discussed. A new transformation of curves, named “Interception”, is introduced and its approximate construction using GeoGebra is described. Some possible applications include geometry, calculus, ordinary differential equations, and military interceptions. PubDate: Sun, 24 Dec 2023 08:15:57 PST

Authors:Mehmet Pakdemirli Abstract: The Fibonacci differential equation is defined with analogy from the Fibonacci difference equation. The linear second order differential equation is solved for suitable initial conditions. The solutions constitute spirals in the polar coordinates. The properties of the spirals with respect to the Fibonacci numbers and the differences between the new spirals and classical spirals are discussed. PubDate: Mon, 20 Nov 2023 11:21:43 PST

Authors:Laurie A. Florio et al. Abstract: A generalized method for solving an undamped second order, linear ordinary differential equation with constant coefficients is presented where the non-homogeneous term of the differential equation is represented by Fourier series and a solution is found through Laplace transforms. This method makes use of a particular partial fraction expansion form for finding the inverse Laplace transform. If a non-homogeneous function meets certain criteria for a Fourier series representation, then this technique can be used as a more automated means to solve the differential equation as transforms for specific functions need not be determined. The combined use of the Fourier series and Laplace transforms also reinforces the understanding of function representation through a Fourier series and its potential limitations, the mechanics of finding the Laplace transform of a differential equation and inverse transforms, the operation of an undamped system, and through programming insight into the practical application of both tools including information on the influence of the number of terms in the series solution. PubDate: Wed, 11 Oct 2023 10:30:57 PDT

Authors:Viktoria Savatorova Abstract: This paper presents an exploration into parameter sensitivity analysis in mathematical modeling using ordinary differential equations (ODEs). Taking the first steps in understanding local sensitivity analysis through the direct differential method and global sensitivity analysis using metrics like Pearson, Spearman, PRCC, and Sobol’, we provide readers with a basic understanding of parameter sensitivity analysis for mathematical modeling using ODEs. As an illustrative application, the system of differential equations modeling population dynamics of several fish species with harvest considerations is utilized. The results of employing local and global sensitivity analysis are compared, shedding light on the strengths and limitations of each approach. The paper serves as a starting point for readers interested in exploring parameter sensitivity in their mathematical models. PubDate: Tue, 10 Oct 2023 02:00:49 PDT