We are approaching the end of week 2 — a short 4-day week because of the 4^{th}of July holiday. Here is a summary of the past few days.

This week, the focus was on *modeling the COVID-19 pandemic*. Here, modeling stands for “mathematical modeling.” (We know from last week’s discussion that the word “model” can have different meanings, depending on the discipline!)

Compartmental Models

The SIR model is the fundamental model of epidemiology. The population of interest is subdivided in three groups of individuals: *S*usceptibles, *I*nfected, and *R*ecovered; each group is assumed to be homogeneous (no spatial or other dependencies) and occupy a compartment. An epidemic is then thought of as a flow through the compartments, from S to I to R. Depending on the problem of interest and the desired level of granularity, the SIR model can be generalized or modified in many ways; well-known variants are the SEIR (including *E*xposed individuals), SIS (accounting for reinfection), and MSEIR (including newborn individuals who inherit temporary immunity from the *M*other). These *compartmental models *are all deterministic; mathematically, they are described by systems of ordinary differential equations (ODEs). They are also known as *conceptual models *— caricatures of the system of interest, in the sense that the mathematical model has been simplified to the point that it is amenable to mathematical analysis but its solution still displays the characteristic features of an epidemic. One characteristic of an epidemic is that the number of infected individuals increases initially, then decreases after reaching a maximum.

On Monday, Linda Allen (Texas Tech) showed us how to formulate stochastic models and how to numerically simulate sample paths using Markov Chain Monte Carlo (MCMC) techniques. Stochastic models work particularly well for small populations; they also provide quantitative information about uncertainties. LInda had prepared a Matlab .m file for the SIR model (also available in Python). The students then followed up in breakout groups on Monday afternoon and Tuesday morning to modify the program for other models like SIS and SEIR.

Network Models

Monday afternoon, we watched (through Watch2Gether) the video presentationby Laurent Hébert-Dufresne (U Vermont) on *network models*. The video had been assigned as homework; the purpose of this afternoon’s session was to discuss details. The video was informative, but we agreed that more time was needed to digest the fine points.

Accounting for Heterogeneity

SIR and similar models, whether deterministic or stochastic, assume homogeneity of the population. The question then arises how to account for heterogeneity. Nancy Rodriguez had prepared four “frameworks” that are popularly used 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. The task was to analyze a technical paper that applied the framework to a particular problem; identify the question(s) that the authors were addressing; discuss the merits, pros and cons of the approach; and assess whether the framework was appropriate to answer the questions. All this was also part of the homework assignment for Tuesday evening, with follow-up discussions on Wednesday.

Science-based Decision Making

Watching a video of a talk on “Mathematical Modeling of COVID-19 in Colorado” by David Bortz (CU Boulder) was the homework assignment for Wednesday evening, in preparation for group discussions and a Q&A session with David on Thursday morning. David is a member of the *Colorado Covid-19 Task Force, *so he could speak from experience how state officials (including the Governor!) used science to make informed decisions about social distancing and wearing masks.

Simulators

Thursday afternoon’s activities were a follow-up on last week’s discussion of simulators. This time, the focus was on figuring out what was “under the hood” of some of the 20 simulators collected by Erik and Punit. Typical questions to consider were: What is the model underlying the simulations? Where are the data coming from? How good is the documentation? Is there enough information to reproduce the results? Once the groups had figured all this out, they had an opportunity to demonstrate the simulators to the camp staff and share their findings with other groups.

In conclusion, I should also mention that Maria led us expertly throughout the week in a daily 15-minute Tai-Chi session, in more or less exotic settings.

Homework

No day goes by without homework; we are expected to think about the mathematics of Covid-19, 24/7. The assignment for the long holiday weekend is to think about questions to ask and, in preparation for next week, watch a seminar talk on “Computational Epidemiology,” by Alessandro Vespignani at the Newton Institute on May 18, 2020.

What Did We Learn This week?

This week, we learned about models as a tool for understanding the evolution of an epidemic, how it starts with one infected individual among a population of susceptible individuals, spreads through contacts of infected and susceptible individuals, reaches a maximum and then descends until the probability of infection becomes too small and the epidemic can no longer sustain itself. The magic number is $R_0$, the *basic reproduction number*–the number of secondary infections caused by a single infected individual introduced into a wholly susceptible population over the course of the infection of this single infected individual. If $R_0<1$, the infection dies out; if $R_0>1$, there is an epidemic.

Looking forward to Week 3, when we focus on data.