AIM/MCRN Summer School on COVID-19: Week 2

Last Thursday marked the end of the second week of the summer school on “Dynamics and Data in the COVID-19 Pandemic,” organized by the American Institute of Mathematics (AIM) and the Mathematics and Climate Research Network (MCRN). This was a short, four-day week because of the Fourth of July holiday. Below is a summary of the week’s activities.

Last week, the program focused on modeling the COVID-19 pandemic. In this context, modeling stands for “mathematical modeling” (we know from previous discussion that the word “model” can have different meanings, depending on the discipline)!

The susceptible-infectious-recovered (SIR) model is the fundamental model of epidemiology. The population of interest is subdivided into three groups of individuals: Susceptible, Infected, and Recovered. Each group is homogeneous—no spatial or other dependencies—and occupies a compartment. One can then think of an epidemic as flowing through the compartments, from S to I to R. Depending on the problem of interest and the desired level of granularity, researchers can generalize or modify the SIR model in many ways; well-known variants include SEIR (accounting for Exposed individuals), SIS (accounting for reinfection), and MSEIR (accounting for newborn individuals who inherit immunity from their Mothers). These compartmental models are all deterministic and mathematically described by ordinary differential equations. They are also known as conceptual models — caricatures of the real world, in the sense that they are simplified to be are amenable to mathematical analysis while retaining an epidemic’s characteristic features. One characteristic of an epidemic is that the number of infected individuals initially increases, then decreases after reaching a maximum.

On Monday, Linda Allen (Texas Tech University) showed participants how to formulate stochastic models and numerically simulate sample paths via Markov chain Monte Carlo techniques. Stochastic models work particularly well for small populations and provide quantitative information about uncertainties. Allen had prepared a MATLAB .m file for the SIR model (also available in Python). Students then convened in breakout groups on Monday afternoon and Tuesday morning to modify the program for other models, like SIS and SEIR.

Later that same day, everyone used Watch2Gether to view Laurent Hébert-Dufresne’s (University of Vermont) virtual presentation on network models. The talk had been assigned as homework, and the purpose of the afternoon session was to discuss details in a group setting. The video was informative, but participants agreed that more time was needed to digest the fine points.

Whether deterministic or stochastic, SIR and similar models assume population homogeneity. The question of how to account for heterogeneity then arises. Nancy Rodriguez (University of Colorado Boulder) prepared four “frameworks" that scientists popularly use to incorporate spatial interactions:

• Agent-based models / interacting-particle systems
• Metapopulation models / patch models 
• Reaction-advection-diffusion equations / partial differential equations
• Integro-differential equations

Each framework was assigned to two groups of students, who were tasked with analyzing a technical paper that applied the framework to a particular problem. They worked to identify the question(s) that the authors were addressing; discuss the merits, pros, and cons of the authors’ approach; and assess whether the framework was an appropriate means of solving the problem. All this was also part of Tuesday evening’s homework assignment; follow-up discussions took place on Wednesday. For some participants, this exercise was their first experience reading a scientific paper.

The homework assignment for Wednesday evening involved watching David Bortz’s (University of Colorado Boulder) talk on “Mathematical Modeling of COVID-19 in Colorado,” in preparation for group discussions and a Q&A session with David on Thursday morning. Bortz is a member of the Colorado Covid-19 Task Force, so he spoke from experience about how state officials (including the governor) use science to make informed decisions about social distancing and mask policies.

Thursday afternoon’s activities served as a follow-up to last week’s discussion of simulators. This time, students focused on determining what was “under the hood” of some collected simulators. Typical questions for consideration were as follows:

• What model underlies the simulations? 
• Where does the data come from? 
• How good is the documentation? 

Once the groups addressed these queries, they demonstrated the simulators to the school staff and shared their findings with one another.

In conclusion, I should also mention that everyone partook in daily 15-minute tai chi sessions (in more or less exotic settings).

Homework

As with any school, no day goes by without homework. Participants are expected to think about the mathematics of COVID-19 all the time. The assignment for the long weekend was to come up with relevant questions and watch a seminar talk on “Computational Epidemiology at the Time of COVID-19” by Alessandro Vespignani (Northeastern University). This presentation took place through the Isaac Newton Institute for Mathematical Sciences on May 18, 2020.

What did we learn last week?

During week two, participants learned about models as a tool for understanding the evolution of an epidemic: how it begins with one infected individual among a population of susceptible individuals, spreads through contact between infected and susceptible individuals, reaches a maximum, and ultimately descends until the probability of infection becomes too small and the epidemic is no longer sustained. The magic number is \(R_0\) — the cumulative number of infected individuals generated through the course of the epidemic by a single infected person.

I look forward to week three, when we will focus on data.