Despite widespread calls for the incorporation of mathematical modeling into the undergraduate biology curriculum, there is lack of a common understanding around the definition of modeling, which inhibits progress. In this paper, we extend the “Rule-of-Four,” initially used in calculus reform efforts, to a framework for models and modeling that is inclusive of varying disciplinary definitions of each. This unifying framework allows us to both build on strengths that each discipline and its students bring, but also identify gaps in modeling activities practiced by each discipline. We also discuss benefits to student learning and interdisciplinary collaboration.

The COVID-19 pandemic has made mathematical epidemiology a topic of critical importance, providing mathematics educators with an unparalleled opportunity. This opportunity is accompanied by a challenge: how do mathematics educators, some of whom have little personal experience with mathematical modeling, teach mathematical epidemiology to their students in courses ranging from precalculus to differential equations, and do so in a way that builds understanding of epidemic disease dynamics as well as mathematical methods? We address this issue with a collection of materials that allow students to conduct virtual experiments with a COVID-19 model to assess the effects of public health policies and community behavior. The materials are designed to require only a bare minimum of coding by students so as to focus students' efforts on interpretation of results.

The standard way of incorporating mass vaccination into a compartment model for an infectious disease is as a spontaneous transition process that applies to the entire susceptible class. The large degree of COVID-19 vaccine refusal, hesitancy, and ineligibility, and initial limitations of supply and distribution require reconsideration of this standard treatment. In this paper, we address these issues for models on endemic and epidemic time scales. On an endemic time scale, we partition the susceptible class into prevaccinated and unprotected subclasses and show that vaccine refusal/hesitancy/ineligibility has a significant impact on endemic behavior, particularly for diseases where immunity is short-lived. On an epidemic time scale, we develop a supply-limited Holling type 3 vaccination model and show that it is an excellent fit to vaccination data. We then extend the Holling model to a COVID-19 scenario in which the population is divided into two risk classes, with the high-risk class being prioritized for vaccination. In both cases, with and without risk stratification, we see significant differences in epidemiological outcomes between the Holling vaccination model and naive models. Finally, we use the new model to explore implications for public health policies in future pandemics.

Summary Treatments in mathematics books of how to select models and fit them to data have generally not been updated to account for improvements in knowledge and computing over the last 40 years. Here, we offer a derivation of the linear least squares formula that requires only precalculus and use it to develop a simple numerical method for fitting a broad class of nonlinear functions, with results that are far superior to methods in common practice. We also discuss the use of the Akaike information criterion (AIC) to choose among competing models that have been fit to a common data set.