The Signal Beat

The signal beat is a blog covering basic topics in the mathematical sciences. The content is organized into series of posts that are short and hopefully suitable for morning coffee.

We begin with some primers on epidemic modeling. No prior background in math or epidemiology is assumed here.

Series: Epidemic models 1

In this series of primers we introduce compartmental models, which are basic to understanding the movement of epidemics through populations (among other things). We start with the simplest models, using them as a “toy playground” for building up concepts and intuitions. At the close, we’ll present the SEIR model, which is a conceptual starting point for understanding models that are actually applied to epidemics like COVID-19.

This material is a teaser for mathematical epidemiology. From this vantage point, the subject is not the medicine of diseases themselves but something more abstract: epidemics as ‘waves’ that propagate through a system of interwoven processes in the population. The processes are things like infection, recovery, and loss of immunity. But rather than being modeled in medical terms, each process is treated as an ensemble of individual events, each occurring with a probability that depends on the state of the epidemic.

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 typical starting point for modeling 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.

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.

Further reading

  • Maia Martcheva, An introduction to mathematical epidemiology, Springer texts in applied mathematics, Volume 61, 2015
  • John Baez, Network theory. See the series of blog articles on this page under the section called Chemical reaction networks, Petri nets and Markov processes.
  • Micah Halter and Evan Patterson, Compositional epidemiological modeling using structured cospans, Algebraic Julia blog, October 17, 2020. This is about some new mathematics-based software than can be used to simulate the dynamics of a pandemic.

Series: A tree leaf model

Part 1 – Introduction – leaf shape, an optimization in nature.

Part 2 – The leaf as a tree of pipes.

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