The Model

Model Overview

This model is a compartmental SIR (susceptible-infected-recovered) model. Compartmental means that we place individuals into distinct compartments, according to some characteristics. We then only track the total number of individuals in each of these compartments. In the simplest model, the only characteristic we track is a person’s infection status. We allow for 3 different stages/compartments:

The SIR model is very basic. It could be extended by introducing further compartments. For instance, we could stratify according to gender, which would give us 2 sets of SIR compartments, one for males and one for females. Some of these extensions are implemented in other apps.

In addition to specifying the compartments of a model, we need to specify the processes/mechanisms determining the changes for each compartment. Broadly speaking, there are processes that increase the number of individuals in a given compartment/stage, and processes that lead to a reduction. Those processes are sometimes called inflows and outflows.

For our system, we specify only 2 processes/flows:

  1. A susceptible individual (S) can become infected by an infectious individual (I) at some rate (for which we use the parameter b). This leads to the susceptible individual leaving the S compartment and entering the I compartment.
  2. An infected individual dies or recovers and enters the recovered/removed (R) compartment at some. This is described by the parameter g in our model.

As with the compartments, we could extend the model and allow other processes to occur. For instance, we could allow for natural births and deaths, waning immunity, deaths due to disease, etc. Some of that will be included in other apps.

Model Representation

For compartmental models (and also often other types of models), it is useful to show a graphical schematic representation of the compartments and processes included in the model. For compartmental models, such a diagram/figure is usually called a flow diagram. Such a diagram consists of a box for each compartment, and arrows pointing in and out of boxes to describe flows and interactions. For the simple SIR model, the flow diagram looks as follows:

Flow diagram for simple SIR model.

Flow diagram for simple SIR model.

Model Implementation

To allow us to simulate this model, we need to implement it on the computer. For that purpose, it is often useful to write the model as mathematical equations (this is not strictly needed, some computer simulation models are never formulated as mathematical models). A very common way (but not the only one) to implement compartmental models such as the simple SIR model is a set of ordinary differential equations. Each compartment/variable gets an equation. The right side of each equation specifies the processes going on in the system and how they change the numbers in each compartment via inflows and outflows. For the model described above, the equations look like this:

S:\[\dot S = -bSI\] I:\[\dot I = bSI - gI\] R:\[\dot R = gI\]

Note: If you don’t see equations but instead gibberish, try opening the app with a different browser. I have found that occasionally, on some computers, the built-in RStudio viewer does not process the equations correctly. Firefox and Chrome seem to work.

Some general notes