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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.

This chapter explores how the elements of responsive and responsible learning were adopted in the first STEM education classroom through collaborative teaching with external experts in a public university in Malaysia. The objectives for the introduction of the course were to enable students to apply the knowledge of mathematical and pedagogical aspects in the teaching of science and mathematics to solve problems in a scientific and systematic manner and to demonstrate the ability to seek new knowledge independently. Three class projects involving collaborators fulfilled 8 weeks of lectures. These included Training of Trainers (ToT), Green Energy Project – Solar Panel and the University Centre for Innovative Delivery and Learning Development (CIDL) focused on Internet of Things (IoT) content. Collaborative teaching in the STEM education classroom has successfully served the specific technical requirements of industry using the design-thinking framework into the university classroom setting.

We develop a lubrication theory-based mathematical model that describes the dynamics of a tear film during blinking and contact lens (CL) wear. The model extends previous work on pre-corneal tear film dynamics during blinking by coupling the partial differential equation for tear film thickness to a dynamic model for CL motion. We explore different models for eyelid motion and also account for possible voluntary and involuntary globe (eyeball) rotation that may accompany blinking. Boundary conditions for mass flux at the eyelids are also adapted to account for the presence and motion of the CL. Our predictions for CL motion compare reasonably with existing data. Away from the eyelids the pre-lens tear film (PrLTF) is shifted, relative to its pre-corneal counterpart, in the direction of CL motion. Near the eyelids, the inflow/outflow of fluid under the eyelids also influences the PrLTF profile. We also compare our PrLTF dynamics to existing in vivo tear film thickness measurements.

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.

In this work, semi-discrete and fully-discrete error estimates are derived for the Biot's consolidation model described using a three-field finite element formulation. The fields include displacements, total stress and pressure. The model is implemented using a backward Euler discretization in time for the fully-discrete scheme and validated for benchmark examples. Computational experiments presented verifies the convergence orders for the lowest order finite elements with discontinuous and continuous finite element appropriation for the total stress.

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.

A new weak Galerkin finite element method, called generalized weak Galerkin method ({g}WG), is introduced for Stokes equations in this paper by using a new definition of the weak gradient. Error estimates in energy norm and $L^2$ norm for the velocity and $L^2$ norm for the pressure are derived for elements with arbitrary combination of polynomials. Some numerical examples are presented to verify the effectiveness, theoretical convergence orders, and robustness of the proposed scheme.

The COVID-19 pandemic has placed epidemiologists, modelers, and policy makers at the forefront of the global discussion of how to control the spread of coronavirus. The main challenges confronting modelling approaches include real-time projections of changes in the numbers of cases, hospitalizations, and fatalities, the consequences of public health policy, the understanding of how best to implement varied non-pharmaceutical interventions and potential vaccination strategies, now that vaccines are available for distribution. Here, we: (i) review carefully selected literature on COVID-19 modeling to identify challenges associated with developing appropriate models along with collecting the fine-tuned data, (ii) use the identified challenges to suggest prospective modeling frameworks through which adaptive interventions such as vaccine strategies and the uses of diagnostic tests can be evaluated, and (iii) provide a novel Multiresolution Modeling Framework which constructs a multi-objective optimization problem by considering relevant stakeholders' participatory perspective to carry out epidemic nowcasting and future prediction. Consolidating our understanding of model approaches to COVID-19 will assist policy makers in designing interventions that are not only maximally effective but also economically beneficial.

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.

Personalized recovery technologies may enable individuals with Substance Use Disorder (SUD) to monitor and manage acute craving and drug use urges in ways that improve drug-seeking decisions in real-time. Direct and indirect regulation of the autonomic nervous system through sensory input monitoring and modulation may enhance control over behavioral decisions and prevent relapse. A personalized sensory support system that monitors neurophysiological reactivity and offers non-pharmacological point-in-time personalized digital interventions may increase awareness of and control over craving reactivity. It is critical to be able to detect these warning signs and intervene early and effectively. The use of wearable technologies that assess point-in-time neurophysiological escalation and shape behavioral response through personalized interventions could be transformative in allowing individuals to better manage their recovery as they transition out of institutions and move back into community settings.

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.

This chapter introduces some novel educational frameworks that provide the opportunity for mathematics educators to not only engage students through mathematical tools to represent, understand and solve real-world problems but also engage them in using tools to make a decision, prediction or solution about a real-world problem. This helps prepare students to become life-long learners who can then go on to pursue state-of-the-art jobs. To create this next-generation cohort of students, however, we must upskill our current educators with pedagogical practices that go beyond just delivering content. In particular, mathematics education must include a variety of learning approaches including experiential learning, inquiry-based learning, challenge-based learning and interdisciplinary problem-based learning. Here the author provides some learning frameworks and benchmark-example mathematical tasks to compare the effectiveness of these approaches. He considers authentic mathematical tasks that incorporate a shared collaborative experience to advance mathematics teaching and learning in the 21st century.

Over the last few decades, there have been significant developments in theoretical, experimental, and clinical approaches to understand the dynamics of cancer cells and their interactions with the immune system. These have led to the development of important methods for cancer therapy including virotherapy, immunotherapy, chemotherapy, targeted drug therapy, and many others. Along with this, there have also been some developments on analytical and computational models to help provide insights into clinical observations. This work develops a new mathematical model that combines important interactions between tumor cells and cells in the immune systems including natural killer cells, dendritic cells, and cytotoxic CD8+ T cells combined with drug delivery to these cell sites. These interactions are described via a system of ordinary differential equations that are solved numerically. A stability analysis of this model is also performed to determine conditions for tumor-free equilibrium to be stable. We also study the influence of proliferation rates and drug interventions in the dynamics of all the cells involved. Another contribution is the development of a novel parameter estimation methodology to determine optimal parameters in the model that can reproduce a given dataset. Our results seem to suggest that the model employed is a robust candidate for studying the dynamics of tumor cells and it helps to provide the dynamic interactions between the tumor cells, immune system, and drug-response systems.

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.

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

Working with a mathematics coach and university researchers in a K-4 lesson study, teachers increase their understanding of student abilities in a fair-share sandwich problem.

Problem formulation is one of the critical first steps in mathematical modeling (MM). This important step requires modelers to identify a problem that requires modeling. Our chapter examines how elementary teachers and students learn to be problem posers as they learn to enact mathematical modeling in their own classrooms. We collected data from teachers implementing MM in their classroom starting from their planning process to the enactment phases of mathematical modeling tasks. Data was collected from lesson studies and teacher interviews exploring the processes of mathematical modeling. We focused on better understanding how teachers involve elementary students in the problem formulation process as they co-construct the problem and how they work through making assumptions, building a solution or a model, and revising their model as they relate back to their problems. We also analyzed the context in which the problem formulation occurred: community-based, school-based, or curriculum-based contexts and how it impacted authentic learning.

In this chapter, we present a novel way to engage students in applied mathematics education and research by combining context provided by the United Nations Sustainable Development Goals that helps to drive content demonstrated via a new mathematical modeling cycle for global problem solving. To illustrate the approach, we select a specific goal in this chapter, namely, health and well-being (SDG-Goal 3) and apply the eight different phases of the mathematical modeling cycle that consists of Observe, Theorize, Formulate, Describe, Analyse, Simulate, Validate and Predict to solve the global problem of controlling the spread of the Zika virus. Along with these phases, we also present the United Nations Sustainable Development Goals as a context to motivate the mathematical modeling cycle as a powerful mechanism to both engage students in a global problem-solving process and also excite them to pursue applied mathematics education and research.

We live in unprecedented times, faced with a pandemic of monumental proportions. Not only has COVID‐19 wreaked havoc across the world, it also has exposed fundamental weaknesses in healthcare systems in a number of countries. We identify challenges and opportunities that COVID‐19 presents by connecting the immediate need to curb the spread of the disease to the United Nations sustainable development goals (SDGs). Considering broad social, political, and economic impacts, we focus on health disparities and the severe effects of the crisis on some populations, especially disadvantaged and underserved ones, and countries relative to others. In particular, we identify key targets for achieving SDG‐3 (Good Health and Wellbeing) that will help strengthen capacities for scaling up healthcare and policy responses to understand, combat, and control COVID‐19.

According to the Centers for Disease Control, opioid overdose rates are rising, and data suggests that both prescription and illicit drugs are significant with respect to the opioid crisis. This is an international problem that affects social and economic welfare as well as public health. The typical framework for mathematical models involves the classic compartmental model involving ordinary differential equations (ODE) that describe phases of infectious diseases. Mathematical modeling for drug addiction as an infectious disease model can be a novel and unique method of approaching this issue, in particular with the addition of technological rehabilitation as a viable method of drug rehabilitation. This project explores two mathematical models, including one describing the dynamics of addiction through over-prescription, and the second describing the dynamics of addiction influenced by social behavior with ODE. In addition, parameters are estimated with AMPL optimization, and machine learning; a rural vs. urban prescription model is studied; a basic reproduction number for social model is derived; efforts to support healthcare providers and education as a control for the opioid crisis are also explored; a numerical simulation of each model is implemented with high-order numerical approximation; and a graphical user interface is developed through MATLAB. We hope to validate the mathematical models with real data and incorporate intervention with rehabilitation technology to further enhance the models to be more predictive., Journal of Student-Scientists' Research, Vol 1 (2019)

In this work, we describe some effective teaching and research practices that can help to integrate mathematics and biology efficiently to enhance student learning at all levels. One of the successful approaches proposed is to employ mathematical modeling that can help transform pedagogical practices. In this regard, we introduce some modeling activities that have been shared with teachers through professional development programs and have been incorporated in the classrooms. We also present how engaging teachers in research experiences in mathematical modeling can help to transform their pedagogical practices and provide opportunities for students to consider pursuing areas at the interface of mathematics and biology.

In this paper, a number of alcohol related studies have been reviewed in light of the strength and challenges of the models developed for their applicability in Tanzanian drinking context. The aim of this paper is to provide a meaningful reference of the current state of the art regarding to modeling the health risks epidemics associated with alcoholic behavior in the community. It also discusses and identify current open problem reflecting on more realistic community and suggesting new research perspectives in the mathematical modeling of alcoholism and its related risks. The peer influence and social cultural practices appear to be two equally important influential aspects of model formulation with opposing effects on the spread of alcoholism. The study also reveals that, different levels of alcohol consumption has differing health effects. In this regard, the study proposes development of an advanced infection models to represent the influence of external motivations on the spreading of alcohol abuse by introducing different population segments with distinct exposition towards and resistance to the influence of drugs as an open relevant problem.

Alcoholism has continually posed health challenges in many communities for decades. In this paper, a more realistic model for health related risks associated with alcoholism is formulated. It considers a population proportion that has social cultural protection from alcohol consumption. In the context of this paper, such protection emanates from religious beliefs. The Next Generation Matrix (NGM) approach is used to compute the basic risk reproduction number. The risk free equilibrium point is proved to be globally asymptotically stable whenever the basic risk reproduction is less than unity and unstable otherwise. The sensitivity analysis of the basic risk reproduction number and numerical simulation results reveal that for effective control of the health risk problem in the community, the deliberate intervention strategies and policies should focus on discouraging alcoholic behaviors on its onset during initiation stage than focusing other population proportions already at risk.