Abstract:

Calculus is a foundation subject in the undergraduate curriculum for most engineering students, who typically struggle with the subsequent abstract thinking and mathematical pure computations. We propose a complete redesign of the subject by introducing the basic notions as tools to attack a topic that has surely impacted their lives: the recent pandemic COVID-19.

As a motivational driving force, we will profit that students have been hearing in the news expressions like peak of the curve, fattening the curve or R0 index, which come from mathematical models applied to epidemiology. The idea is to explain the meaning of these terminology by using the easiest of these models (the basic SIR model created by Kermack & McKendrick in 1927) as the storyline of a Calculus course.

The main ideas that we will develop using this model as an excuse are as follows:
• The model divides the population in 3 basic compartments: susceptible, infectious, and recovered. With the goal to study the flow of people between the different categories, we analyse the daily change in the number of infected individuals. By making the time interval closer and closer to zero, we end up with the notion of derivative as instantaneous rate of change.
• This leads to a system of differential equations, and the students will learn how to use specific tools (Mathematica, Python…) to represent the evolution of the disease. Accordingly, they will construct and visualize “the curve”, which was so present in the media. We will seize the opportunity to teach numerical methods to solve differential equations and compare the predictions of the model to real data.
• Once depicted the curves, we can identify the so-called peak as the absolute maximum and argue why it was so important to have the hope of reaching it.
• The next step is to ask the students what should happen so that the disease spreads out. They should then discover the relation between the sign of the derivative and the tendency (increasing or decreasing) of the function. Then related concepts like inflection point, concavity… will spontaneously show up.
• The above analysis will naturally lead to the famous R0 index, which was used to decide whether some regions keep or not confined and measures the number of contagions produced by the patient zero. Later the example of how a disease disseminates with R0=2 will serve to introduce the notion of exponential growth (as opposite to linear behaviour) and, as by-product, the exponential function.
• To answer questions related to the cumulative number of infected people for a certain time interval, the definite integral will finally pop up.

In short, we will present how to design a lecture series to exploit COVID-19 as an excuse to create a meaningful learning of Calculus. In addition, we will encourage critical thinking of the students by debating what is the exact application of these models, and why, despite not providing exact predictions, they are still valuable to generate different scenarios and as help to take decisions based on data. Finally, we will explore actual applications of these models to study the spread of information in social networks (including viral marketing), or to detect fake news.

Keywords:

Calculus, SIR model, meaningful learning.