David A. Tanzer, July 18, 2020, in unit Epidemic Models 1.

General idea of reactions

Last time we surveyed some compartmental models in epidemiology. Now it’s time to look deeper, as the compartments are only part of a model.

The models also contain reactions, which connect the compartments and cause them to change over time; they are processes which ‘stir’ and transform the contents of the compartments. A reaction proceeds in small steps. In each step some individuals are drawn from containers, they interact and morph into new forms, and then get redeposited to other containers.

The reactions are actual processes that take place in populations. For example, take the SIR model, with compartments Susceptible, Infected and Recovered. Here there are two reaction processes, infection and recovery, each of which stirs the compartments in a specific way. A step of the infection reaction is an event in which one susceptible person gets infected; a step of the recovery reaction is an event in which one infected person recovers.

The whole process can involve a many reaction steps, with multiple reactions occurring at the same time. This is the norm with epidemic models, as multiple infection and recovery events are spread out in space and take place at the same time.

The reaction processes get interwoven into a dynamic whole system; they are intertwined to make a ‘stew’. That’s because events in one reaction can affect the events in another. For example, as more individuals recover and become immune, there become fewer chances for infections to occur, and this causes the infection process slows down. The reactions form a network.

Note there is a deep analogy between epidemic reaction networks and chemical reaction networks, which is suggested by the metaphor of the stew. At this level of generality, the only difference is that one involves transformations of individuals and medical conditions, while the other involves transformations of molecules. This is, however, just an analogy, and we won’t be needing to fire up the bunsen burners in order to go forward with our understanding of epidemic models.

Next time we will look into the workings of the SIR reaction network.

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