The Signal Beat

By David A. Tanzer

This blog will cover a variety of topics in mathematical science. We start with two series of primers on epidemic models. No prior background in mathematics or epidemiology is assumed. Articles are kept short; suitable for morning coffee.

Series: Epidemic models 1

In this primer series on epidemic models, we introduce compartmental models, which are basic to understanding the movement of epidemics through populations. We start with the most rudimentary models, using them as a “toy playground” for building up concepts and intuitions. At the close, we’ll present the SEIR model, which is widely applied to epidemics such as COVID-19.

This material is a teaser for the subject of mathematical epidemiology. It’s not about the medicine of diseases themselves but rather about epidemics as ‘dynamic waves’ which propagate through a system of interwoven population processes, such as infection, recovery and loss of immunity.

Part 1 – A first look at compartmental models. Compartments refer to subpopulations, such as susceptible, infected and recovered individuals.

Part 2 – A menu of compartmental models. Models are named for the compartments they contain. The most common one used for COVID is SEIR, which stands for SusceptibleExposedInfectedRecovered.

Part 3 – General idea of reactions. Here, ‘reactions’ are processes, like infection and recovery, which change the health status of individuals and shuttle them between compartments.

Part 4 – The SIR model. This is a fundamental model, with compartments Susceptible, Infected, and Recovered, and two reactions, infection and recovery. In this post, we show the ‘wiring diagram’ for the model and look into the structure of the recovery reaction.

Part 5 – The SIR model (cont’d). Here we complete the introduction to SIR by looking into the structure of infection.

Part 6 – A diversity of compartmental models. The ideas we learned from the SIR model can easily be extended to a diversity of other useful models.

Part 7 – The SEIR model. This one applies to a broad class of epidemics, including COVID-19.

Series: Epidemic models 2

Here we approach the dynamics of epidemic reaction networks. By the end of the series, we will see how the motion of an epidemic can be approximated with equations derived from the ‘wiring diagram’ of the model. Much of the discussion is qualitative, with just a tad of calculus involved in the rate equations.

Part 1 – The rates matter.

Part 2 – What is meant by the rate of a reaction?

Part 3 – Analysis of reaction rates: first approach.

Part 4 – Digression: solving the popcorn rate equation.

Part 5 – Analysis of reaction rates: second approach.

Part 6 – Digression: reaction rates in chemistry.

Part 7 – The dynamics of SIR.

Part 8 – Continuous vs. discrete flow.

Part 9 – Continuous and discrete flows in epidemic models.

Reference:

  • Maia Martcheva, An Introduction to Mathematical Epidemiology, Springer Texts in Applied Mathematics, Vol. 61, 2015.

Series: The language of probability spaces and random variables (TODO)

Series: The language of random processes (TODO)

Copyright © 2020, David A. Tanzer. All Rights Reserved.