Historical Context for Evolutionary Epidemic Modeling
Paper Referenced: http://www.sciencedirect.com/science/article/pii/0040580987900141
In the above paper from 1985, the author formulates a mathematical model to represent the spread of “evolutionary epidemics” which are distinguished from classical epidemics the same way SIRS and the SIR Epidemic Models are distinguished in Chapter 21. The author argues that evolutionary epidemics are able to re-infect previously infected patients (nodes) through variations in the genetic makeup of the disease. This is why you need to get a flu shot periodically, not just once. Later in the paper, the author cites the period for influenza infection oscillations as being around one year to a year and a half (which we know is what gives us the “flu season” every winter. The author pays particular attention to a factor he calls epsilon, which communicates the ratio of the infection period to the length of time before re-infection. Varying this ratio lets us see an effect that looks like a damped oscillator until the percent of the population infected reaches an equilibrium.
The paper cited and the content of this course are different approaches to viewing the same phenomenon, and some parallels shine through, like the classification of re-infectious diseases vs SIR modeled diseases. Both models account for the oscillatory infection cycles possible with SIRS diseases. While the paper chose to illustrate a “well-behaved” model where the infection rate stabilized at a certain value, the model did allow for chaotic behavior like we see for various values of c in Figure 21.7 of the course text. So both the paper’s model and our course’s model did allow for some form of well-behaved oscillatory behavior, but the similar behavior was achieved by altering differently defined but related parameters. When I say the parameters are related I mean to say that the fraction of long-range links (“c” in the course text) and the ratio of infectious period to time to reinfection (“epsilon” in the paper) are related in the sense that the more long-range links a person has the more an epidemic will behave as a wave, whereas a well-defined epsilon (as opposed to an irrational ratio) will lead to a more periodic level of infections.
While the fancy mathematical treatment of the spread of epidemics is interesting, I found the class model built on the concepts of networks, gives a much simpler description of the spread of epidemics while keeping information contained in the more complex mathematical model. I think this example goes to show the simplifying power of networked models.