By David A. Tanzer, July 8, 2020.
This is the first of a series of blog-style tutorials on epidemic modeling. It is hoped that these will be accessible to the general population of intelligent human beings.
The grim curves that we see in the papers show things like the number of daily infections, and the number of daily deaths. Epidemic models allow these curves to be predicted. The models depend upon parameters, such as virus transmissibility and infectious period; they also depend on parameters having to do with mitigation efforts, such as the degree of social distancing. With a computer simulation of the model, one can vary the parameters that we have control over, such as social distancing, and then run the model, to get the curves for expected mortalities and infections. So it can be used as a tool for pubic health policy planning.
In this article, we introduce compartmental models, which are a fundamental, ubiquitous model for the epidemic spread of infectious disease.
In this type of model, the population is divided into compartments, with one compartment for each relevant health status.
In the simplest model, called SI, everyone in the population is considered to be susceptible or infected. The compartment Susceptible contains all susceptible individuals, and Infected contains all infected individuals.
Let’s put this model to work immediately, with a simplified, ‘toy’ model. Note: toy models are essential, as they can show the core principles of a theory, in a way that is uncluttered by detail.
Suppose the following model:
The population had 8 individuals.
On the first day of the epidemic, there is one infected person, and seven susceptible people.
On each day, each infected person “finds” a susceptible person and infects them.
Once infected, a person remains infected.
Based on the above information, trace out how the epidemic will evolve:
What will eventually happen to the population?
Starting from day 1, how many infected and susceptible people will there be on each successive day?
In other words, what are the specific dynamics of this epidemic?
Since individuals never become uninfected, eventually the whole population becomes infected. Initially, the size of compartment Infected is 1, and the size of Susceptible is 7, Eventually, the size of Infected becomes 8, and the size of Susceptible becomes 0.
On day 1, Infected = 1, Susceptible = 7.
On day 2, Infected = 2, Susceptible = 6. The one infected person from day 1 infected a susceptible person. This newly infected person got moved out of the compartment Susceptible, and into Infected. So the size of Infected went up by 1, and the size of Susceptible went down by 1.
On day 3, Infected = 4, Susceptible = 4. The 2 infected people from the previous day infected 2 new people.
On day 4, Infected = 8, Susceptible = 0. The 4 infected people infected the remaining 4 susceptible people.
On day 5, Infected = 8, Susceptible = 0. No more change, as the epidemic has completely spread.
On day 6, Infected = 8, Susceptible = 0.
…
Here are the curves for this epidemic:
[Todo: show as a graph]
In the next article, will look at more realistic models, which contain more compartments.
I, David Tanzer:
Like ACT and strive to make things happen here at the Functorial.
Manage the Azimuth forum and wiki.
Am employed in NYC in a combined role of quant programmer, tech. lead and project manager.
Have a PhD in CS from Courant.
Have a B.A. in math from U. Penn.
Practice jazz guitar and singing.
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