David A. Tanzer, July 10, 2020
A first look at compartmental models
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 aim to predict these curves. The models depend both on natural parameters such as infectiousness and the duration of the infectious period, as well as on social parameters like the degree of distancing. With a computer simulation of the model, we can vary the social parameters and run the model to get curves for expected mortalities and infections. In this way, the models are practical tools for public health policy planning.
Here we introduce the idea of compartmental models, which are a fundamental and ubiquitous model for the epidemic spread of infectious disease.
Compartmental models – starting with the simplest, called SI
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 can be very useful, as they can show core principles without the clutter of detail.
Consider this scenario:
- The population has eight individuals.
- On the first day of the epidemic, there is one infected person, and seven susceptible people.
- On each day, each infected person interacts with a susceptible person and infects them.
- Once infected, a person remains infected.
Based on the above information, trace out how the epidemic will evolve:
- Starting from day 1, how many infected and susceptibles will there be on each successive day?
- What will eventually happen to the population?
In other words: calculate the dynamics of this miniature epidemic.
Since individuals never become uninfected, the whole population eventually becomes infected. Initially, the size of compartment Infected is 1, and the size of Susceptible is 7, Eventually, the size of Infected maxes out at 8, and Susceptible goes down to zero.epidemic1
- Day 1: Infected = 1, Susceptible = 7.
- Day 2: Infected = 2, Susceptible = 6. The one infected person infected a susceptible.
- Day 3: Infected = 4, Susceptible = 4. Two infected people infected two susceptibles.
- Day 4: Infected = 8, Susceptible = 0. Four infected people infected four susceptibles.
- Day 5: Infected = 8, Susceptible = 0. No change, from this point on.
- Day 6: Infected = 8, Susceptible = 0.
The next step
This was a simplified example, for the simplest model, SI. Because SI only has compartments for susceptible and infected, it can’t express conditions such as recovery, death or loss of immunity.
Next time, we will look at extensions to SI to cover these more realistic cases.
One step at a time, the plot will thicken.
Copyright © 2020, David A. Tanzer. All Rights Reserved.