**Instructor:** James Nolen

This special-topics course will focus on mathematical modeling and analysis related to the COVID-19 Coronavirus pandemic, which has spread rapidly around the world. The mathematical content will include selected topics from ordinary differential equations (nonlinear systems) and from probability (Markov chains), as well as data analysis tools. We will study current mathematical models that are being used by epidemiologists and mathematicians to understand the spread of the virus and the potential effect of mitigation strategies. If time permits, we will also discuss mathematical modeling and analysis of other infectious diseases (e.g. influenza, HIV) and non-infectious diseases (e.g. cancer growth). The course is ideal for students interested in mathematics, public health and/or medicine, or mathematical modeling. The course will be a unique opportunity to learn some of the very many ways that mathematics and mathematical modeling can be used to understand contemporary scientific problems and the complex world around us. Students will apply these tools to understand real data that has been generated during this epidemic, and they will be able to understand and analyze current research in this area.

**Prerequisites:** Students should have taken a Linear Algebra course (either math 216 or 221). A previous course in Probability (either math 230 or 340) is recommended, but students concurrently enrolled in 230 or 340 will be welcome. Talk to the instructor if you are concerned about the background. A previous course on differential equations is NOT required; that content will be introduced during the course. There will be some computational exercises, to be done with either MATLAB or Python. Previous experience with either MATLAB or Python will be benificial, but not strictly required if students are willing to learn along the way (e.g. sample codes will be provided).

**Assignments and Evaluation:** There will be regular readings, problem sets, group projects and exercises. There will be a final project and presentation, but no final exam.

**Days/Time:** Mondays/Wednesdays 1:45-3:00. This will be a ``hybrid" course, including options for either in-person participation or online-only participation. There are two sections, which distinguish who will participate in-person from those who will participate only online. During some weeks, students will meet once per week on either Monday or Wednesday, depending on their section: the online-only section will meet on Monday, while the in-person group will meet on Wednesday. On other weeks, all students (i.e. both sections) will meet online as one cohort twice during the week (i.e. on both Monday and Wednesday) and there will be online interaction between the two groups. For this reason, all students need to keep both the Monday and Wednesday periods free in their schedule.

**Course content:** The course material and activities will be organized around several scientific papers on various aspects of the COVID-19 pandemic, and other diseases. For example, related to COVID-19, specific topics we will study through mathematical modeling include understanding the transmissibility of the virus, the efficacy of lockdown and social-distancing measures, the efficacy of contact-tracing, estimating the herd-immunity threshold, short-term forecasting, estimating epidemiological parameters, etc. We also will discuss some models for within-host dynamics of certain diseases (e.g. HIV, cancer growth). Along the way, we will cover the necessary the mathematical background material needed to formulate and understand these models. So, the readings will include both mathematical sources and papers from the primary scientific literature, even recently-completed research articles. Specific mathematical topics to be covered include:

- Ordinary differential equations
- basic theory for linear and nonlinear systems of differential equations
- qualitative behavior of solutions
- stability/instability of equilibria
- dependence on parameters, bifurcations
- compartmental models (e.g. SIR and many variants) and their analysis

- Stochastic growth models
- branching processes
- stochastic network models

- Parameter estimation and data assimilation
- maximum likelihood estimation
- uncertainty quantification, Bayesian posterior estimation

More details about the course can be found on the course Sakai page, which is currently open to all Duke Sakai users. (Look in the Resources folder):

Do you have **questions** about the class? Contact Prof. Nolen