David A. Tanzer, August 24, 2020, in unit Epidemic Models 2.

Continuous and discrete flows in epidemic models

The standard compartmental models for SIR, SEIR etc. use the rate equations for the dynamics, and hence assume a continuous flow model.

This works well, particularly due to the fact that population sizes in epidemics are “large”, on the order of millions of individuals, and the law of large numbers kicks in, which tells us that we won’t be far off using a continuous flow approximation.

On the other hand, when population counts are small, e.g. for studying the transmission of disease within a hospital ward, the continuous model is only useful for determining the average rather than the actual behavior.

The actual behavior then becomes dominated by “stochastic” i.e. random, effects, which is the subject of a large field, called stochastic reaction networks. That is a subject for another series.

Outlook

We reached a milestone, by understanding equations for some compartmental models. You now have all the concepts needed in order to construct the rate equations, for example for the SEIR model which applies to COVID.

In these primer series on epidemic models, we’ve been speaking in fairly loose “hand-wavy” terms about things like randomness. A good next step would be to work thorough some of the basic language which is used to understand random variables, like the number of popped kernels after one minute of cooking, and random processes, like the cooking of a pot of kernels.

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