It looks like you're new here. If you want to get involved, click one of these buttons!

- All Categories 2.4K
- Chat 502
- Study Groups 21
- Petri Nets 9
- Epidemiology 4
- Leaf Modeling 2
- Review Sections 9
- MIT 2020: Programming with Categories 51
- MIT 2020: Lectures 20
- MIT 2020: Exercises 25
- Baez ACT 2019: Online Course 339
- Baez ACT 2019: Lectures 79
- Baez ACT 2019: Exercises 149
- Baez ACT 2019: Chat 50
- UCR ACT Seminar 4
- General 73
- Azimuth Code Project 110
- Statistical methods 4
- Drafts 10
- Math Syntax Demos 15
- Wiki - Latest Changes 3
- Strategy 113
- Azimuth Project 1.1K
- - Spam 1
- News and Information 148
- Azimuth Blog 149
- - Conventions and Policies 21
- - Questions 43
- Azimuth Wiki 718

Options

This blog will cover various topics in mathematical science. We begin with some primers on epidemic modeling. No prior background in math or epidemiology is assumed here. These articles are kept short; suitable for morning coffee.

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 *Susceptible* - *Exposed* - *Infected* - *Recovered*.

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.

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.

- 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.

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

hello world×