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In this paper, we introduce novel instructional approaches to engage students in using modelling with data to motivate and teach differential equations. Specifically, we introduce a pedagogical framework that will execute instructional modules to teach different solution techniques for differential equations through repositories and notebook environments during real-time instruction. Each of these teaching modules employs a literate programming approach that uses the notebook environment to explain the concepts in anatural language, such as English, interspersed with snippets of macros and traditional source code on a web browser. The pedagogical approach employed is reproducible and leads to openaccess material for students to motivate and teach differential equations efficiently. We will share examples of this framework applied to teaching advanced concepts such as machine learning and neural network approaches for solving ordinary and partial differential equations as well as estimating parameters in these equations for given datasets. More details of the work can be accessed from https://aoguedao.github.io/teaching-ml-diffeq.

This article details the development of design principles to support teachers in planning for a Community-Based Mathematical Modeling task with a focus on social justice in the elementary grades. By reflecting on the dilemmas we encountered in the design and enactment of the tasks, we developed five design principles that allowed us to address issues of social justice as well as attend to powerful mathematical ideas to bring awareness and take action around a local problem. Through our article, we hope to share with mathematics teacher educators design principles to help plan for tasks with pre- and in-service teachers that prioritize connecting mathematics to social issues and empower both teachers and students to take action to make a positive impact in the community.

A powerful way to build students' mathematical power and agency is through Teaching Mathematics for Social Justice (TMfSJ), where students read and write the world with mathematics (Gutstein, 2005). Reading the world with mathematics uses mathematics to understand relations of power, resource inequities, and disparate opportunities between different social groups and to understand explicit discrimination based on race, class, gender, language, and other differences, while writing the world with mathematics involves taking action (Gutstein & Peterson, 2013). In this article, the authors share one teacher's story of how she used her students' interests to uncover reasons for the economic inequalities and fair access to sports at their local high school.

More than half of the coronavirus disease 19 (COVID-19) related mortality rates in the United States and Europe are associated with long-term-care facilities (LTCFs) such as old-age organizations, nursing homes, and disability centers. These facilities are considered most vulnerable to spread of an pandemic like COVID-19 because of multiple reasons including high density of elderly population with a diverse range of medical requirements, limited resources, nursing activities/medications, and the role of external visitors. In this study, we aim to understand the role of visitor’s family members and specific interventions (such as use of face masks and restriction of visiting hours) on the dynamics of infection in a community using a mathematical model. The model considers two types of social contexts (community and LTCFs) with three different groups of interacting populations (non-mobile community individuals, mobile community individuals, and long-term facility residents). The goal of this work is to compare the outbreak burden between different centre of disease control (CDC) planning scenarios, which capture distinct types of intensity of diseases spread in LTCF observed during COVID-19 outbreak. The movement of community mobile members is captured via their average relative times in and out of the long-term facilities to understand the strategies that would work well in these facilities the CDC planning scenarios. Our results suggest that heterogeneous mixing worsens epidemic scenario as compared to homogeneous mixing and the epidemic burden is hundreds times greater for community spread than within the facility population.

This article details the design and implementation of a professional development model called Learning Trajectory-Based Lesson Study focused on issues of equity, identity, and agency. We developed the Vertical Articulation to Unpack the Learning Trajectory (VAULT) tool to orient teachers' instructional planning toward an asset-based view of students' mathematics competencies. We examined teachers' use of the VAULT to plan, implement, and debrief on student strategies for one spatial reasoning task in elementary, middle, and high school classrooms. The VAULT facilitated intentional planning for a progression of anticipated strategies and equitable access to instruction. Teachers demonstrated an asset-based view of all student thinking independent of grade-level expectations.

As potential strategies to control the spread of COVID-19, governments all across the globe have implemented interventions such as lockdowns and confinement. While these strategies have helped to control the spread, there have also been evidence of widespread increase in Domestic Violence (DV) which is often under-reported. In this work, we have developed two new models that will help study the relationship between lockdowns, the spread of COVID-19 and DV in the hope of mitigating the social problems that follow such drastic measures. Two different models, in increasing level of complexity have been employed to simulate the effect of the lockdown strategy in the spread of COVID-19 and DV. One of the models simulates the spread of DV under three different lockdown scenarios: one long period, two and three shorter intervals that comprise the same interval of time since onset of the Pandemic. Another model studies the interaction between COVID-19 and DV during confinement in relation to the length of the lockdowns. Our findings indicate multiple lockdowns are safer for DV but not necessarily for controlling spread of COVID-19. We also present a derivation of the basic reproduction number for the model involving the interaction between COVID-19 and DV.

In late 2019, the emergence of COVID-19 in Wuhan, China, led to the implementation of stringent measures forming the zero-COVID policy aimed at eliminating transmission. Zero-COVID policy basically aimed at completely eliminating the transmission of COVID-19. However, the relaxation of this policy in late 2022 reportedly resulted in a rapid surge of COVID-19 cases. The aim of this work is to investigate the factors contributing to this outbreak using a new SEIR-type epidemic model with time-dependent level of immunity. Our model incorporates a time-dependent level of immunity considering vaccine doses administered and time-post-vaccination dependent vaccine efficacy. We find that vaccine efficacy plays a significant role in determining the outbreak size and maximum number of daily infected. Additionally, our model considers under-reporting in daily cases and deaths, revealing their combined effects on the outbreak magnitude. We also introduce a novel Physics Informed Neural Networks (PINNs) approach which is extremely useful in estimating critical parameters and helps in evaluating the predictive capability of our model.

In this paper, a stochastic epidemiological model is presented as an extension of a compartmental SEIR model with random perturbations to analyze the dynamics of the COVID-19 pandemic in the city of Bogotá D.C., Colombia. This model incorporates the spread of COVID-19 impacted by social behaviors in the population and allows for projecting the number of infected, recovered, and deceased individuals considering the mitigation measures, namely confinement and partial relaxed restrictions. Also, the role of randomness using the concept of Brownian motion is emphasized to explain the behavior of the population. Computational experiments for the stochastic model with random perturbations were performed, and the model is validated through numerical simulations for actual data from Bogotá D.C.

In this work, the dynamics of the spread of COVID-19 is considered in the presence of both human-to-human transmission as well as environment-to-human transmission. Specifically, we expand and modify traditional epidemiological model for COVID-19 by incorporating a compartment to study the dynamics of pathogen concentration in the environmental reservoir, for instance concentration of droplets in closed spaces. We perform a mathematical analysis for the model proposed including an endemic equilibrium analysis as well as a next-generation approach both of which help to derive the basic reproduction number. We also study the e˚cacy of wearing a facemask through this model. Another important contribution of this work is the introduction to physics informed deep learning methods (PINNs) to study the dynamics. We propose this as an alternative to traditional numerical methods for solving system of differential equations used to describe dynamics of infectious diseases. Our results show that the proposed PINNs approach is a reliable candidate for both solving such systems and for helping identify important parameters that control the disease dynamics.

With the increasing popularity of active teaching methods, universities have become more interested in changing instruction from direct lecture to interactive engagement. The George Mason University (Mason, Virginia, USA) Biology Department is achieving this goal through: 1) faculty participation in the SIMPLE project; 2) faculty and student involvement in Discipline-Based Education Research projects; 3) participating in the Learning Assistant program for undergraduate students as scholars; and 4) the Accelerator Program. These approaches are discussed in this paper. Also, interactive teaching strategies, which the biology faculty and graduate teaching assistants use at Mason, are documented in the form of design memos; these memos are introduced in this paper

The sandwich problem described in this article would likely be considered more appropriate for third grade or above. However, as members of the research team for a lesson study involving teachers from kindergarten through fourth grade, the authors saw kindergarten students rise to the challenge. The authors invite readers to review two scenarios, and imagine themselves observing each one. They ask readers to examine the student work and the dialogue provided and ask themselves the following questions: (1) What knowledge is revealed about the students? (2) How might you respond to groups 1 and 2? (3) What might be the learning goal? (4) What questions might you pose? and (5) How would you interpret and respond to student thinking?

As COVID-19 cases continue to rise globally, many researchers have developed mathematical models to help capture the dynamics of the spread of COVID-19. Specifically, the compartmental SEIR model and its variations have been widely employed. These models differ in the type of compartments included, nature of the transmission rates, seasonality, and several other factors. Yet, while the spread of COVID-19 is largely attributed to a wide range of social behaviors in the population, several of these SEIR models do not account for such behaviors. In this project, we consider novel SEIR-based models that incorporate various behaviors. We created a baseline model and explored incorporating both explicit and implicit behavioral changes. Furthermore, using the Next Generation Matrix method, we derive a basic reproduction number, which indicates the estimated number of secondary cases by a single infected individual. Numerical simulations for the various models we made were performed and user-friendly graphical user interfaces were created. In the future, we plan to expand our project to account for the use of face masks, age-based behaviors and transmission rates, and mixing patterns.

In this paper, we consider compartmental disease transmission models and discuss the importance of determining model parameters that provide an insight into disease transmission and prevalence. After a brief review and comparison of the performance of some heuristic approaches, the paper introduces three approaches including an optimization approach, a physics informed deep learning and a statistical inference method to estimate parameters and analyse disease transmission. The deep learning framework utilizes the hidden physics of infectious diseases and infer the latent quantities of interest in the model by approximating them using deep neural networks. The performance of the deep learning method is validated against representative small and big data sets corresponding to a well-known benchmark example and the results indicate that deep learning is a viable candidate to determine model parameters. The paper also presents the need for researchers to understand different types of dependencies exhibited in a typical data set and discovering the most appropriate approaches for statistical inference. Specifically, in this work we apply a time-series inferential method with a variety of statistical models. Our results indicate the efficiency and importance of statistical inference methods for researchers to understand and analyse time-series data to make confident predictions.

Youth that live in public housing communities with high social and financial needs are at risk of getting involved in illegal drug trafficking gangs that are controlled by adults. This social disease spreads like an epidemic in these densely populated sectors and metropolitan area. In this work, a model based on SIR disease dynamics is used to study the spread of gangs in vulnerable youth and adult sub-populations. Three types of mixing patterns govern interaction between the groups: proportionate, preferred and like-with-like. This new model is analysed presenting formulations for the reproduction number, sensitivity analysis, and stability analysis for the like-with-like mixing. Insights gained from simulation results on the sensitivity of the model to parameters show the relevance of the activity parameter over the reproduction number when heterogeneous mixing is present.

In this article, we provide some useful perspectives and experiences in mentoring students in undergraduate research (UR) in mathematical modeling using differential equations. To engage students in this topic, we present a systematic approach to the creation of rich problems from real-world phenomena; present mathematical models that are derived from calculus, but can be suitably modified to solve more complex problems; and finally provide pointers on how to train students through the UR process that can help to impact student's future academic and non-academic careers.

In this work, we review and share some useful technology for mathematics educators that can help to enhance their pedagogical practices to motivate students to improve their further understanding of differential equations after an initial exposure to the topic in an undergraduate class. In particular, we present some innovative ways to employ technology tools to motivate the learning of students and their appreciation of the applications of differential equations to real-world problems. The objective is to engage the students in exploration to solve differential equations using technology. This work presents a benchmark example involving modelling epidemics as a real-world problem and how to engage students who have basic familiarity with differential equations through literate programming. Visualization of the results using graphical user interface is also presented. The readers are also exposed to a novel free network computing laboratory called NCLab which is applied to a variety of applications involving differential equations.

Intracranial aneurysms (IAs) are localized enlargements of cerebral blood vessels that cause substantial rates of mortality and morbidity in humans. The rupture possibility of these aneurysms is a critical medical challenge for physicians during treatment planning. This treatment planning while assessing the rupture potential of aneurysms becomes more complicated when they are constrained by an adjacent structure such as optic nerve tissues or bones, which is not widely studied yet. In this work, we considered and studied a constitutive model to investigate the bio-mechanical response of image-based patient-specific IA data using cardiovascular structural mechanics equations. We performed biomechanical modeling and simulations of four different patient-specific aneurysms’ data (three middle cerebral arteries and one internal carotid artery) to assess the rupture potential of those aneurysms under a plane contact constraint. Our results suggest that aneurysms with plane contact constraints produce less or almost similar maximum wall effective stress compared to aneurysms with no contact constraints. In our research findings, we observed that a plane contact constraint on top of an internal carotid artery might work as a protective wall due to the 16.6% reduction in maximum wall effective stress than that for the case where there is no contact on top of the aneurysm.

The Zika arbovirus transmitted by the Aedes aegypti mosquitoes has been shown to be capable of infecting humans via two routes: the bites of infected vectors and through sexual contacts involving infected and non-infected persons. There is no treatment and current prevention or mitigating efforts rely on the use of the Centers for Disease Control and Prevention recommendations including the use of insecticide-treated bed nets (ITN) and indoor residual spraying (IRS). In this work, we investigate via a mathematical model, the role of ITN and IRS as methods for limiting the impact of Zika transmission. We introduce a model that builds on classical SEIR epidemiological single outbreak models. We compute the basic and control reproduction numbers and the final epidemic size in the presence of control measures ITN and IRS. We derive a gross estimate for the rate of sexual transmission, during the initial stages of the outbreak, in terms of prior estimates of the basic reproduction number from related albeit not sexually transmitted arboviral diseases.

Even years after experiencing a physical trauma, individuals describe trauma-specific distress, as trauma triggers present as sensory reminders that initiate physiological reactions at time of exposure. Mobile technologies offer tremendous potential in helping individuals who have experienced trauma manage symptoms as they transition out of hospital care and move back into their communities where they are at risk of trauma trigger exposure. A personalized wearable device, tailored to a patient-specific diagnosis (e.g., PTSD) with programmable neurophysiological behavioral risk set-points, could be a useful tool in helping individuals monitor symptomology. When this type of monitoring device is also connected to a personalized recovery cue intervention on a smartwatch or phone, and activated when the wearable sensor detects heightened risk, there is the opportunity for in-the-moment symptom management. In this study we sought to understand the value for trauma survivors of using this type of personalized mobile recovery support system. Study participants were all trauma survivors or family members of survivors who were involved in the Trauma Survivors Network. A semi-structured interview was conducted with participants to understand perceptions on the utility, sensory experiences, and innovation insights of a mobile recovery sensory support system overall, and about the recovery cue intervention most specifically. Results from participant interviews inform the further development of our mobile recovery support system model in significant ways, suggesting that three components must be included: 1) Recovery cues; 2) Relationships (connecting to supportive network); and 3) Regulation (neurophysiological regulation and behavioral risk reduction). Experience Framework This article is associated with the Innovation & Technology lens of The Beryl Institute Experience Framework. (http://bit.ly/ExperienceFramework) Access other PXJ articles related to this lens. Access other resources related to this lens

This case study focuses on a team of teachers and students in a Lesson Study, focused on using mathematical modeling (MM) to make significant decisions to design and plan for a sustainable edible garden in their community. We examined (a) how teachers develop students’ capacity to engage in mathematical modeling, while attending to equitable teaching practices; and (b) how teachers’ view of teaching through mathematics modeling changed after unit implementation. We found that teachers were deliberate in employing specific structures, routines, and tools to attend to equitable participation, when eliciting student thinking in the modeling process. We found that teachers’ view of mathematics modeling changed as they recognized how MM allowed for (a) integration of important mathematics concepts while giving students ownership of the mathematics; (b) opportunity to assess both content and 21st century process skills; and (c) positive energy that came from both students and teachers when teaching through the use of mathematical modeling. A promising strategy for preparing our youth for rigorous mathematics and skills to solve ill-structured problems is by integrating mathematical modeling in early elementary grades to develop critical 21st century skills and a productive disposition towards problem posing and problem solving.

This study examines elementary- and middle-grade teachers' understanding of the mathematical learning progression as they participated in a 6-month professional learning project. Teachers participated in a professional development project that consisted of a 1-week summer content-focused institute with school-based follow-up Lesson Study cycles in the fall that focused on the vertical articulation of algebraic concepts across grade levels. The following research examines how a vertical team of teachers from multiple grades designed, taught and learned from the Lesson Study cycle. The video analysis from the research lessons and the teachers' reflections revealed teachers' developing vertical connections and representational fluency from their planning, teaching, observation and debriefs. In addition, Lesson Study afforded teachers opportunities to deepen their understanding of the mathematical learning progression through observation and analysis of students' thinking through a situated school-based professional development experience.

With the increasing popularity of active teaching methods, universities have become more interested in changing instruction from direct lecture to interactive engagement. The George Mason University (Mason, Virginia, USA) Biology Department is achieving this goal through: 1) faculty participation in the SIMPLE project; 2) faculty and student involvement in Discipline-Based Education Research projects; 3) participating in the Learning Assistant program for undergraduate students as scholars; and 4) the Accelerator Program. These approaches are discussed in this paper. Also, interactive teaching strategies, which the biology faculty and graduate teaching assistants use at Mason, are documented in the form of design memos; these memos are introduced in this paper.

As much as ever before, mathematics teachers are searching for ways to connect mathematics to real-life scenarios within STEM contexts. As students develop skill in proportional reasoning, they examine graphical representations of linear functions, learn to associate "slope" with "steepness" and rate of change, and develop skill in calculating slopes of lines. Slope and rate of change, however, do not exist only in math class. Engineering applications associated with ramps, staircases, and roller coasters rely on analyses of slope and rate of change. In science class, students learn about velocity and speed, each of which connects to slope and rate of change. Technology can be used to support students as they explore slope and rate of change in these and other contexts. Given these connections, slope is an ideal topic with which to bring STEM into the mathematics classroom. This article presents a series of science- and engineering-related activities addressing rate of change that occurred in a middle-grades public school math class. As students worked through the activities, they developed an understanding of slope and rate of change within STEM contexts. The activities were developed as part of Expeditions in Science, Technology, Engineering, and Education through Mathematics (ESTEEM), a professional development program funded by the Virginia State Department of Education and held at George Mason University in Fairfax, Virginia. The two activities helped develop students' thinking about rate of change and slope within STEM-related contexts. With exposure to hands-on activities like the staircase problem, students come to understand how math can be used to solve problems in engineering. This article presents one example of the many ways that teachers can bring science and engineering connections into math class. As teachers think of these ideas, they need to continue to share them so that we develop not only a mathematically literate but also a STEM-literate citizenry.

The rupture potential of intracranial aneurysms is an important medical question for physicians. While most intracranial (brain) aneurysms are asymptomatic, the quantification of rupture potential of both symptomatic and asymptomatic lesions is an active area of research. Furthermore, an intracranial aneurysm constrained by an optic nerve tissue might be a scenario for a physician to deal with during the treatment process. In this work, we developed a computational model of an idealized intracranial saccular aneurysm constrained by a rigid nerve tissue to investigate the impact of constrained nerve tissues on the dynamics of aneurysms. A comparative parametric study for constraints of varying length on aneurysm surface was considered. Our computational results demonstrated the impact of contact constraints on the level of stress near the fundus and provided insight on when these constraints can be protective and when they can be destructive. The results show that lesions with long contact constraints generated higher stress (0.116 MPa), whereas lesions without constraints generated less stress (0.1 MPa) at the fundus, which indicated that lesions with nerve constraints can be protective and less likely to rupture than the lesions without constraints. Moreover, lesions with point load on the fundus generated the highest stress (18.15 MPa) and, hence, they can be destructive.

Understanding chemical transport in blood flow involves coupling the chemical transport process with flow equations describing the blood and plasma in the membrane wall. In this work, we consider a coupled two-dimensional model with transient Navier-Stokes equation to model the blood flow in the vessel and Darcy's flow to model the plasma flow through the vessel wall. The advection-diffusion equation is coupled with the velocities from the flows in the vessel and wall, respectively to model the transport of the chemical. The coupled chemical transport equations are discretized by the finite difference method and the resulting system is solved using the additive Schwarz method. Development of the model and related analytical and numerical results are presented in this work.

Solving complex coupled processes involving fluid-structure-thermal interactions is a challenging problem in computational sciences and engineering. Currently there exist numerous public-domain and commercial codes available in the area of Computational Fluid Dynamics (CFD), Computational Structural Dynamics (CSD) and Computational Thermodynamics (CTD). Different groups specializing in modelling individual process such as CSD, CFD, CTD often come together to solve a complex coupled application. Direct numerical simulation of the non-linear equations for even the most simplified fluid-structure-thermal interaction (FSTI) model depends on the convergence of iterative solvers which in turn rely heavily on the properties of the coupled system. The purpose of this paper is to introduce a flexible multilevel algorithm with finite elements that can be used to study a coupled FSTI. The method relies on decomposing the complex global domain, into several local sub-domains, solving smaller problems over these sub-domains and then gluing back the local solution in an efficient and accurate fashion to yield the global solution. Our numerical results suggest that the proposed solution methodology is robust and reliable.

This paper examines the experiences of two elementary teachers’ implementation of mathematical modeling in their classrooms and how the enactment by the teachers and the engagement by students exhibited their creativity, critical thinking, collaboration and communication skills. In particular, we explore the questions: (1) How can phases of mathematical modeling as a process serve as a venue for exhibiting students’ critical 21st century skills? (2) What were some effective pedagogical practices teachers used as they implemented mathematical modeling with elementary students and how did these promote students’ 21st century skills? We propose that mathematical modeling provides space for teachers and students to have a collective experience through the iterative process of making sense of and building knowledge of important mathematical ideas while engaging in the critical 21st century skills necessary in our complex modern world.

The enlargement and rupture of intracranial and abdominal aortic aneurysms constitutes a major medical problem. It has been suggested that enlargement and rupture are due to mechanical instabilities of the associated complex fluid-solid interaction in the lesions. In this paper, we examine a coupled fluid-structure mathematical model for a cylindrical geometry representing an idealized aneurysm using both analytical and numerical techniques. A stability analysis for this subclass of aneurysms is presented. It is shown that this subclass of aneurysms is dynamically stable both with and without a viscoelastic contribution to the arterial wall.

The problem of effcient modeling and computation of the nonlinear interaction of fluid with a solid undergoing nonlinear deformation has remained a challenging problem in computational science and engineering. Direct numerical simulation of the non-linear equations, governing even the most simplified fluid-structure interaction model depends on the convergence of iterative solvers which in turn relies heavily on the properties of the coupled system. The purpose of this paper is to introduce a distributed multilevel algorithm with finite elements that offers the flexibility and efficiency to study coupled problems involving fluid-structure interaction. Our numerical results suggest that the proposed computational methodology for solving coupled problems involving fluid-structure interaction is reliable and robust.