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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 6 - Digression: reaction rates in chemistry.
Part 7 - The dynamics of SIR.
Part 8 - Continuous vs. discrete flow.
Copyright © 2020, David A. Tanzer. All Rights Reserved.