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A simple numerical solution procedure – namely the method of lines combined with an off-the-shelf ordinary differential equation (ODE) solver – was shown in previous work to provide efficient, mass-conservative solutions to the pressure-head form of Richards' equation. We implement such a solution in our model openRE. We developed a novel method to quantify the boundary fluxes that reduce water balance errors without negative impacts on model runtimes – the solver flux output method (SFOM). We compare this solution with alternatives, including the classic modified Picard iteration method and the Hydrus 1D model. We reproduce a set of benchmark solutions with all models. We find that Celia's solution has the best water balance, but it can incur significant truncation errors in the simulated boundary fluxes, depending on the time steps used. Our solution has comparable runtimes to Hydrus and better water balance performance (though both models have excellent water balance closure for all the problems we considered). Our solution can be implemented in an interpreted language, such as MATLAB or Python, making use of off-the-shelf ODE solvers. We evaluated alternative SciPy ODE solvers that are available in Python and make practical recommendations about the best way to implement them for Richards' equation. There are two advantages of our approach: (i) the code is concise, making it ideal for teaching purposes; and (ii) the method can be easily extended to represent alternative properties (e.g., novel ways to parameterize the K(ψ) relationship) and processes (e.g., it is straightforward to couple heat or solute transport), making it ideal for testing alternative hypotheses.

The susceptible-infected-recovered (SIR) model is perhaps the most basic epidemiological model for the evolution of disease spread within a population. Because of its direct representation of fundamental physical quantities, a true solution to an SIR model possesses a number of qualitative properties, such as conservation of the total population or positivity or monotonicity of its constituent populations, that may only be guaranteed to hold numerically under step-size restrictions on the solver. Operator-splitting methods with order greater than two require backward sub-steps in each operator, and the effects of these backward sub-steps on the step-size restrictions for guarantees of qualitative correctness of numerical solutions are not well studied. In this study, we analyze the impact of backward steps on step-size restrictions for guaranteed qualitative properties by applying third- and fourth-order operator-splitting methods to the SIR epidemic model. We find that it is possible to provide step-size restrictions that guarantee qualitative property preservation of the numerical solution despite the negative sub-steps, but care must be taken in the choice of the method. Results such as this open the door for the design and application of high-order operator-splitting methods to other mathematical models in general for which qualitative property preservation is important.

Background: Mental and physical health are both important for overall health. Mental health includes emotional, psychological, and social well-being; however, it is often difficult to monitor remotely. The objective of this scoping review is to investigate studies that focus on mental health and stress detection and monitoring using PPG-based wearable sensors. Methods: A literature review for this scoping review was conducted using the PRISMA (Preferred Reporting Items for the Systematic Reviews and Meta-analyses) framework. A total of 290 studies were found in five medical databases (PubMed, Medline, Embase, CINAHL, and Web of Science). Studies were deemed eligible if non-invasive PPG-based wearables were worn on the wrist or ear to measure vital signs of the heart (heart rate, pulse transit time, pulse waves, blood pressure, and blood volume pressure) and analyzed the data qualitatively. Results: Twenty-three studies met the inclusion criteria, with four real-life studies, eighteen clinical studies, and one joint clinical and real-life study. Out of the twenty-three studies, seventeen were published as journal-based articles, and six were conference papers with full texts. Because most of the articles were concerned with physiological and psychological stress, we decided to only include those that focused on stress. In twelve of the twenty articles, a PPG-based sensor alone was used to monitor stress, while in the remaining eight papers, a PPG sensor was used in combination with other sensors. Conclusion: The growing demand for wearable devices for mental health monitoring is evident. However, there is still a significant amount of research required before wearable devices can be used easily and effectively for such monitoring. Although the results of this review indicate that mental health monitoring and stress detection using PPG is possible, there are still many limitations within the current literature, such as a lack of large and diverse studies and ground-truth methods, that need to be addressed before wearable devices can be globally useful to patients.

We show that a simple numerical solution procedure – namely the method of lines combined with an off-the-shelf ODE solver – can provide efficient, mass conservative solutions to the pressure-head form of Richards’ equation. We present a novel method to quantify the boundary fluxes that reduces water balance errors without negative impacts on model runtimes. We compare this solution with alternatives, including the classic Modified Picard Iteration method of Celia et al. (1990) and the Hydrus 1D model (Šimůnek et al., 2005, 2016). We reproduce a set of benchmark solutions with all models. We find that Celia’s solution has the best water balance, but it can incur significant truncation errors in the simulated boundary fluxes, depending on the time steps used. Our solution has comparable run-times to Hydrus and better water balance performance (though both models have excellent water balance closure for all the problems we considered). Our solution can be implemented in an interpreted language, such as MATLAB or Python, making use of off-the-shelf ODE solvers. We investigated alternative scipy ODE solvers in Python and make practical recommendations about the best way to implement them for Richards’ equation. There are two advantages of our approach: i) the code is short and simple, making it ideal for teaching purposes; and ii) the method can be easily extended to represent alternative properties (e.g., novel ways to parameterize the K(ψ) relationship) and processes (e.g., it is straightforward to couple heat or solute transport), making it ideal for testing alternative hypotheses.